[
    {
        "id": 1,
        "task_id": 473,
        "test_case_id": 1,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "05:50\n05:44\n",
        "output": "00:06\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 2,
        "task_id": 473,
        "test_case_id": 2,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "00:00\n01:00\n",
        "output": "23:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 3,
        "task_id": 473,
        "test_case_id": 3,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "00:01\n00:00\n",
        "output": "00:01\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 4,
        "task_id": 473,
        "test_case_id": 4,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "23:59\n23:59\n",
        "output": "00:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 5,
        "task_id": 473,
        "test_case_id": 5,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "23:44\n23:55\n",
        "output": "23:49\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 6,
        "task_id": 473,
        "test_case_id": 6,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "00:00\n13:12\n",
        "output": "10:48\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 7,
        "task_id": 473,
        "test_case_id": 7,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "12:00\n23:59\n",
        "output": "12:01\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 8,
        "task_id": 473,
        "test_case_id": 8,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "12:44\n12:44\n",
        "output": "00:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 9,
        "task_id": 473,
        "test_case_id": 9,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "05:55\n07:12\n",
        "output": "22:43\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 10,
        "task_id": 473,
        "test_case_id": 10,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "07:12\n05:55\n",
        "output": "01:17\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 11,
        "task_id": 473,
        "test_case_id": 11,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "22:22\n22:22\n",
        "output": "00:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 12,
        "task_id": 473,
        "test_case_id": 12,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "22:22\n22:23\n",
        "output": "23:59\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 13,
        "task_id": 473,
        "test_case_id": 13,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "23:24\n23:23\n",
        "output": "00:01\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 14,
        "task_id": 473,
        "test_case_id": 14,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "00:00\n00:00\n",
        "output": "00:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 15,
        "task_id": 473,
        "test_case_id": 15,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "23:30\n00:00\n",
        "output": "23:30\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 16,
        "task_id": 473,
        "test_case_id": 16,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "01:00\n00:00\n",
        "output": "01:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 17,
        "task_id": 473,
        "test_case_id": 17,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "05:44\n06:00\n",
        "output": "23:44\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 18,
        "task_id": 473,
        "test_case_id": 18,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "00:00\n23:59\n",
        "output": "00:01\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 19,
        "task_id": 473,
        "test_case_id": 19,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "21:00\n01:00\n",
        "output": "20:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 20,
        "task_id": 473,
        "test_case_id": 20,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "21:21\n12:21\n",
        "output": "09:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 21,
        "task_id": 473,
        "test_case_id": 21,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "12:21\n21:12\n",
        "output": "15:09\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 22,
        "task_id": 473,
        "test_case_id": 22,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "12:33\n23:33\n",
        "output": "13:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 23,
        "task_id": 473,
        "test_case_id": 23,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "07:55\n05:53\n",
        "output": "02:02\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 24,
        "task_id": 473,
        "test_case_id": 24,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "19:30\n02:00\n",
        "output": "17:30\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 25,
        "task_id": 473,
        "test_case_id": 25,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "21:30\n02:00\n",
        "output": "19:30\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 26,
        "task_id": 473,
        "test_case_id": 26,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "19:30\n09:30\n",
        "output": "10:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 27,
        "task_id": 473,
        "test_case_id": 27,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "13:08\n00:42\n",
        "output": "12:26\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 28,
        "task_id": 473,
        "test_case_id": 28,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "13:04\n09:58\n",
        "output": "03:06\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 29,
        "task_id": 473,
        "test_case_id": 29,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "21:21\n23:06\n",
        "output": "22:15\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 30,
        "task_id": 473,
        "test_case_id": 30,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "20:53\n10:23\n",
        "output": "10:30\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 31,
        "task_id": 473,
        "test_case_id": 31,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "12:59\n00:45\n",
        "output": "12:14\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 32,
        "task_id": 473,
        "test_case_id": 32,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "12:39\n22:21\n",
        "output": "14:18\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 33,
        "task_id": 473,
        "test_case_id": 33,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "21:10\n13:50\n",
        "output": "07:20\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 34,
        "task_id": 473,
        "test_case_id": 34,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "03:38\n23:46\n",
        "output": "03:52\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 35,
        "task_id": 473,
        "test_case_id": 35,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "03:48\n00:41\n",
        "output": "03:07\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 36,
        "task_id": 473,
        "test_case_id": 36,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "07:43\n12:27\n",
        "output": "19:16\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 37,
        "task_id": 473,
        "test_case_id": 37,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "03:23\n08:52\n",
        "output": "18:31\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 38,
        "task_id": 473,
        "test_case_id": 38,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "16:04\n10:28\n",
        "output": "05:36\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 39,
        "task_id": 473,
        "test_case_id": 39,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "12:53\n08:37\n",
        "output": "04:16\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 40,
        "task_id": 473,
        "test_case_id": 40,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "13:43\n17:23\n",
        "output": "20:20\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 41,
        "task_id": 473,
        "test_case_id": 41,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "00:00\n00:01\n",
        "output": "23:59\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 42,
        "task_id": 473,
        "test_case_id": 42,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "10:10\n01:01\n",
        "output": "09:09\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 43,
        "task_id": 473,
        "test_case_id": 43,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "10:05\n00:00\n",
        "output": "10:05\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 44,
        "task_id": 473,
        "test_case_id": 44,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "09:09\n00:00\n",
        "output": "09:09\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 45,
        "task_id": 473,
        "test_case_id": 45,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "09:10\n00:01\n",
        "output": "09:09\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 46,
        "task_id": 473,
        "test_case_id": 46,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "23:24\n00:28\n",
        "output": "22:56\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 47,
        "task_id": 473,
        "test_case_id": 47,
        "question": "George woke up and saw the current time s on the digital clock. Besides, George knows that he has slept for time t. \n\nHelp George! Write a program that will, given time s and t, determine the time p when George went to bed. Note that George could have gone to bed yesterday relatively to the current time (see the second test sample). \n\n\n-----Input-----\n\nThe first line contains current time s as a string in the format \"hh:mm\". The second line contains time t in the format \"hh:mm\" — the duration of George's sleep. It is guaranteed that the input contains the correct time in the 24-hour format, that is, 00 ≤ hh ≤ 23, 00 ≤ mm ≤ 59.\n\n\n-----Output-----\n\nIn the single line print time p — the time George went to bed in the format similar to the format of the time in the input.\n\n\n-----Examples-----\nInput\n05:50\n05:44\n\nOutput\n00:06\n\nInput\n00:00\n01:00\n\nOutput\n23:00\n\nInput\n00:01\n00:00\n\nOutput\n00:01\n\n\n\n-----Note-----\n\nIn the first sample George went to bed at \"00:06\". Note that you should print the time only in the format \"00:06\". That's why answers \"0:06\", \"00:6\" and others will be considered incorrect. \n\nIn the second sample, George went to bed yesterday.\n\nIn the third sample, George didn't do to bed at all.",
        "solutions": "[\"s = input()\\nt = input()\\na, b = int(s[:2]), int(s[3:])\\nc, d = int(t[:2]), int(t[3:])\\na -= c\\nb -= d\\nif b < 0:\\n    a -= 1\\n    b = 60 + b\\nif a < 0:\\n    a = 24 + a\\nif a < 10:\\n    print(0, end = '')\\nprint(a, ':', end = '', sep = '')\\nif b < 10:\\n    print(0, end = '')\\nprint(b)\\n\", \"a, b = list(map(int, input().split(':')))\\nc, d = list(map(int, input().split(':')))\\nt = a * 60 + b - c * 60 - d\\n\\nif t < 0:\\n    t += 60 * 24\\na, b = t // 60, t % 60\\n\\nprint(str(a // 10) + str(a % 10) + ':' + str(b // 10) + str(b % 10))\\n\", \"h2, m2 = list(map(int, input().split(':')))\\nh_sleep, m_sleep = list(map(int, input().split(':')))\\n\\nt2 = m2 + h2 * 60\\nt_sleep = m_sleep + h_sleep * 60\\n\\nt2 -= t_sleep\\nif t2 < 0:\\n    t2 += 60 * 24\\n\\nprint('0' * (2 - len(str(t2 // 60))) + str(t2 // 60) + ':' + '0' * (2 - len(str(t2 % 60))) + str(t2 % 60))\\n\", \"def ui(a):\\n    a = str(a)\\n    return '0'*(2-len(a))+a\\na,b = map(int,input().split(':'))\\nc,d = map(int,input().split(':'))\\nt1 = 60*a+b\\nt2 = 60*c+d\\nif t1>=t2:\\n    dt = t1-t2\\nelse:\\n    dt = 1440-t2+t1\\nprint(ui(dt//60),':',ui(dt%60),sep='')\", \"s=input()\\n\\nwh=int(s[0]+s[1])\\nwm=int(s[3]+s[4])\\n\\ns=input()\\n\\nth=int(s[0]+s[1])\\ntm=int(s[3]+s[4])\\n\\ntm+=th*60\\n\\nwm+=wh*60\\n\\nwm-=tm\\nif(wm<0):\\n    wm=1440+wm\\nansh=str(wm//60)\\nansm=str(wm%60)\\nif(len(ansh)<2):\\n    ansh='0'+ansh\\nif(len(ansm)<2):\\n    ansm='0'+ansm\\nprint(ansh+\\\":\\\"+ansm)\\n\", \"def conv(s):\\n\\th = str(s // 60)\\n\\tm = str(s % 60)\\n\\tif len(h) < 2:\\n\\t\\th = '0' + h\\n\\tif len(m) < 2:\\n\\t\\tm = '0' + m\\n\\treturn h + ':' + m\\n\\ns = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\ns = s[0] * 60 + s[1]\\nt = t[0] * 60 + t[1]\\np = s - t\\nif p < 0:\\n\\tp += 24 * 60\\nprint(conv(p))\\n\", \"h1, m1 = list(map(int, input().split(':')))\\nh2, m2 = list(map(int, input().split(':')))\\nif (m2 > m1):\\n\\tm1 += 60\\n\\th1 -= 1\\nh1 -= h2\\nm1 -= m2\\nif (h1 < 0):\\n\\th1 += 24\\nh = ''\\nm = ''\\nif (h1 < 10):\\n\\th = '0'\\nh += str(h1)\\nif (m1 < 10):\\n\\tm = '0'\\nm += str(m1)\\nprint(h, m, sep = ':', end = '\\\\n')\\n\", \"s = input()\\np = input()\\nt1 = int(s[len(s) - 2:len(s)]) - int(p[len(p) - 2:len(p)])\\nk = 0\\nif t1 < 0:\\n    t1 += 60\\n    k = 1\\nt0 = int(s[:2]) - int(p[:2]) - k\\nwhile t0 < 0:\\n    t0 += 24\\nt0 %= 24\\nres = str(t0) + ':'\\nif len(res) < 3:\\n    res = '0' + res\\nss = '0' + str(t1)\\nss = ss[-2:]\\nres += ss\\nprint(res)\", \"s=input().strip()\\np=input().strip()\\nans=int(s[:2])*60+int(s[3:])-int(p[:2])*60-int(p[3:])\\nans=ans%1440\\nt=('0' if ans//60<10 else '') + str(ans//60) + ':' + ('0' if ans%60<10 else '')+str(ans%60)\\nprint(t)\", \"s = input()\\nt1 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\ns = input()\\nt2 = (10 * int(s[0]) + int(s[1])) * 60 + (10 * int(s[3]) + int(s[4]))\\nt1 = (t1 - t2 + 24 * 60) % (24 * 60)\\nx1 = t1 // 60\\nx2 = t1 % 60\\nprint(x1 // 10, end = '')\\nprint(x1 % 10, end = '')\\nprint(':', end = '')\\nprint(x2 // 10, end = '')\\nprint(x2 % 10)\\n\", \"s = input().split(\\\":\\\")\\nt = input().split(\\\":\\\")\\n\\nh = int(s[0]) - int(t[0])\\n\\nm = int(s[1]) - int(t[1])\\nwhile m < 0:\\n    m += 60\\n    h -= 1\\nwhile h < 0:\\n    h += 24\\n\\n\\nmstr = ''\\nif m < 10:\\n    mstr = '0' + str(m)\\nelse:\\n    mstr = str(m)\\n\\nhstr = ''\\nif h < 10:\\n    hstr = '0' + str(h)\\nelse:\\n    hstr = str(h)\\n\\nprint(hstr + ':' + mstr)\\n\\n\", \"s1=input().strip()\\ns2=input().strip()\\nh1=int(s1[:2])\\nh2=int(s2[:2])\\nm1=int(s1[3:])\\nm2=int(s2[3:])\\nh3=h1-h2\\nm3=m1-m2\\nif m3<0:\\n    m3=60+m3\\n    h3-=1\\nif h3<0:\\n    h3=24+h3\\nprint((2-len(str(h3)))*'0'+str(h3)+':'+(2-len(str(m3)))*'0'+str(m3))\\n\", \"s = list(map(int,input().split(':')))\\ns = (24+s[0])*60+s[1]\\nt = list(map(int,input().split(':')))\\nt = t[0]*60+t[1]\\np = (s - t) % (24 * 60)\\np = '{:02}:{:02}'.format(p//60, p % 60)\\nprint(p)\\n\", \"h,m = [int(x) for x in input().split(\\\":\\\")]\\nhdiff, mdiff = [int(x) for x in input().split(\\\":\\\")]\\n\\nm1=m-mdiff\\nh1=h-hdiff\\nif m1<0:\\n  m1=60+m1\\n  h1=h1-1\\nif h1<0:\\n  h1=24+h1\\nh1=str(h1)\\nm1=str(m1)\\nprint(h1.zfill(2)+\\\":\\\"+m1.zfill(2))\", \"from sys import stdin\\nfrom datetime import datetime, timedelta\\n\\ndef main():\\n\\ts = stdin.readline().strip()\\n\\tt = stdin.readline().strip()\\n\\n\\ts = datetime.strptime(s, '%H:%M')\\n\\tt = datetime.strptime(t, '%H:%M')\\n\\n\\tt = timedelta(hours = t.hour, minutes = t.minute)\\n\\n\\tp = s - t\\n\\n\\tprint('{:02}:{:02}'.format(p.hour, p.minute))\\n\\ndef __starting_point(): main()\\n__starting_point()\", \"a = input()\\nq1, q2, q3, q4 = int(a[0]), int(a[1]), int(a[3]), int(a[4])\\nb = input()\\nw1, w2, w3, w4 = int(b[0]), int(b[1]), int(b[3]), int(b[4])\\nww = (q3*10 + q4) + (q1*10 + q2)*60\\nqq = (w3*10 + w4) + (w1*10 + w2)*60\\ns = \\\"\\\"\\nif ww == qq:\\n    print(\\\"00:00\\\")\\nelif (ww - qq) > 0:\\n    if(ww - qq) // 60 > 9:\\n        d = str((ww - qq)//60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq) % 60 > 9:\\n        d = str((ww - qq)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq)%60)\\n        s += d \\n    print(s)\\nelse:\\n    if(ww - qq + 1440) // 60 > 9:\\n            d = str((ww - qq+ 1440)//60)\\n            s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq+ 1440)//60)\\n        s += d\\n    s += \\\":\\\"\\n    if (ww - qq + 1440) % 60 > 9:\\n        d = str((ww - qq + 1440)%60)\\n        s += d\\n    else:\\n        s += \\\"0\\\"\\n        d = str((ww - qq + 1440)%60)\\n        s += d   \\n    print(s)\\n\\n\", \"from datetime import *\\na = input().split(\\\":\\\")\\nb = input().split(\\\":\\\")\\nx = datetime(1,1,3,int(a[0]),int(a[1]))\\ny = timedelta(hours=int(b[0]),minutes=int(b[1]))\\nz = str(x - y).split(' ')[1].split(\\\":\\\")\\nprint(\\\"%s:%s\\\"%(z[0],z[1]))\\n\", \"current = input()\\nsleep = input()\\n\\nhour_current = current[:current.index(':')]\\nmin_current = current[current.index(':')+1:]\\n\\nhour_sleep = sleep[:sleep.index(':')]\\nmin_sleep = sleep[sleep.index(':')+1:]\\n\\ncurrent_in_min = int(hour_current)*60 + int(min_current)\\nsleep_in_min = int(hour_sleep)*60 + int(min_sleep)\\n\\nbed_time = current_in_min - sleep_in_min\\nif bed_time < 0 :\\n    bed_time = 24*60 + bed_time\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nelse:\\n    out_hour = bed_time // 60\\n    out_min = bed_time % 60\\nif out_hour < 10 :\\n    out_hour = \\\"0\\\"+str(out_hour)\\nif out_min < 10 :\\n    out_min = \\\"0\\\" + str(out_min)\\nprint(str(out_hour) + \\\":\\\" + str(out_min))\\n\", \"s1=input()\\ns2=input()\\nh1,m1=s1.split(':')\\nh2,m2=s2.split(':')\\nm=int(m1)-int(m2)\\nh=int(h1)-int(h2)-1*(m<0)\\nprint(str(h%24).zfill(2),str(m%60).zfill(2),sep=':')\\n\\n\\n\", \"import sys\\nimport math\\nimport heapq\\nimport random\\nimport collections\\nimport datetime\\n\\ndef main():\\n    # sys.stdin = open('input.txt', 'r')\\n    # sys.stdout = open('output.txt', 'w')\\n\\n    t1 = list(map(int, input().strip().split(':')))\\n    t2 = list(map(int, input().strip().split(':')))\\n\\n    t1_ = datetime.timedelta(hours=t1[0], minutes=t1[1])\\n    t2_ = datetime.timedelta(hours=t2[0], minutes=t2[1])\\n\\n    t3_ = t1_- t2_\\n    s = str(t3_)\\n\\n    if len(s[-8:-3]) == 4:\\n        print('0'+s[-8:-3])\\n\\n    elif len(s[-8:-3]) == 5 and s[-8] == ' ':\\n        print('0'+s[-7:-3])\\n\\n    else:\\n        print(s[-8:-3])\\n\\n    # sys.stdin.close()\\n    # sys.stdout.close()\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nout = ''\\nif int(ans/60) < 10:\\n    out = out + '0' + str(int(ans/60))\\nelse:\\n    out = str(int(ans/60))\\nout = out + ':'\\nif int(ans%60) < 10:\\n    out = out + '0' + str(ans%60)\\nelse:\\n    out = out + str(ans%60)\\nprint(out)\", \"[e1,e2] = map(int,input().split(':'))\\n[s1,s2] = map(int,input().split(':'))\\n\\ntotal1 = e1*60 + e2;\\ntotal2 = s1*60 + s2;\\nans = 0\\nif total1 >= total2 :\\n    ans = total1 - total2\\nelse:\\n    ans = 1440 - total2 + total1\\nprint('{:02}:{:02}'.format(int(ans/60),int(ans%60)))\", \"def ss(a):\\n    if a > 9:\\n        return str(a)\\n    else:\\n        return '0'+str(a)\\na = [int(x) for x in input().split(':')]\\nb = [int(x) for x in input().split(':')]\\nh = a[0] - b[0]\\nm = a[1] - b[1]\\nhh = 0\\nwhile m < 0:\\n    m = 60 - abs(m)\\n    hh += 1\\nh -= hh\\nwhile h < 0:\\n    h = 24 - abs(h)\\nprint(ss(h),':',ss(m),sep = '')\", \"s = list(map(int, input().split(':')))\\nt = list(map(int, input().split(':')))\\n\\np = [0, 0]\\nif s[0] - t[0] < 0:\\n    p[0] = 24 - (t[0] - s[0])\\nelse:\\n    p[0] = s[0] - t[0]\\n\\nif s[1] - t[1] < 0:\\n    p[1] = 60 - (t[1] - s[1])\\n    if p[0] == 0:\\n        p[0] = 23\\n    else:\\n        p[0] -= 1\\nelse:\\n    p[1] = s[1] - t[1]\\n\\nif p[0] < 10:\\n    print('0', p[0], sep = '', end = ':')\\nelse:\\n    print(p[0], end = ':')\\nif p[1] < 10:\\n    print('0', p[1], sep = '')\\nelse:\\n    print(p[1])\\n\"]",
        "difficulty": "interview",
        "input": "10:00\n01:00\n",
        "output": "09:00\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/387/A"
    },
    {
        "id": 48,
        "task_id": 3170,
        "test_case_id": 1,
        "question": "From her friends, Theta learned a $2$-player card game called Trash. Trash is played with a standard $52$-card deck consisting of Ace, $2$ to $10$, Jack, Queen, and King in each of the $4$ suits. Each player is dealt $10$ cards face down that are arranged in $2$ rows for each player as shown in the figure. The first row represents slots for an ace, a $2$, a $3$, a $4$, and a $5$ of any suit, the second row represents slots for $6$, $7$, $8$, $9$, $10$ of any suit. The first player who can fill all $10$ of their slots with a card of the appropriate kind wins.\n\nPlayers take turns to play Trash. At the beginning of each turn, a player draws from the drawing pile, which initially contains all cards that have not been dealt face-down. If the player can use the card drawn to fill one of their slots that has not been filled, they uncover and remove the face-down card from the slot and put the drawn card face up in the slot. Then, the player repeats their turn with the card uncovered (instead of drawing a new card from the drawing pile) until they either fill all slots or uncover a card that cannot fill a slot.\n\nIf a player draws or uncovers a card that cannot fill any slot (such as a Queen or King), or if a player draws or uncovers a card that corresponds to a slot they’ve already filled, that card is put on the discard pile and the player loses their turn. Then, the other player takes their turn or turns by first drawing from the drawing pile, then placing cards as long as they fill slots. Note that in this version of Trash, a player may never take any cards from the discard pile. Whichever player first fills all of their slots wins.\n\nA special case is the Jack, which acts as a wildcard. Whenever a player draws (or uncovers) a Jack, they can use the Jack to fill any of their remaining slots.\n\nTheta quickly discovers that it matters how you use any Jacks you draw, namely so that you minimize the chance of being left with an unfilled slot. If there are multiple optimal choices for slots in which to place a Jack, she chooses the lowest-numbered slot (Ace counts as $1$). Most of her friends, however, use a simpler strategy of always using their Jacks to fill the lowest-numbered unfilled slot, regardless of which cards were already placed or discarded.\n\nWrite a program that determines who wins in a game between Theta and one of her friends! Note that Theta bases her decision on only those cards that are turned face-up in a slot and the cards that have been discarded onto the discard pile, but not the face-down cards or the cards remaining in the drawing pile.\n\n-----Input-----\nInput consists of a single string denoting the shuffled deck, which consists of $52$ characters, with each of A, J, Q, K, 2, 3, 4, 5, 6, 7, 8, 9, and T appearing exactly $4$ times. Since the suit of a card does not matter for the game, it is not given in the input. 2 - 9 stand for cards $2$ to $9$, T stands for $10$, and A, J, Q, and K stand for Ace, Jack, Queen, and King.\n\nThe first $10$ cards in the deck are dealt face-down to Theta, with the first card in her Ace/$1$ slot, the second card in her $2$ slot, and so on. The next $10$ cards are dealt to her friend, with the $11^{\\texttt{th}}$ in the Ace/$1$ slot, and so on. The $21^{\\texttt{st}}$ card in the deck is the first card drawn. Neither player is allowed to look at their face-down cards. Theta starts the game.\n\nYou are guaranteed that one player will be able to win before the deck runs out of cards.\n\n-----Output-----\nIf Theta wins this game with her strategy, output “Theta wins”. Otherwise, output “Theta loses”. Do not add a period to the output.\n\n-----Examples-----\nSample Input 1:\n23456789TJ23456789TJA89Q66JK37T2A4AQK3AK5T8Q24K97JQ5\nSample Output 1:\nTheta wins\n\nSample Input 2:\n89724TJTA67K4J87Q8T6Q7J2324T558KA99A3KA356QJ6523QK49\nSample Output 2:\nTheta wins",
        "solutions": "",
        "difficulty": "competition",
        "input": "23456789TJ23456789TJA89Q66JK37T2A4AQK3AK5T8Q24K97JQ5\n",
        "output": "Theta wins\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/trash"
    },
    {
        "id": 49,
        "task_id": 3170,
        "test_case_id": 2,
        "question": "From her friends, Theta learned a $2$-player card game called Trash. Trash is played with a standard $52$-card deck consisting of Ace, $2$ to $10$, Jack, Queen, and King in each of the $4$ suits. Each player is dealt $10$ cards face down that are arranged in $2$ rows for each player as shown in the figure. The first row represents slots for an ace, a $2$, a $3$, a $4$, and a $5$ of any suit, the second row represents slots for $6$, $7$, $8$, $9$, $10$ of any suit. The first player who can fill all $10$ of their slots with a card of the appropriate kind wins.\n\nPlayers take turns to play Trash. At the beginning of each turn, a player draws from the drawing pile, which initially contains all cards that have not been dealt face-down. If the player can use the card drawn to fill one of their slots that has not been filled, they uncover and remove the face-down card from the slot and put the drawn card face up in the slot. Then, the player repeats their turn with the card uncovered (instead of drawing a new card from the drawing pile) until they either fill all slots or uncover a card that cannot fill a slot.\n\nIf a player draws or uncovers a card that cannot fill any slot (such as a Queen or King), or if a player draws or uncovers a card that corresponds to a slot they’ve already filled, that card is put on the discard pile and the player loses their turn. Then, the other player takes their turn or turns by first drawing from the drawing pile, then placing cards as long as they fill slots. Note that in this version of Trash, a player may never take any cards from the discard pile. Whichever player first fills all of their slots wins.\n\nA special case is the Jack, which acts as a wildcard. Whenever a player draws (or uncovers) a Jack, they can use the Jack to fill any of their remaining slots.\n\nTheta quickly discovers that it matters how you use any Jacks you draw, namely so that you minimize the chance of being left with an unfilled slot. If there are multiple optimal choices for slots in which to place a Jack, she chooses the lowest-numbered slot (Ace counts as $1$). Most of her friends, however, use a simpler strategy of always using their Jacks to fill the lowest-numbered unfilled slot, regardless of which cards were already placed or discarded.\n\nWrite a program that determines who wins in a game between Theta and one of her friends! Note that Theta bases her decision on only those cards that are turned face-up in a slot and the cards that have been discarded onto the discard pile, but not the face-down cards or the cards remaining in the drawing pile.\n\n-----Input-----\nInput consists of a single string denoting the shuffled deck, which consists of $52$ characters, with each of A, J, Q, K, 2, 3, 4, 5, 6, 7, 8, 9, and T appearing exactly $4$ times. Since the suit of a card does not matter for the game, it is not given in the input. 2 - 9 stand for cards $2$ to $9$, T stands for $10$, and A, J, Q, and K stand for Ace, Jack, Queen, and King.\n\nThe first $10$ cards in the deck are dealt face-down to Theta, with the first card in her Ace/$1$ slot, the second card in her $2$ slot, and so on. The next $10$ cards are dealt to her friend, with the $11^{\\texttt{th}}$ in the Ace/$1$ slot, and so on. The $21^{\\texttt{st}}$ card in the deck is the first card drawn. Neither player is allowed to look at their face-down cards. Theta starts the game.\n\nYou are guaranteed that one player will be able to win before the deck runs out of cards.\n\n-----Output-----\nIf Theta wins this game with her strategy, output “Theta wins”. Otherwise, output “Theta loses”. Do not add a period to the output.\n\n-----Examples-----\nSample Input 1:\n23456789TJ23456789TJA89Q66JK37T2A4AQK3AK5T8Q24K97JQ5\nSample Output 1:\nTheta wins\n\nSample Input 2:\n89724TJTA67K4J87Q8T6Q7J2324T558KA99A3KA356QJ6523QK49\nSample Output 2:\nTheta wins",
        "solutions": "",
        "difficulty": "competition",
        "input": "89724TJTA67K4J87Q8T6Q7J2324T558KA99A3KA356QJ6523QK49\n",
        "output": "Theta wins\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/trash"
    },
    {
        "id": 50,
        "task_id": 3170,
        "test_case_id": 3,
        "question": "From her friends, Theta learned a $2$-player card game called Trash. Trash is played with a standard $52$-card deck consisting of Ace, $2$ to $10$, Jack, Queen, and King in each of the $4$ suits. Each player is dealt $10$ cards face down that are arranged in $2$ rows for each player as shown in the figure. The first row represents slots for an ace, a $2$, a $3$, a $4$, and a $5$ of any suit, the second row represents slots for $6$, $7$, $8$, $9$, $10$ of any suit. The first player who can fill all $10$ of their slots with a card of the appropriate kind wins.\n\nPlayers take turns to play Trash. At the beginning of each turn, a player draws from the drawing pile, which initially contains all cards that have not been dealt face-down. If the player can use the card drawn to fill one of their slots that has not been filled, they uncover and remove the face-down card from the slot and put the drawn card face up in the slot. Then, the player repeats their turn with the card uncovered (instead of drawing a new card from the drawing pile) until they either fill all slots or uncover a card that cannot fill a slot.\n\nIf a player draws or uncovers a card that cannot fill any slot (such as a Queen or King), or if a player draws or uncovers a card that corresponds to a slot they’ve already filled, that card is put on the discard pile and the player loses their turn. Then, the other player takes their turn or turns by first drawing from the drawing pile, then placing cards as long as they fill slots. Note that in this version of Trash, a player may never take any cards from the discard pile. Whichever player first fills all of their slots wins.\n\nA special case is the Jack, which acts as a wildcard. Whenever a player draws (or uncovers) a Jack, they can use the Jack to fill any of their remaining slots.\n\nTheta quickly discovers that it matters how you use any Jacks you draw, namely so that you minimize the chance of being left with an unfilled slot. If there are multiple optimal choices for slots in which to place a Jack, she chooses the lowest-numbered slot (Ace counts as $1$). Most of her friends, however, use a simpler strategy of always using their Jacks to fill the lowest-numbered unfilled slot, regardless of which cards were already placed or discarded.\n\nWrite a program that determines who wins in a game between Theta and one of her friends! Note that Theta bases her decision on only those cards that are turned face-up in a slot and the cards that have been discarded onto the discard pile, but not the face-down cards or the cards remaining in the drawing pile.\n\n-----Input-----\nInput consists of a single string denoting the shuffled deck, which consists of $52$ characters, with each of A, J, Q, K, 2, 3, 4, 5, 6, 7, 8, 9, and T appearing exactly $4$ times. Since the suit of a card does not matter for the game, it is not given in the input. 2 - 9 stand for cards $2$ to $9$, T stands for $10$, and A, J, Q, and K stand for Ace, Jack, Queen, and King.\n\nThe first $10$ cards in the deck are dealt face-down to Theta, with the first card in her Ace/$1$ slot, the second card in her $2$ slot, and so on. The next $10$ cards are dealt to her friend, with the $11^{\\texttt{th}}$ in the Ace/$1$ slot, and so on. The $21^{\\texttt{st}}$ card in the deck is the first card drawn. Neither player is allowed to look at their face-down cards. Theta starts the game.\n\nYou are guaranteed that one player will be able to win before the deck runs out of cards.\n\n-----Output-----\nIf Theta wins this game with her strategy, output “Theta wins”. Otherwise, output “Theta loses”. Do not add a period to the output.\n\n-----Examples-----\nSample Input 1:\n23456789TJ23456789TJA89Q66JK37T2A4AQK3AK5T8Q24K97JQ5\nSample Output 1:\nTheta wins\n\nSample Input 2:\n89724TJTA67K4J87Q8T6Q7J2324T558KA99A3KA356QJ6523QK49\nSample Output 2:\nTheta wins",
        "solutions": "",
        "difficulty": "competition",
        "input": "6Q4K476722745A9A9875A2TT3JA6K5K34JKQQTQ235T9868J893J\n",
        "output": "Theta loses\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/trash"
    },
    {
        "id": 51,
        "task_id": 3339,
        "test_case_id": 1,
        "question": "Evenland used to be a normal country. Then Steven became ruler and now everything must be done as he wishes. For some odd reason he is obsessed with the number two. Everything must be even in his country, hence he even changed the name to Evenland. \n\nThe other day, Steven was driving through his country when he noticed that, at some intersections, an odd number of roads meet. Naturally, some roads must now be destroyed in order to make the number of roads even at every intersection.\n\nYou are in charge of this project. You start wondering: in how many ways can this project be carried out? In other words, in how many ways can you select a set of roads to destroy so that all intersections become even? The resulting road network does not have to be connected, so for instance, one possible way is to destroy all roads.\n\n-----Input-----\nThe first line of the input contains two integers $N$ and $M$, where $1\\leq N, M\\leq 100000$. $N$ denotes the number of intersections in Evenland and $M$ is the number of roads. $M$ lines follow, each contains two space separated integers $a$, $b$ indicating that there is a road between intersections $a$ and $b$. You may assume that $1\\leq a, b\\leq N$ and $a\\not= b$. There might be more than one road connecting a pair of intersections.\n\n-----Output-----\nOutput one line with one integer – the number of ways of making all intersections even. Since this number might be big, output the remainder modulo $1000000009$.\n\n-----Examples-----\nSample Input:\n4 5\n1 2\n1 3\n1 4\n2 3\n2 4\nSample Output:\n4",
        "solutions": "",
        "difficulty": "competition",
        "input": "4 5\n1 2\n1 3\n1 4\n2 3\n2 4\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/evenland"
    },
    {
        "id": 52,
        "task_id": 3339,
        "test_case_id": 2,
        "question": "Evenland used to be a normal country. Then Steven became ruler and now everything must be done as he wishes. For some odd reason he is obsessed with the number two. Everything must be even in his country, hence he even changed the name to Evenland. \n\nThe other day, Steven was driving through his country when he noticed that, at some intersections, an odd number of roads meet. Naturally, some roads must now be destroyed in order to make the number of roads even at every intersection.\n\nYou are in charge of this project. You start wondering: in how many ways can this project be carried out? In other words, in how many ways can you select a set of roads to destroy so that all intersections become even? The resulting road network does not have to be connected, so for instance, one possible way is to destroy all roads.\n\n-----Input-----\nThe first line of the input contains two integers $N$ and $M$, where $1\\leq N, M\\leq 100000$. $N$ denotes the number of intersections in Evenland and $M$ is the number of roads. $M$ lines follow, each contains two space separated integers $a$, $b$ indicating that there is a road between intersections $a$ and $b$. You may assume that $1\\leq a, b\\leq N$ and $a\\not= b$. There might be more than one road connecting a pair of intersections.\n\n-----Output-----\nOutput one line with one integer – the number of ways of making all intersections even. Since this number might be big, output the remainder modulo $1000000009$.\n\n-----Examples-----\nSample Input:\n4 5\n1 2\n1 3\n1 4\n2 3\n2 4\nSample Output:\n4",
        "solutions": "",
        "difficulty": "competition",
        "input": "2 1\n1 2\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/evenland"
    },
    {
        "id": 53,
        "task_id": 66,
        "test_case_id": 1,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "10 3 2\n",
        "output": "3/10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 54,
        "task_id": 66,
        "test_case_id": 2,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "7 1 2\n",
        "output": "3/7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 55,
        "task_id": 66,
        "test_case_id": 3,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1 1 1\n",
        "output": "1/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 56,
        "task_id": 66,
        "test_case_id": 4,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "5814 31 7\n",
        "output": "94/2907\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 57,
        "task_id": 66,
        "test_case_id": 5,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "94268 813 766\n",
        "output": "765/94268\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 58,
        "task_id": 66,
        "test_case_id": 6,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "262610 5583 4717\n",
        "output": "2358/131305\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 59,
        "task_id": 66,
        "test_case_id": 7,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "3898439 96326 71937\n",
        "output": "71936/3898439\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 60,
        "task_id": 66,
        "test_case_id": 8,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "257593781689876390 32561717 4411677\n",
        "output": "7914548537/257593781689876390\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 61,
        "task_id": 66,
        "test_case_id": 9,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "111319886766128339 7862842484895022 3003994959686829\n",
        "output": "3003994959686828/111319886766128339\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 62,
        "task_id": 66,
        "test_case_id": 10,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "413850294331656955 570110918058849723 409853735661743839\n",
        "output": "409853735661743838/413850294331656955\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 63,
        "task_id": 66,
        "test_case_id": 11,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "3000000000000000000 2999999999999999873 2999999999999999977\n",
        "output": "23437499999999999/23437500000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 64,
        "task_id": 66,
        "test_case_id": 12,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "9 6 1\n",
        "output": "1/9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 65,
        "task_id": 66,
        "test_case_id": 13,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "32 9 2\n",
        "output": "3/32\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 66,
        "task_id": 66,
        "test_case_id": 14,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "976 5 6\n",
        "output": "41/244\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 67,
        "task_id": 66,
        "test_case_id": 15,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "5814 31 7\n",
        "output": "94/2907\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 68,
        "task_id": 66,
        "test_case_id": 16,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "94268 714 345\n",
        "output": "689/94268\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 69,
        "task_id": 66,
        "test_case_id": 17,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "262610 5583 4717\n",
        "output": "2358/131305\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 70,
        "task_id": 66,
        "test_case_id": 18,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "3898439 96326 71937\n",
        "output": "71936/3898439\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 71,
        "task_id": 66,
        "test_case_id": 19,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "54682301 778668 253103\n",
        "output": "253102/54682301\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 72,
        "task_id": 66,
        "test_case_id": 20,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "329245015 1173508 8918834\n",
        "output": "1173507/329245015\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 73,
        "task_id": 66,
        "test_case_id": 21,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "321076647734423976 7 7\n",
        "output": "1/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 74,
        "task_id": 66,
        "test_case_id": 22,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "455227494055672047 92 28\n",
        "output": "19792499741550983/455227494055672047\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 75,
        "task_id": 66,
        "test_case_id": 23,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "595779167455745259 6954 8697\n",
        "output": "205511958419723/595779167455745259\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 76,
        "task_id": 66,
        "test_case_id": 24,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1000000000000000000 1000000000 2000000000\n",
        "output": "1/2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 77,
        "task_id": 66,
        "test_case_id": 25,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "462643382718281828 462643382718281507 462643382718281701\n",
        "output": "33045955908448679/33045955908448702\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 78,
        "task_id": 66,
        "test_case_id": 26,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "4000000000000000000 9999999999999997 99999999999999999\n",
        "output": "2499999999999999/1000000000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 79,
        "task_id": 66,
        "test_case_id": 27,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "4003000100004000000 9999999099999999 99999999999999999\n",
        "output": "4999999549999999/2001500050002000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 80,
        "task_id": 66,
        "test_case_id": 28,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "4903000100004000000 58997960959949999 99933992929999999\n",
        "output": "29498980479974999/2451500050002000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 81,
        "task_id": 66,
        "test_case_id": 29,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "257593781689876390 32561717 4411677\n",
        "output": "7914548537/257593781689876390\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 82,
        "task_id": 66,
        "test_case_id": 30,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "111319886766128339 7862842484895022 3003994959686829\n",
        "output": "3003994959686828/111319886766128339\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 83,
        "task_id": 66,
        "test_case_id": 31,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "413850294331656955 570110918058849723 409853735661743839\n",
        "output": "409853735661743838/413850294331656955\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 84,
        "task_id": 66,
        "test_case_id": 32,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "232 17 83\n",
        "output": "2/29\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 85,
        "task_id": 66,
        "test_case_id": 33,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "5496272 63 200\n",
        "output": "13765/2748136\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 86,
        "task_id": 66,
        "test_case_id": 34,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "180 174 53\n",
        "output": "13/45\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 87,
        "task_id": 66,
        "test_case_id": 35,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1954 190 537\n",
        "output": "189/1954\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 88,
        "task_id": 66,
        "test_case_id": 36,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "146752429 510 514\n",
        "output": "571199/146752429\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 89,
        "task_id": 66,
        "test_case_id": 37,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "579312860 55 70\n",
        "output": "10344881/144828215\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 90,
        "task_id": 66,
        "test_case_id": 39,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "95 19 19\n",
        "output": "1/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 91,
        "task_id": 66,
        "test_case_id": 40,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "404 63 441\n",
        "output": "31/202\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 92,
        "task_id": 66,
        "test_case_id": 41,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "5566 4798 4798\n",
        "output": "1/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 93,
        "task_id": 66,
        "test_case_id": 43,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "763 358 358\n",
        "output": "1/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 94,
        "task_id": 66,
        "test_case_id": 44,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "85356138 7223 482120804\n",
        "output": "3611/42678069\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 95,
        "task_id": 66,
        "test_case_id": 45,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "674664088 435395270 5\n",
        "output": "9/674664088\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 96,
        "task_id": 66,
        "test_case_id": 46,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "762200126044291557 370330636048898430 6\n",
        "output": "17/762200126044291557\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 97,
        "task_id": 66,
        "test_case_id": 47,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "917148533938841535 47 344459175789842163\n",
        "output": "28/183429706787768307\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 98,
        "task_id": 66,
        "test_case_id": 48,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "360212127113008697 877228952036215545 5259\n",
        "output": "5258/360212127113008697\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 99,
        "task_id": 66,
        "test_case_id": 49,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "683705963104411677 89876390 116741460012229240\n",
        "output": "539258339/683705963104411677\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 100,
        "task_id": 66,
        "test_case_id": 50,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "573003994959686829 275856334120822851 1319886766128339\n",
        "output": "3959660298385016/573003994959686829\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 101,
        "task_id": 66,
        "test_case_id": 52,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "19 1 19\n",
        "output": "1/19\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 102,
        "task_id": 66,
        "test_case_id": 53,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "576 18 32\n",
        "output": "1/16\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 103,
        "task_id": 66,
        "test_case_id": 54,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "9540 10 954\n",
        "output": "1/477\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 104,
        "task_id": 66,
        "test_case_id": 55,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "101997840 6 16999640\n",
        "output": "1/8499820\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 105,
        "task_id": 66,
        "test_case_id": 56,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "955944 1278 748\n",
        "output": "1/639\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 106,
        "task_id": 66,
        "test_case_id": 57,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "482120804 66748 7223\n",
        "output": "1/66748\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 107,
        "task_id": 66,
        "test_case_id": 58,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "370330636048898430 61721772674816405 6\n",
        "output": "1/61721772674816405\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 108,
        "task_id": 66,
        "test_case_id": 59,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "344459175789842163 7328918633826429 47\n",
        "output": "1/7328918633826429\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 109,
        "task_id": 66,
        "test_case_id": 60,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "877228952036215545 166805277055755 5259\n",
        "output": "1/55601759018585\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 110,
        "task_id": 66,
        "test_case_id": 61,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "116741460012229240 1298911316 89876390\n",
        "output": "1/649455658\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 111,
        "task_id": 66,
        "test_case_id": 62,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "275856334120822851 209 1319886766128339\n",
        "output": "1/1319886766128339\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 112,
        "task_id": 66,
        "test_case_id": 63,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "413850294331656955 1 413850294331656955\n",
        "output": "1/413850294331656955\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 113,
        "task_id": 66,
        "test_case_id": 64,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "54682301 778668 253103\n",
        "output": "253102/54682301\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 114,
        "task_id": 66,
        "test_case_id": 65,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "329245015 3931027 6443236\n",
        "output": "357366/29931365\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 115,
        "task_id": 66,
        "test_case_id": 66,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "321076647734423976 7 8\n",
        "output": "1672274206950125/13378193655600999\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 116,
        "task_id": 66,
        "test_case_id": 67,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "455227494055672047 71 60\n",
        "output": "6411654845854559/455227494055672047\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 117,
        "task_id": 66,
        "test_case_id": 68,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "595779167455745259 9741 9331\n",
        "output": "61162012885196/595779167455745259\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 118,
        "task_id": 66,
        "test_case_id": 69,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "6470 80 160\n",
        "output": "327/647\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 119,
        "task_id": 66,
        "test_case_id": 70,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "686325 828 1656\n",
        "output": "114511/228775\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 120,
        "task_id": 66,
        "test_case_id": 71,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "4535304 2129 4258\n",
        "output": "755973/1511768\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 121,
        "task_id": 66,
        "test_case_id": 72,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "40525189 6365 12730\n",
        "output": "20265394/40525189\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 122,
        "task_id": 66,
        "test_case_id": 73,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "675297075 25986 51972\n",
        "output": "112553659/225099025\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 123,
        "task_id": 66,
        "test_case_id": 74,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "5681598412 75376 226128\n",
        "output": "1893897375/5681598412\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 124,
        "task_id": 66,
        "test_case_id": 75,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "384118571739435733 619773000 1859319000\n",
        "output": "128039524053435733/384118571739435733\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 125,
        "task_id": 66,
        "test_case_id": 76,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "391554751752251913 625743359 1877230077\n",
        "output": "130518250652782079/391554751752251913\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 126,
        "task_id": 66,
        "test_case_id": 77,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "390728504279201198 625082797 1250165594\n",
        "output": "195364252413988195/390728504279201198\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 127,
        "task_id": 66,
        "test_case_id": 78,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "389902265396085075 624421544 1248843088\n",
        "output": "64983710976697837/129967421798695025\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 128,
        "task_id": 66,
        "test_case_id": 79,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "734812071040507372 857211800 2571635400\n",
        "output": "61234339274051543/183703017760126843\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 129,
        "task_id": 66,
        "test_case_id": 80,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1 1 2\n",
        "output": "0/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 130,
        "task_id": 66,
        "test_case_id": 81,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "3 1 4\n",
        "output": "0/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 131,
        "task_id": 66,
        "test_case_id": 82,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "8 2 3\n",
        "output": "3/8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 132,
        "task_id": 66,
        "test_case_id": 83,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "64 32 16\n",
        "output": "1/2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 133,
        "task_id": 66,
        "test_case_id": 84,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1 1 1000000000\n",
        "output": "0/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 134,
        "task_id": 66,
        "test_case_id": 85,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1000000000 1 1\n",
        "output": "1/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 135,
        "task_id": 66,
        "test_case_id": 86,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1000000000 1000000000 1000000000\n",
        "output": "1/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 136,
        "task_id": 66,
        "test_case_id": 87,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1000000000 2 4\n",
        "output": "1/2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 137,
        "task_id": 66,
        "test_case_id": 88,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1000000000 123 456\n",
        "output": "6579023/1000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 138,
        "task_id": 66,
        "test_case_id": 89,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "1000000000 123123 654\n",
        "output": "24851/1000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 139,
        "task_id": 66,
        "test_case_id": 90,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "123456 123 456\n",
        "output": "215/30864\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 140,
        "task_id": 66,
        "test_case_id": 91,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "123456 1234567 123\n",
        "output": "61/61728\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 141,
        "task_id": 66,
        "test_case_id": 92,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "314159265 271 8281\n",
        "output": "37939/314159265\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 142,
        "task_id": 66,
        "test_case_id": 93,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "11071994 4231 1324\n",
        "output": "2647/11071994\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 143,
        "task_id": 66,
        "test_case_id": 95,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "15485221 1259 90863\n",
        "output": "1258/15485221\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 144,
        "task_id": 66,
        "test_case_id": 96,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "5000000000000000000 4999999999999999837 4999999999999999963\n",
        "output": "1249999999999999959/1250000000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 145,
        "task_id": 66,
        "test_case_id": 97,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "4000000000000000000 3999999999999999691 3999999999999999887\n",
        "output": "399999999999999969/400000000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 146,
        "task_id": 66,
        "test_case_id": 98,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "999999999999999999 999999999999999709 999999999999999737\n",
        "output": "333333333333333236/333333333333333333\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 147,
        "task_id": 66,
        "test_case_id": 99,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "799999999999999999 799999999999999969 799999999999999991\n",
        "output": "799999999999999968/799999999999999999\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 148,
        "task_id": 66,
        "test_case_id": 100,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "812312312312312222 812312312312311897 812312312312312029\n",
        "output": "406156156156155948/406156156156156111\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 149,
        "task_id": 66,
        "test_case_id": 101,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "500000000000000000 499999999999999927 499999999999999931\n",
        "output": "249999999999999963/250000000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 150,
        "task_id": 66,
        "test_case_id": 102,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "555555555555555555 555555555555555083 555555555555555229\n",
        "output": "50505050505050462/50505050505050505\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 151,
        "task_id": 66,
        "test_case_id": 103,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "199419941994199419 199419941994199369 199419941994199391\n",
        "output": "66473313998066456/66473313998066473\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 152,
        "task_id": 66,
        "test_case_id": 104,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "145685485411238588 145685485411238483 145685485411238573\n",
        "output": "72842742705619241/72842742705619294\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 153,
        "task_id": 66,
        "test_case_id": 105,
        "question": "Vector Willman and Array Bolt are the two most famous athletes of Byteforces. They are going to compete in a race with a distance of L meters today.\n\n [Image] \n\nWillman and Bolt have exactly the same speed, so when they compete the result is always a tie. That is a problem for the organizers because they want a winner. \n\nWhile watching previous races the organizers have noticed that Willman can perform only steps of length equal to w meters, and Bolt can perform only steps of length equal to b meters. Organizers decided to slightly change the rules of the race. Now, at the end of the racetrack there will be an abyss, and the winner will be declared the athlete, who manages to run farther from the starting point of the the racetrack (which is not the subject to change by any of the athletes). \n\nNote that none of the athletes can run infinitely far, as they both will at some moment of time face the point, such that only one step further will cause them to fall in the abyss. In other words, the athlete will not fall into the abyss if the total length of all his steps will be less or equal to the chosen distance L.\n\nSince the organizers are very fair, the are going to set the length of the racetrack as an integer chosen randomly and uniformly in range from 1 to t (both are included). What is the probability that Willman and Bolt tie again today?\n\n\n-----Input-----\n\nThe first line of the input contains three integers t, w and b (1 ≤ t, w, b ≤ 5·10^18) — the maximum possible length of the racetrack, the length of Willman's steps and the length of Bolt's steps respectively.\n\n\n-----Output-----\n\nPrint the answer to the problem as an irreducible fraction [Image]. Follow the format of the samples output.\n\nThe fraction [Image] (p and q are integers, and both p ≥ 0 and q > 0 holds) is called irreducible, if there is no such integer d > 1, that both p and q are divisible by d.\n\n\n-----Examples-----\nInput\n10 3 2\n\nOutput\n3/10\n\nInput\n7 1 2\n\nOutput\n3/7\n\n\n\n-----Note-----\n\nIn the first sample Willman and Bolt will tie in case 1, 6 or 7 are chosen as the length of the racetrack.",
        "solutions": "[\"from fractions import gcd\\n\\nt,w,b = map(int,input().split())\\nper = w*b//gcd(w,b)\\ncan = (t//per+1)*min(w,b)-1\\nif t%per<min(w,b):\\n    can-=min(w,b)\\n    can+=t%per+1\\ng = gcd(can,t)\\ncan//=g\\nt//=g\\nprint(str(can)+\\\"/\\\"+str(t))\", \"\\nfrom fractions import gcd\\na, b, c = list(map(int, input().split(' ')))\\nl = b * c // gcd(b, c)\\nb, c = min(b, c), max(b, c)\\n\\n## 0...b-1 ##\\nmults = a // l\\nrem = a - l * mults + 1\\n\\nnum = mults * (b)\\nrem = min(b, rem)\\nx = num + rem - 1\\n\\ng = gcd(x, a)\\nprint(str(x//g) + '/' + str(a//g))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef lcm(a,b):\\n    return (a*b)//gcd(a,b)\\nt,w,b = map(int,input().split())\\nlc = lcm(w,b)\\nmn = 0\\nif w > b:\\n    mn = b\\nelse:\\n    mn = w\\nans = mn*(t//lc+1)-1\\nval = (t//lc)*lc + mn - 1\\nif t - val < 0:\\n    ans += t-val\\ng = gcd(ans,t)\\nans //= g\\nt //= g\\nprint(ans,end=\\\"\\\")\\nprint(\\\"/\\\",end=\\\"\\\")\\nprint(t)\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nmn = min(w, b)\\nnk = w // gcd(w, b) * b\\nfirst = (t  + 1) // nk * mn - 1\\ne = first + min(mn, (t + 1) % nk)\\nnd = gcd(e, t)\\nprint(e // nd, \\\"/\\\", t // nd, sep = \\\"\\\")\\n\", \"from fractions import gcd\\n\\nt, w, b = map(int, input().split())\\nnok = w * b // gcd(w, b)\\nans = (t // nok) * min(w, b) + min(w - 1, b - 1, t % nok)\\nans, t = ans // gcd(ans, t), t // gcd(ans, t)\\nprint (ans, t, sep = \\\"/\\\")\\n\", \"def gcd(a, b):\\n    while (b > 0):\\n        a, b = b, a % b\\n    return a\\n\\ninp = [int(i) for i in input().split(' ')]\\nt = inp[0]\\nw = inp[1]\\nb = inp[2]\\nnok = w * b // gcd(w, b)\\nans = t // nok * min(w, b) - 1\\ntmp = t % nok\\nans += min(tmp + 1, min(w, b))\\ng = gcd(ans, t)\\nprint(ans // g, t // g, sep='/')\\n\", \"import math\\nimport sys\\n\\ndef Cmmdc(a, b):\\n\\tr = 0\\n\\twhile b > 0:\\n\\t\\tr = a % b\\n\\t\\ta = b\\n\\t\\tb = r\\n\\treturn a\\n\\nlst = list(map(int, input().split()))\\n\\nt = auxT = lst[0]\\nn = lst[1]\\nm = lst[2]\\n\\nif n > m:\\n\\taux = n\\n\\tn = m\\n\\tm = aux\\n\\ncmmdc = Cmmdc(n, m)\\ncmmmc = n * m // cmmdc\\n\\nfav = (t // cmmmc) * n\\nt = t % cmmmc\\nfav += min(n - 1, t)\\n\\nt = auxT\\nc = Cmmdc(fav, t)\\nfav //= c\\nt //= c\\n\\nprint(str(fav) + \\\"/\\\" + str(t) + \\\"\\\\n\\\")\\n\", \"import sys\\nfrom math import *\\nsys.setrecursionlimit(100000000)\\n\\ndef pgcd(a,b):\\n    while b!=0:\\n        a,b=b,a%b\\n    return a\\n\\ndef ppcm(a,b):\\n    if (a==0) or (b==0):\\n        return 0\\n    else:\\n        return (a*b)//pgcd(a,b)\\n\\nt,w,b=map(int,input().split())\\na=ppcm(w,b)\\nx=min(w,b)\\ny=t//a\\nv=x*y+min(x,t%a+1)-1\\n\\nw=pgcd(v,t)\\nif v==0:print(\\\"0/1\\\")\\nelse:\\n\\tprint(v//w,end=\\\"\\\")\\n\\tprint(\\\"/\\\",end=\\\"\\\")\\n\\tprint(t//w)\", \"t, w, b = list(map(int, input().split()))\\n\\n\\ndef NOK(a, b):\\n    m = a*b\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return m // (a+b)\\n\\ndef NOD(a, b):\\n    while a != 0 and b != 0:\\n        if a > b:\\n            a %= b\\n        else:\\n            b %= a\\n    return (a+b)\\n\\n\\nif w == 1 or b == 1:\\n    res = t // max (w, b)\\nelse:\\n    k = NOK(w, b)\\n    ost = max(0, min(w, b) - 1 - t % k)\\n    res = (t // k + 1) * min(w,b) - 1 - ost\\n\\nm = NOD(t, res)\\n\\nprint(str(res // m) + '/' + str(t // m))\\n\", \"def gcd(a,b):\\n    if b == 0: return a\\n    return gcd(b, a%b)\\n\\ndef lcm(a,b):\\n    return a//gcd(a,b)*b\\n\\nt,w,b = map(int,input().split())\\np = min(w,b)\\nlc = lcm(w,b)\\nkol = t//lc\\nret = kol*p\\nzv = t%lc\\nret += min(zv, p-1)\\ng = gcd(ret, t)\\nret//=g\\nt//=g\\nprint(ret,'/',t,sep=\\\"\\\")\\n\", \"l, n, m = map(int, input().split())\\nfrom fractions import gcd\\nlcm = lambda x, y: x // gcd(x,y) * y\\nu = lcm(n,m)\\nv = min(n,m)\\na = (l//u) * v + min(v, l%u+1) - 1    \\nprint(a//gcd(a,l),'/',l//gcd(a,l),sep='')\\n\\n\", \"#!/usr/bin/env python3\\nimport math\\nt, a, b = list(map(int,input().split()))\\nl= a * b // math.gcd(a,b)\\np = (t // l) * min(a,b) + min(t % l, min(a,b) - 1)\\nq = t\\nr = math.gcd(p, q)\\nprint('{}/{}'.format(p//r, q//r))\\n\", \"#!/usr/bin/env python3\\n\\ndef gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\ndef main():\\n    L, a, b = list(map(int, input().split()))\\n    c = a // gcd(a, b) * b\\n    m = min(a, b)\\n    if c <= m:\\n        print('1/1')\\n    else:\\n        p = L // c * m\\n        p += min(L % c, m - 1)\\n        g = gcd(p, L)\\n        print('%d/%d' % (p // g, L // g))\\n\\nmain()\", \"def gcd (a, b) :\\n\\twhile (b) :\\n\\t\\ta %= b\\n\\t\\ta, b = b, a\\n\\treturn a;\\n\\n\\nt, w, b = map(int, input().split())\\ng = w * b // gcd(w, b)\\nres = 0\\nminh = min(w, b)\\nres += (t // g + 1) * minh - 1\\ncorrect = (t // g) * g + minh - 1\\nif (correct > t) :\\n\\tres -= correct - t\\ny = gcd(res, t)\\nprint(res // y, \\\"/\\\", t // y, sep = \\\"\\\")\", \"from fractions import gcd\\n\\nt, a, b = map(int, input().split())\\n\\nif a > b:\\n    a, b = b, a\\n\\nlcm = a * b // gcd(a, b)\\ncnt = t // lcm\\nlst = lcm * cnt\\n\\nans = cnt * a + a - 1 \\n\\nif lst + a > t + 1:\\n    ans -= lst + a - t - 1\\n\\nnum = ans\\nden = t\\n\\ng = gcd(num, den)\\nnum //= g\\nden //= g\\n\\nprint(num, den, sep='/')\\n\\n\", \"from fractions import gcd\\n\\ndef lcm(a, b):\\n    return (a * b) // gcd(a, b)\\n\\n\\ndef __starting_point():\\n    t, w, b = list(map(int, input().split()))\\n\\n    l = lcm(w, b)\\n    m = min(w, b)\\n\\n    count = t // l\\n    result = count * m\\n    result += (m - 1) # 1 to m-1\\n\\n    diff = max(count*l + m - t - 1, 0)\\n    result -= diff\\n\\n    g = gcd(result, t)\\n\\n    print('{}/{}'.format(result//g, t//g))\\n\\n\\n\\n__starting_point()\", \"t, w, b = map(int, input().split())\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nadd = min(w, b) - 1\\nl = lcm(w, b)\\n\\ncnt = t // l\\n\\nans = add + cnt + cnt * add\\nans -= max(0, l * cnt + add - t)\\n\\ng = gcd(ans, t)\\nif g != 0:\\n    ans //= g\\n    t //= g\\n\\nprint(ans, end='')\\nprint('/', end='')\\nprint(t)\\n\", \"3\\n\\ndef gcd(a, b):\\n\\tif b == 0:\\n\\t\\treturn a\\n\\treturn gcd(b, a%b)\\n\\nx = input()\\nx = [int(_) for _ in x.split()]\\n# print(x)\\n\\nt = x[0]\\nw = x[1]\\nb = x[2]\\n\\nx = gcd(w, b)\\nk = min(w,b)\\n\\nlcm = (w*b)//x\\n\\nalpha = t//lcm\\n\\nans = alpha*(k)\\n\\nl = alpha*lcm + k- 1\\n\\nif l <= t :\\n\\tans += k\\nelse:\\n\\tans += t - (alpha*lcm) + 1\\n\\nans -= 1\\n\\ngg = gcd(ans, t)\\nans = ans//gg\\nt = t//gg\\n\\nprint(str(ans)+\\\"/\\\"+str(t))\", \"# your code goes here\\n\\n[t, w, b] = [int(x) for x in input().split()]\\n\\ndef gcd(a, b):\\n    if (b==0):\\n        return a\\n    else:\\n        return gcd(b, a%b)\\n\\nd = w*b // gcd(w, b)\\nm = min(w, b)\\n\\ndint = t // d\\n\\ncount = m * dint\\n\\ncount += m - 1\\n    \\nd = dint * d + m - 1\\n\\nif (d > t):\\n    count -= (d - t)\\n\\ngcdtcnt = gcd(t, count)\\nt = t // gcdtcnt\\ncount = count // gcdtcnt\\n\\nprint(count, '/', t, sep='')\", \"#! /usr/bin/python\\n\\nfrom fractions import gcd\\n\\nt, w, b = list(map(int, input().split()))\\n\\nif w == b:\\n    print('1/1')\\nelse:\\n    wb = w * b // gcd(w, b)\\n    m = min(w, b)\\n    n = t // wb * m - 1 + min(t % wb + 1, m)\\n    g = gcd(n, t)\\n    print(\\\"%d/%d\\\" % (n // g, t // g))\\n\", \"t, a, b = list(map(int, input().split()))\\n\\ndef gcd(a, b):\\n    if (b == 0):\\n        return a\\n    return gcd(b, a % b)\\n\\ndef lcm(a, b):\\n    return a // gcd(a, b) * b\\n\\nl = lcm(a, b)\\nlast = t // l * l\\nans = t // l * min(a, b)\\nans += min(a, b, t - last + 1)\\nans -= 1\\ng = gcd(ans, t)\\nans //= g\\nt //= g\\nans = str(ans) + '/' + str(t)\\nprint(ans)\\n\\n\", \"def gcd(a, b):\\n    return a if b == 0 else gcd(b, a % b)\\n\\n\\nt, w, b = [int(i) for i in input().split()]\\nmi = min(w, b)\\nlcm = w * b // gcd(w, b)\\n\\np = t // lcm * mi + min(mi - 1, t % lcm)\\nq = t\\nprint(\\\"{0}/{1}\\\".format(p // gcd(p, q), q // gcd(p, q)))\", \"t, w, b = list(map(int, input().split()))\\n\\ndef gcd(x, y):\\n    while (x != 0 and y != 0):\\n        if (x < y):\\n            x, y = y, x\\n        x %= y\\n    return x + y;\\n\\nans = (t // (w * b // gcd(w, b)) - 1) * min(w, b)\\nans += min(t - (t // (w * b // gcd(w, b))) * (w * b // gcd(w, b)) + 1, min(b, w))\\nans += min(w, b) - 1;\\nprint(ans // gcd(ans, t), end = \\\"/\\\")\\nprint(t // gcd(ans, t))\", \"from fractions import gcd\\ns = list(map(int, input().split()))\\nt = s[0]\\na = s[1]\\nb = s[2]\\nnod = gcd(a, b)\\nnok = a * b // nod\\nans = t // nok * min(a, b) + min(a, b) - 1\\nans -= max(0, ((t // nok) * nok) + min(a, b) - 1 - t)\\ngc = gcd(ans, t)\\nans //= gc\\nt //= gc\\nprint(str(int(ans)) + '/' + str(int(t)))\", \"from fractions import gcd\\nt, w, b = list(map(int, input().split()))\\nif w > b:\\n    w, b = b, w\\nl = w * b // gcd(w, b)\\nc = t // l\\nans = c * w + (min((t + 1) - l * c, w)) - 1\\ng = gcd(ans, t)\\nprint('{}/{}'.format(ans // g, t // g))\\n\"]",
        "difficulty": "interview",
        "input": "314159265358979323 314159265358979167 314159265358979213\n",
        "output": "314159265358979166/314159265358979323\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/592/C"
    },
    {
        "id": 154,
        "task_id": 643,
        "test_case_id": 1,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n",
        "output": "4\n10\n0\n-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 155,
        "task_id": 643,
        "test_case_id": 2,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "8\n0 1 0 1\n0 2 1 2\n0 3 1 1\n1 2 0 1\n1 2 1 1\n2 2 0 1\n3 3 1 2\n4 4 1 1\n",
        "output": "0\n2\n-1\n-1\n-1\n-1\n3\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 156,
        "task_id": 643,
        "test_case_id": 3,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n1 1000000000 1 2\n1 1000000000 1 2\n1 1000000000 1 2\n1 1000000000 1 2\n1 1000000000 1 2\n",
        "output": "999999998\n999999998\n999999998\n999999998\n999999998\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 157,
        "task_id": 643,
        "test_case_id": 4,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n999999999 1000000000 1 1000000000\n999999999 1000000000 1 1000000000\n999999999 1000000000 1 1000000000\n999999999 1000000000 1 1000000000\n999999999 1000000000 1 1000000000\n",
        "output": "999999998000000000\n999999998000000000\n999999998000000000\n999999998000000000\n999999998000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 158,
        "task_id": 643,
        "test_case_id": 5,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n0 1000000000 999999999 1000000000\n0 1000000000 999999999 1000000000\n0 1000000000 999999999 1000000000\n0 1000000000 999999999 1000000000\n0 1000000000 999999999 1000000000\n",
        "output": "999999999000000000\n999999999000000000\n999999999000000000\n999999999000000000\n999999999000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 159,
        "task_id": 643,
        "test_case_id": 6,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "1\n999999999 1000000000 1 2\n",
        "output": "999999998\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 160,
        "task_id": 643,
        "test_case_id": 9,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "1\n3 999999990 1 1000000000\n",
        "output": "2000000010\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 161,
        "task_id": 643,
        "test_case_id": 10,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n1 1 1 1\n",
        "output": "4\n10\n0\n-1\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 162,
        "task_id": 643,
        "test_case_id": 11,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n9999999 10000000 1 1000000000\n9999999 10000000 1 1000000000\n9999999 10000000 1 1000000000\n9999999 10000000 1 1000000000\n9999999 10000000 1 1000000000\n",
        "output": "9999998990000000\n9999998990000000\n9999998990000000\n9999998990000000\n9999998990000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 163,
        "task_id": 643,
        "test_case_id": 12,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "1\n0 1000000000 999999999 1000000000\n",
        "output": "999999999000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 164,
        "task_id": 643,
        "test_case_id": 13,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n1 1000000000 999999999 1000000000\n1 1000000000 999999999 1000000000\n1 1000000000 999999999 1000000000\n1 1000000000 999999999 1000000000\n1 1000000000 999999999 1000000000\n",
        "output": "999999998000000000\n999999998000000000\n999999998000000000\n999999998000000000\n999999998000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 165,
        "task_id": 643,
        "test_case_id": 14,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n1 1000000000 999999999 1000000000\n2 1000000000 999999999 1000000000\n3 1000000000 999999999 1000000000\n4 1000000000 999999999 1000000000\n5 1000000000 999999999 1000000000\n",
        "output": "999999998000000000\n999999997000000000\n999999996000000000\n999999995000000000\n999999994000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 166,
        "task_id": 643,
        "test_case_id": 16,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n999999997 999999998 2 999999999\n999999997 999999998 2 999999999\n999999997 999999998 2 999999999\n999999997 999999998 2 999999999\n999999997 999999998 2 999999999\n",
        "output": "499999997500000003\n499999997500000003\n499999997500000003\n499999997500000003\n499999997500000003\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 167,
        "task_id": 643,
        "test_case_id": 18,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n99999997 999999998 999999998 999999999\n99999996 999999997 999999997 999999999\n99999997 999999998 999999998 999999999\n99999996 999999997 999999997 999999999\n99999997 999999998 999999998 999999999\n",
        "output": "899999999100000001\n449999999550000002\n899999999100000001\n449999999550000002\n899999999100000001\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 168,
        "task_id": 643,
        "test_case_id": 23,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "1\n1 1000000000 999999999 1000000000\n",
        "output": "999999998000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 169,
        "task_id": 643,
        "test_case_id": 24,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "4\n1 1000000000 999999999 1000000000\n999999999 1000000000 1 1000000000\n1 2 0 1\n0 1 0 1\n",
        "output": "999999998000000000\n999999998000000000\n-1\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 170,
        "task_id": 643,
        "test_case_id": 25,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "1\n1 1000000000 1 2\n",
        "output": "999999998\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 171,
        "task_id": 643,
        "test_case_id": 26,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n1 982449707 1 2\n1 982449707 1 2\n1 982449707 1 2\n1 982449707 1 2\n1 982449707 1 2\n",
        "output": "982449705\n982449705\n982449705\n982449705\n982449705\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 172,
        "task_id": 643,
        "test_case_id": 27,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n13 900000007 900000007 900000009\n13 900000007 900000007 900000009\n13 900000007 900000007 900000009\n13 900000007 900000007 900000009\n13 900000007 900000007 900000009\n",
        "output": "405000000449999966\n405000000449999966\n405000000449999966\n405000000449999966\n405000000449999966\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 173,
        "task_id": 643,
        "test_case_id": 28,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "1\n5 10 0 1\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 174,
        "task_id": 643,
        "test_case_id": 30,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n0 999999999 999999999 1000000000\n0 999999999 999999999 1000000000\n0 999999999 999999999 1000000000\n0 999999999 999999999 1000000000\n0 999999999 999999999 1000000000\n",
        "output": "999999998000000001\n999999998000000001\n999999998000000001\n999999998000000001\n999999998000000001\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 175,
        "task_id": 643,
        "test_case_id": 32,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n999999999 1000000000 1 9999\n999999999 1000000000 1 9999\n999999999 1000000000 1 9999\n999999999 1000000000 1 9999\n999999999 1000000000 1 9999\n",
        "output": "9997999990001\n9997999990001\n9997999990001\n9997999990001\n9997999990001\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 176,
        "task_id": 643,
        "test_case_id": 33,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "5\n999999997 1000000000 3 1000000000\n999999997 1000000000 3 1000000000\n999999997 1000000000 3 1000000000\n999999997 1000000000 3 1000000000\n999999997 1000000000 3 1000000000\n",
        "output": "333333332000000000\n333333332000000000\n333333332000000000\n333333332000000000\n333333332000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 177,
        "task_id": 643,
        "test_case_id": 36,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "1\n999999998 999999999 1 10\n",
        "output": "8999999981\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 178,
        "task_id": 643,
        "test_case_id": 38,
        "question": "You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.\n\nYour favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?\n\n\n-----Input-----\n\nThe first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.\n\nEach of the next t lines contains four integers x, y, p and q (0 ≤ x ≤ y ≤ 10^9; 0 ≤ p ≤ q ≤ 10^9; y > 0; q > 0).\n\nIt is guaranteed that p / q is an irreducible fraction.\n\nHacks. For hacks, an additional constraint of t ≤ 5 must be met.\n\n\n-----Output-----\n\nFor each test case, output a single integer equal to the smallest number of submissions you have to make if you want your success rate to be equal to your favorite rational number, or -1 if this is impossible to achieve.\n\n\n-----Example-----\nInput\n4\n3 10 1 2\n7 14 3 8\n20 70 2 7\n5 6 1 1\n\nOutput\n4\n10\n0\n-1\n\n\n\n-----Note-----\n\nIn the first example, you have to make 4 successful submissions. Your success rate will be equal to 7 / 14, or 1 / 2.\n\nIn the second example, you have to make 2 successful and 8 unsuccessful submissions. Your success rate will be equal to 9 / 24, or 3 / 8.\n\nIn the third example, there is no need to make any new submissions. Your success rate is already equal to 20 / 70, or 2 / 7.\n\nIn the fourth example, the only unsuccessful submission breaks your hopes of having the success rate equal to 1.",
        "solutions": "[\"def solve(x, y, p, q):\\n    if p == 0: return 0 if x == 0 else -1\\n    pp = (x - 1) // p + 1 if p != 0 else 0\\n    L = max((y - 1) // q + 1, pp) - 1\\n    L = max(L, -1)\\n    z = y - x\\n    INF = L + 10 ** 10\\n    R = INF\\n    while R - L > 1:\\n        M = (L + R) >> 1\\n        cur = q * M\\n        curp = p * M\\n        curz = cur - curp\\n        dl = cur - y\\n        if curp >= x and curz >= z:\\n            R = M\\n        else:\\n            L = M\\n        #print(L, R)\\n    if R == INF:\\n        return -1\\n    return R * q - y\\n\\nread = lambda: map(int, input().split())\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = read()\\n    print(solve(x, y, p, q))\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\nclass solution:\\n    def __init__(self, a=0, b=0):\\n        self.x = a\\n        self.y = b\\n\\ndef eu (a, b, sol):\\n    if a == 0:\\n        sol.x = 0\\n        sol.y = 1\\n        return b\\n    sol2 = solution()\\n    d = eu (b%a, a, sol2)\\n    sol.x = sol2.y - (b // a) * sol2.x\\n    sol.y = sol2.x\\n    return d\\n\\n\\ndef find_any_solution (a, b, c, sol):\\n    g = eu(abs(a), abs(b), sol)\\n    if c % g != 0:\\n        return -1\\n    sol.x *= c // g\\n    sol.y *= c // g\\n    if (a < 0):\\n        sol.x *= -1\\n    if (b < 0):\\n        sol.y *= -1\\n    return g\\n\\n\\ndef shift_solution (sol, a, b, cnt):\\n    sol.x += cnt * b\\n    sol.y -= cnt * a\\n\\n\\ndef find_all_solution(a, b, c, minx, maxx, miny, maxy):\\n    sol = solution()\\n    g = find_any_solution(a, b, c, sol)\\n    if g == -1:\\n        return (-1, -1)\\n\\n    a //= g\\n    b //= g\\n\\n    sign_a = 1\\n    if a < 0:\\n        sign_a = -1\\n    sign_b = 1\\n    if b < 0:\\n        sign_b = -1\\n\\n    shift_solution(sol, a, b, (minx - sol.x) // b)\\n    if sol.x < minx:\\n        shift_solution (sol, a, b, sign_b)\\n    if sol.x > maxx:\\n        return (-1, -1)\\n    lx1 = sol.x\\n\\n    shift_solution (sol, a, b, (maxx - sol.x) // b)\\n    if sol.x > maxx:\\n        shift_solution (sol, a, b, -sign_b)\\n    rx1 = sol.x\\n\\n    shift_solution (sol, a, b, - (miny - sol.y) // a)\\n    if sol.y < miny:\\n        shift_solution (sol, a, b, -sign_a)\\n    if sol.y > maxy:\\n        return (-1, -1)\\n    lx2 = sol.x\\n\\n    shift_solution (sol, a, b, - (maxy - sol.y) // a)\\n    if sol.y > maxy:\\n        shift_solution (sol, a, b, sign_a)\\n    rx2 = sol.x\\n\\n    if lx2 > rx2:\\n        lx2, rx2 = rx2, lx2\\n    lx = max (lx1, lx2)\\n    rx = min (rx1, rx2)\\n\\n    if lx > rx:\\n        return (-1, -1)\\n    return (lx, rx)\\n\\ndef solve():\\n    s = input().split()\\n    x = int(s[0])\\n    y = int(s[1])\\n    p = int(s[2])\\n    q = int(s[3])\\n\\n    # x, y, p, q = 3, 10, 1, 2\\n \\n    if p == 0:\\n        if x == 0:\\n            return 0\\n        else:\\n            return -1\\n    if q == p:\\n        if x == y:\\n            return 0\\n        return -1\\n    if p * y - q * x == 0:\\n        return 0\\n\\n    a = q - p\\n    b = -p\\n    c = p * y - q * x\\n\\n    ans1, ans2 = find_all_solution(a, b, c, 0, int(10 ** 20), 0, int(10 ** 20))\\n\\n    ansy1 = (p * y - q * x - (q - p) * ans1) // (-p)\\n    ansy2 = (p * y - q * x - (q - p) * ans2) // (-p)\\n\\n    # print(x, y)\\n\\n    sum1 = int(10 ** 25)\\n    if ans1 >= 0 and ansy1 >= 0 and (x + ans1) * q == (y + ansy1 + ans1) * p:\\n        sum1 = min(sum1, ans1 + ansy1)\\n\\n    if ans2 >= 0 and ansy2 >= 0 and (x + ans2) * q == (y + ansy2 + ans2) * p:\\n        sum1 = min(sum1, ans2 + ansy2)\\n    if sum1 == int(10 ** 25):\\n        return -1\\n    return sum1\\n# print(solve())\\n\\nt = int(input())\\nfor i in range(t):\\n    print(solve())\", \"t = int(input())\\nfor i in range(t):\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == q:\\n        if x != y:\\n            print(-1)\\n        else:\\n            if x == y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        continue\\n    if p == 0:\\n        if x == 0:\\n            if y == 0:\\n                print(1)\\n            else:\\n                print(0)\\n        else:\\n            print(-1)\\n        continue\\n    \\n    k = max((y - x + q - p - 1) // (q - p), (x + p - 1) // p) \\n    print(k * q - y)\", \"def gcd(a, b):\\n\\tif a == 0:\\n\\t\\treturn [b, 0, 1]\\n\\td = gcd(b % a, a)\\n\\treturn [d[0], d[2] - (b // a) * d[1], d[1]]\\n\\nt = int(input())\\nwhile t > 0:\\n\\tt -= 1\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tif p == q:\\n\\t\\tif x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\tif p == 0:\\n\\t\\tif x == 0:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\ta = p - q\\n\\tb = p\\n\\tc = q * x - p * y\\n\\tg, xa, ya = gcd(abs(a), abs(b))\\n\\tif c % g != 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\txa *= c // g\\n\\t\\tya *= c // g\\n\\t\\tif a < 0:\\n\\t\\t\\txa = -xa\\n\\t\\tif b < 0:\\n\\t\\t\\tya = -ya\\n\\t\\tif xa < 0:\\n\\t\\t\\tgaps = (-xa + (b // g) - 1) // (b // g)\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\tif ya < 0:\\n\\t\\t\\tgaps = (-ya + (-a // g) - 1) // (-a // g)\\t\\t\\n\\t\\t\\txa += gaps * (b // g)\\n\\t\\t\\tya -= gaps * (a // g)\\n\\t\\t#print(xa, ya, a, b, c)\\n\\t\\tif xa < 0 or ya < 0:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\taddon = min(xa // (b // g), ya // (-a // g))\\n\\t\\t\\txa -= addon * (b // g)\\n\\t\\t\\tya += addon * (a // g)\\n\\t\\t\\tprint(xa + ya)\\n\", \"import sys\\ndef de(x, y):\\n\\tif (x % y == 0):\\n\\t\\t return x // y\\n\\treturn x // y + 1\\n\\ndef euc(a, b):\\n\\tif (b == 0):\\n\\t\\treturn 1, 0\\n\\tx, y = euc(b, a % b)\\n\\treturn -y, -x - y * (a // b)\\n\\n\\n\\ndef solve( x, y, p, q):\\n\\tif (q == p):\\n\\t\\tif (x == y):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\tif (p == 0):\\n\\t\\tif (x == 0):\\n\\t\\t\\t return 0\\n\\t\\treturn -1\\n\\t\\n\\t\\t\\n\\ta0, b0 = euc(p, q)\\n\\tg = a0 * p - q * b0\\n\\tc = x * q - p * y\\n\\tif (c % g):\\n\\t\\treturn -1\\n\\ta0 = a0 * (c // g)\\n\\tb0 = b0 * (c // g)\\n\\tt1 = a0 // q\\n\\t\\n\\tt = max(de(b0 - a0, q - p), de(-b0, p))\\n\\ta = a0 + q * t\\n\\tb = b0 + p * t\\n\\n\\treturn a\\n\\n\\n#sys.stdin = open('input.txt', 'r')\\n\\n\\nt = int(input())\\nfor it in range(t):\\n\\tx, y, p, q = list(map(int, input().split()))\\n\\tprint(solve(x, y, p, q))\\n\", \"def xgcd(a, b):\\n    if b == 0:\\n        return 1, 0\\n    x1, y1 = xgcd(b, a % b)\\n    return y1, x1 - (a // b) * y1\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\nt = int(input())\\nfor kek in range(t):\\n    x, y, p, q = map(int, input().split())\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    v = p * y - q * x\\n    g = gcd(q - p, p)\\n    k0, l0 = xgcd(q - p, p)\\n    l0 *= -1\\n    g1 = v // g\\n    k0 *= g1\\n    l0 *= g1\\n\\n    xx = (-k0 + p - 1) // p\\n    \\n    xx = max(xx, (-l0 + q - p - 1) // (q - p))\\n    \\n    l = l0 + xx * (q - p)\\n    k = k0 + xx * p;\\n\\n    print(k + l)\", \"def gcd(a, b):\\n    if a == 0:\\n        return b, 0, 1\\n\\n    g, x1, y1 = gcd(b % a, a);\\n\\n    y = x1;\\n    x = y1 - (b // a) * x1;\\n    return g, x, y;\\n\\ndef comp(da, db, t, a, b):\\n    ra = a + da * t\\n    rb = b + db * t\\n\\n    # assert(ra.v0 >= 0);\\n    # assert(ra.v1 >= 0);\\n    # assert(rb.v0 >= 0);\\n    # assert(rb.v1 >= 0);\\n    return ra + rb\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    if p == q:\\n        if x == y:\\n            print(0)\\n        else:\\n            print(-1)\\n        return\\n\\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        return;\\n\\n    r = x * q - y * p;\\n    g, a, b = gcd(p - q, p);\\n    if r % g != 0:\\n        print(-1)\\n        return\\n\\n    a *= r // g;\\n    b *= r // g;\\n\\n    da = p;\\n    db = q - p;\\n    minT = -10**18;\\n    minT = max(minT, ((-a + (da - 1)) // da));\\n    minT = max(minT, ((-b + (db - 1)) // db));\\n\\n    t = minT;\\n    rr = comp(da, db, t, a, b);\\n    print(rr)\\n\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"def gcd(a, b):\\n\\tif (a == 0):\\n\\t\\treturn (b, 0, 1)\\n\\n\\t(g, x, y) = gcd(b % a, a)\\n\\treturn (g, y - (b // a) * x, x) \\n\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\t(x, y, p, q) = list(map(int, input().split()))\\n\\tval = p * y - q * x\\n\\t(g, a, b) = gcd(q, -p)\\n\\n\\tif (val % g != 0):\\n\\t\\tprint(-1)\\n\\t\\tcontinue\\n\\n\\ta *= val // g\\n\\tb *= val // g\\n\\n\\tda = abs(p // g)\\n\\tdb = abs(q // g)\\n\\n\\tif (a < 0):\\n\\t\\tif (da == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(a) + da - 1) // da\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\n\\tif (b < 0):\\n\\t\\tif (db == 0):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tmul = (abs(b) + db - 1) // db;\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tif (a > b):\\n\\t\\tif (da == db):\\n\\t\\t\\tprint(-1)\\n\\t\\t\\tcontinue\\n\\n\\t\\tdiff = a - b\\n\\t\\tstep = db - da\\n\\t\\tmul = (diff + step - 1) // step\\n\\t\\ta += mul * da\\n\\t\\tb += mul * db\\n\\n\\tmul = b // db;\\n\\tif (da != 0):\\n\\t\\tmul = min(mul, a // da)\\n\\n\\tif (da != db):\\n\\t\\tmul = min(mul, (b - a) // (db - da))\\n\\n\\tprint(b - db * mul) \\n\", \"INF = 10**40\\n\\ndef gcd_ex(A, B):\\n    if A == 0:\\n        return B, 0, 1\\n    g, a1, b1 = gcd_ex(B % A, A)\\n    a = b1 - (B // A) * a1\\n    b = a1\\n    return g, a, b\\n\\ndef round_down(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return a // b\\n    return -((a + b - 1) // b)\\n\\ndef round_up(a, b):\\n    s = (a < 0) != (b < 0)\\n    a = abs(a)\\n    b = abs(b)\\n    if not s:\\n        return (a + b - 1) // b\\n    return -(a // b)\\n\\ndef solve():\\n    x, y, p, q = list(map(int, input().split()))\\n\\n    A = q\\n    B = -p\\n    C = p * y - q * x\\n\\n    g, a0, b0 = gcd_ex(A, -B)\\n    b0 *= -1\\n\\n    if C % g != 0:\\n        print(-1)\\n        return\\n    a0 *= C // g\\n    b0 *= C // g\\n\\n    k_up1 = 0\\n    if B == 0:\\n        if -a0 > 0:\\n            print(-1)\\n            return\\n        k_up1 = INF\\n    else:\\n        k_up1 = round_down(-a0, B // g)\\n\\n    k_down2 = 0\\n    k_up2 = 0\\n    if A + B == 0:\\n        if b0 - a0 < 0:\\n            print(-1)\\n            return\\n        k_down2 = -INF\\n        k_up2 = INF\\n    elif A + B > 0:\\n        k_down2 = -INF\\n        k_up2 = round_down(b0 - a0, (A + B) // g)\\n    else:\\n        k_down2 = round_up(b0 - a0, (A + B) // g)\\n        k_up2 = INF\\n\\n    k_down = k_down2\\n    k_up = min(k_up1, k_up2)\\n    if k_down > k_up:\\n        print(-1)\\n        return\\n\\n    if k_up == INF:\\n        raise Exception\\n    b = b0 - (A // g) * k_up\\n\\n    print(b)\\n\\ndef main():\\n    t = int(input())\\n    for i in range(t):\\n        solve()\\n\\nmain()\\n\", \"t = int(input())\\n\\nfor i in range(t):\\n    x, y, p, q = [int(i) for i in input().split()]\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n\\n    l = 0\\n    r = 10000000000\\n\\n    while l < r:\\n        t = (l + r) // 2\\n\\n        c1 = p * t - x\\n        c2 = q * t - y - c1\\n\\n        if c1 >= 0 and c2 >= 0:\\n            r = t\\n        else:\\n            l = t + 1\\n\\n    if r == 10000000000:\\n        print(-1)\\n    else:\\n        print(q * l - y)\\n\", \"def gcd(a, b):\\n    if b > a:\\n        return gcd(b, a)\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\ndef works(a, b, c, d, x):\\n    return b <= d * x and a <= c * x and d * x - b >= c * x - a\\n\\ndef solve():\\n    a, b, c, d = list(map(int, input().rstrip().split()))\\n    if c == d == 1 and not a / b == 1:\\n        print(-1)\\n        return\\n    if c == 0 and not a == 0:\\n        print(-1)\\n        return\\n    g = gcd(c, d)\\n    c //= g\\n    d //= g\\n    low = -1\\n    high = 1000000000000\\n    while low + 1 < high:\\n        mid = (low + high) // 2\\n        if works(a, b, c, d, mid):\\n            high = mid\\n        else:\\n            low = mid\\n    print(d * high - b)\\n\\ndef __starting_point():\\n    t = int(input())\\n    for _ in range(t):\\n        solve()\\n\\n__starting_point()\", \"# cook your code here\\ndef solve():\\n    x, y, p, q = map(int, input().split())\\n    \\n    if p == 0:\\n        if x == 0:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    if p == q:\\n        if x == y:\\n            print(0)\\n            return\\n        else:\\n            print(-1)\\n            return\\n        \\n    var1 = ( p + x - 1 ) // p;\\n    var2 = ((y - x) + (q - p) - 1) // (q - p);\\n    max1 = max(var1, var2);\\n    solution = (max1 * q) - y;\\n    \\n    print(solution)\\n    return\\n\\n\\ndef main():\\n    n = int(input())\\n    for i in range (n):\\n        solve()\\n        \\nmain()\", \"for i in range(int(input())):\\n    x, y, p, q = map(int, input().split())\\n    print(-(x > 0) if not p else -(y > x) if p == q else max((x - 1) // p, (y - x - 1) // (q - p)) * q + q - y)\", \"import math\\nimport sys\\nt=int(input())\\nfor j in range(1,(t+1)):\\n    x,y,p,q=(list(map(int,input().split())))\\n    f1=0\\n    if(p!=q and p!=0):\\n            tmp=max(((y-x)+(q-p)-1)//(q-p),(x+p-1)//p)\\n            #print(tmp)\\n            #include<FU*k> test case\\n            print((q*tmp)-y)\\n            f1=1\\n           \\n    if(p==0 and x==0):\\n            print(0)\\n    elif(p==q and x==y):\\n            print(0)\\n    elif (not f1):\\n            print(-1)\\n        \\n        \\n\\n\\n    \\n\", \"for case in range(int(input())):\\n    x,y,p,q = map(int, input().split())\\n\\n    lo = 0\\n    hi = 10**10\\n    while lo < hi:\\n        mid = lo + (hi - lo) // 2\\n         \\n        np,nq = mid*p, mid*q\\n        if nq >= y and np >= x:\\n            if nq - y >= np - x:\\n                hi = mid\\n            else:\\n                lo = mid + 1\\n        else:\\n            lo = mid + 1\\n     \\n    print(lo * q - y if lo != 10**10 else -1)\", \"for _ in range(int(input())):\\n\\n  x,y,p,q=list(map(int,input().split()))\\n\\n  l,r,res=0,10**18,-1\\n\\n  while l<=r:\\n\\n    mid=(l+r)//2\\n\\n    a,b=p*mid-x,q*mid-y\\n\\n    if a<=b and a>-1 and b>-1:res=b;r=mid-1\\n\\n    else :l=mid+1\\n\\n  print(res)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"R = lambda: map(int, input().split())\\nmx = 10**9 + 7\\nt = int(input())\\nfor i in range(t):\\n    a, b, p, q = R()\\n    l, r = 1, mx\\n    while l < r:\\n        k = (l + r) // 2\\n        x, y = k * p - a, k * q - b\\n        if 0 <= x <= y and y >= 0:\\n            r = k\\n        else:\\n            l = k + 1\\n    if r >= mx:\\n        print(\\\"-1\\\")\\n    else:\\n        print(r * q - b)\", \"N = int(input())\\nimport math\\nfor _ in range(N):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    if (p == 0) or (p==q):\\n        if (x*q == p*y):\\n            print(0)\\n        else:\\n            print(-1)\\n        continue\\n    n = math.ceil((y-x)/(q-p))\\n    n = max(n,math.ceil(x/p))\\n    n = max(n,math.ceil(y/q))\\n    print(n*q-y)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x,y,p,q = [int(x) for x in input().split()]\\n    print(math.ceil(max((y-x)/(q-p),x/p,y/q))*q-y) if p%q else print(0) if (x*q == p*y) else print(-1)\\n\", \"#https://codeforces.com/problemset/problem/773/A\\n\\nt = int(input())\\nfor i in range(t):\\n    x, y, p, q = list(map(int, input().split()))\\n    left = -1\\n    right = 10**9\\n    r = right\\n    while left + 1 < right:\\n        t = (left + right) // 2\\n        if p*t >= x and q*t - p*t >= y - x:\\n            right = t\\n        else:\\n            left = t\\n    if not (p*r >= x and q*r - p*r >= y - x):\\n        print(-1)\\n    else:\\n        print(q*right - y)\\n    \\n\", \"from math import ceil\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int, input().split()))\\n    if p == q and x != y:\\n        print(-1)\\n    elif p == q:\\n        print(0)\\n    elif p == 0 and x == 0:\\n        print(0)\\n    elif p == 0:\\n        print(-1)\\n    else:\\n        a = y // q\\n        r = y % q\\n        g1 = (x - p * a + p - 1) // p\\n        g2 = ceil((x - r - p * a) / (p - q))\\n        print(max(g1, g2) * q - r)\\n\", \"import math\\nfor _ in range(int(input())):\\n    x, y, p, q = list(map(int,input().split()))\\n    if p == 0 or  p == q :\\n        print(0 if  x * q == p * y else  -1 )\\n        continue    \\n    n = math.ceil((y - x) / (q - p) )\\n    n = max(n, math.ceil(x / p))\\n    n = max(n, math.ceil(y / q))\\n    print(n * q - y )\\n\", \"T = int(input())\\n\\nfor t in range(T):\\n  x, y, p, q = map(int, input().split())\\n  if p == q and x != y:\\n      print(-1)\\n      continue\\n\\n  INF = 10000000000\\n  lb = -1\\n  ub = INF\\n  mod_y = (q - y % q) % q\\n\\n  while ub - lb > 1:\\n    mid = (ub + lb) // 2\\n    diff = mod_y + q * mid\\n    bunbo = y + diff\\n    bunshi = p * bunbo // q\\n    if bunshi >= x and bunshi - x <= diff:\\n      ub = mid\\n    else:\\n      lb = mid\\n  if ub == INF:\\n    print(-1)\\n  else:\\n    print(mod_y + q * ub)\", \"def chec(a,b,p,q,mid):\\n    if p*r>=a and (q-p)*mid>=b-a:\\n        return True\\n    else:\\n        return False\\ndef check(np, nq):\\n  return np >= a and nq >= b and (np - a <= nq - b)    \\nfor _ in range(int(input())):\\n    a,b,p,q = list(map(int,input().split()))\\n    l=0\\n    r=10000000000\\n    if check(p*r,q*r)==False:\\n        print(-1)\\n        continue\\n    while l<=r:\\n        mid = l +(r-l)//2\\n        if check(p*mid,q*mid):\\n            r=mid-1\\n        else:\\n            l=mid+1\\n    print(l*q-b)\\n\", \"def div(a, b):\\n\\treturn (a+b-1)//b\\n\\nfor t in range(int(input())):\\n\\tx,y,p,q = map(int,input().split())\\n\\tif q == 1:\\n\\t\\tif p == 0 and x == 0 or p == 1 and x == y:\\n\\t\\t\\tprint(0)\\n\\t\\telse:\\n\\t\\t\\tprint(-1)\\n\\telse:\\n\\t\\tz = max(max(div(x,p),div(y,q)),max(div(y-x,q-p),0))\\n\\t\\tprint(z*q-y)\"]",
        "difficulty": "interview",
        "input": "4\n1 1000000000 999999999 1000000000\n1 1000000000 999999999 1000000000\n1 1000000000 999999999 1000000000\n1 1000000000 999999999 1000000000\n",
        "output": "999999998000000000\n999999998000000000\n999999998000000000\n999999998000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/773/A"
    },
    {
        "id": 179,
        "task_id": 1076,
        "test_case_id": 1,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "2 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 180,
        "task_id": 1076,
        "test_case_id": 2,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "3 2\n",
        "output": "332748127\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 181,
        "task_id": 1076,
        "test_case_id": 3,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "14 9\n",
        "output": "969862773\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 182,
        "task_id": 1076,
        "test_case_id": 4,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "20 20\n",
        "output": "445919622\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 183,
        "task_id": 1076,
        "test_case_id": 5,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1 2000000\n",
        "output": "2000002\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 184,
        "task_id": 1076,
        "test_case_id": 6,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "2000000 1\n",
        "output": "520971794\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 185,
        "task_id": 1076,
        "test_case_id": 7,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "2000000 2000000\n",
        "output": "507914970\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 186,
        "task_id": 1076,
        "test_case_id": 8,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1999922 1999943\n",
        "output": "255234262\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 187,
        "task_id": 1076,
        "test_case_id": 9,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1999968 1999941\n",
        "output": "499830706\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 188,
        "task_id": 1076,
        "test_case_id": 10,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1999913 1999918\n",
        "output": "667185390\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 189,
        "task_id": 1076,
        "test_case_id": 11,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1999992 1999927\n",
        "output": "334593851\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 190,
        "task_id": 1076,
        "test_case_id": 12,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "156914 1840237\n",
        "output": "51145361\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 191,
        "task_id": 1076,
        "test_case_id": 13,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "240338 1389081\n",
        "output": "827396276\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 192,
        "task_id": 1076,
        "test_case_id": 14,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "323762 937926\n",
        "output": "100358499\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 193,
        "task_id": 1076,
        "test_case_id": 15,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1439890 486770\n",
        "output": "507402218\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 194,
        "task_id": 1076,
        "test_case_id": 16,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "7 1867300\n",
        "output": "194362425\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 195,
        "task_id": 1076,
        "test_case_id": 17,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "5 1159248\n",
        "output": "112139371\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 196,
        "task_id": 1076,
        "test_case_id": 18,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "9 708093\n",
        "output": "858055671\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 197,
        "task_id": 1076,
        "test_case_id": 19,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1492240 6\n",
        "output": "942167012\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 198,
        "task_id": 1076,
        "test_case_id": 20,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "542960 4\n",
        "output": "215021492\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 199,
        "task_id": 1076,
        "test_case_id": 21,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "1659088 8\n",
        "output": "682055147\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 200,
        "task_id": 1076,
        "test_case_id": 22,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "400 400\n",
        "output": "983285438\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 201,
        "task_id": 1076,
        "test_case_id": 23,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "90 181\n",
        "output": "814868350\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 202,
        "task_id": 1076,
        "test_case_id": 24,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "314 33\n",
        "output": "581491843\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 203,
        "task_id": 1076,
        "test_case_id": 25,
        "question": "zscoder has a deck of $n+m$ custom-made cards, which consists of $n$ cards labelled from $1$ to $n$ and $m$ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. \n\nInitially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $S$ which is initially empty. \n\nEvery second, zscoder draws the top card from the deck.   If the card has a number $x$ written on it, zscoder removes the card and adds $x$ to the set $S$.  If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $n+m$ cards to form a new deck (hence the new deck now contains all cards from $1$ to $n$ and the $m$ jokers). Then, if $S$ currently contains all the elements from $1$ to $n$, the game ends. Shuffling the deck doesn't take time at all. \n\nWhat is the expected number of seconds before the game ends? We can show that the answer can be written in the form $\\frac{P}{Q}$ where $P, Q$ are relatively prime integers and $Q \\neq 0 \\bmod 998244353$. Output the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Input-----\n\nThe only line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^{6}$).\n\n\n-----Output-----\n\nOutput a single integer, the value of $(P \\cdot Q^{-1})$ modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n\nOutput\n5\n\nInput\n3 2\n\nOutput\n332748127\n\nInput\n14 9\n\nOutput\n969862773\n\n\n\n-----Note-----\n\nFor the first sample, it can be proven that the expected time before the game ends is $5$ seconds.\n\nFor the second sample, it can be proven that the expected time before the game ends is $\\frac{28}{3}$ seconds.",
        "solutions": "[\"import sys\\nimport math\\n\\nQ = 998244353\\nn, m = list(map(int, sys.stdin.readline().strip().split()))\\n\\nz = [1, 1]\\nf = [1, 1]\\nfor i in range (0, n):\\n    f[0] = f[0] * (n - i) % Q\\n    f[1] = f[1] * (n + m - i) % Q\\n    z = [(z[0]*f[1]+z[1]*f[0]) % Q, (z[1]*f[1]) % Q]\\nans = [z[0] * (m+1), z[1]]\\nfor i in range (2, n + 1):\\n    ans = [(ans[0] * i * z[1] + ans[1] * m * z[0]) % Q, (ans[1] * i * z[1]) % Q]\\ny = ans[1]\\nans = ans[0]\\nq = Q - 2\\nwhile q > 0:\\n    if q % 2 == 1:\\n        ans = (ans * y) % Q\\n    q = q // 2\\n    y = (y * y) % Q\\nprint(ans)\\n\", \"nn = 2002002\\nP = 998244353\\nfa = [1] * (nn+1)\\nfainv = [1] * (nn+1)\\nfor i in range(nn):\\n    fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:\\n    fainv[i] = fainv[i+1] * (i+1) % P\\n\\nN, M = list(map(int, input().split()))\\ninvM1 = pow(M + 1, P - 2, P)\\npre = 1\\nfor i in range(2, N + 2):\\n    nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1)\\n    de = invM1 * fainv[i-1] % P * fa[i-2] % P\\n    pre = nu * de % P\\nprint(pre)\\n\", \"nn = 2002002;P = 998244353;fa = [1] * (nn+1);fainv = [1] * (nn+1)\\nfor i in range(nn):fa[i+1] = fa[i] * (i+1) % P\\nfainv[-1] = pow(fa[-1], P-2, P)\\nfor i in range(nn)[::-1]:fainv[i] = fainv[i+1] * (i+1) % P\\nN, M = map(int, input().split());invM1 = pow(M + 1, P - 2, P);pre = 1\\nfor i in range(2, N + 2):nu = ((M + i - 1) + (i - 1) * pre) * (M + 1) + M * (N - i + 1);de = invM1 * fainv[i-1] % P * fa[i-2] % P;pre = nu * de % P\\nprint(pre)\"]",
        "difficulty": "interview",
        "input": "42 77\n",
        "output": "715341123\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1392/H"
    },
    {
        "id": 204,
        "task_id": 1119,
        "test_case_id": 1,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "1 1 1\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 205,
        "task_id": 1119,
        "test_case_id": 2,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "3 1 4\n",
        "output": "370000006\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 206,
        "task_id": 1119,
        "test_case_id": 3,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "1000 123456 654321\n",
        "output": "977760856\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 207,
        "task_id": 1119,
        "test_case_id": 4,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "305 337309 378395\n",
        "output": "174667130\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 208,
        "task_id": 1119,
        "test_case_id": 5,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "108 531040 908573\n",
        "output": "145579983\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 209,
        "task_id": 1119,
        "test_case_id": 6,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "575 39377 68346\n",
        "output": "899189133\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 210,
        "task_id": 1119,
        "test_case_id": 7,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "66 199449 266025\n",
        "output": "27912582\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 211,
        "task_id": 1119,
        "test_case_id": 8,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "781 817338 452871\n",
        "output": "711597307\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 212,
        "task_id": 1119,
        "test_case_id": 9,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "99 534023 117289\n",
        "output": "29694885\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 213,
        "task_id": 1119,
        "test_case_id": 10,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "156 78149 46740\n",
        "output": "114906561\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 214,
        "task_id": 1119,
        "test_case_id": 11,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "57 339480 774350\n",
        "output": "622654301\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 215,
        "task_id": 1119,
        "test_case_id": 12,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "270 967166 795005\n",
        "output": "530539317\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 216,
        "task_id": 1119,
        "test_case_id": 13,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "628 446579 365440\n",
        "output": "214808787\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 217,
        "task_id": 1119,
        "test_case_id": 14,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "97 119368 2062\n",
        "output": "2436614\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 218,
        "task_id": 1119,
        "test_case_id": 15,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "757 869978 224540\n",
        "output": "921904658\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 219,
        "task_id": 1119,
        "test_case_id": 16,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "892 777143 664073\n",
        "output": "527873013\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 220,
        "task_id": 1119,
        "test_case_id": 17,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "177 2501 570142\n",
        "output": "779148936\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 221,
        "task_id": 1119,
        "test_case_id": 18,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "908 879494 944888\n",
        "output": "114377456\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 222,
        "task_id": 1119,
        "test_case_id": 19,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "734 32585 49636\n",
        "output": "684730644\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 223,
        "task_id": 1119,
        "test_case_id": 20,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "38 592277 400426\n",
        "output": "499077928\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 224,
        "task_id": 1119,
        "test_case_id": 21,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "192 42070 61266\n",
        "output": "904814024\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 225,
        "task_id": 1119,
        "test_case_id": 22,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "78 535199 331023\n",
        "output": "684367478\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 226,
        "task_id": 1119,
        "test_case_id": 23,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "842 171735 282219\n",
        "output": "948183028\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 227,
        "task_id": 1119,
        "test_case_id": 24,
        "question": "You are given three integers k, p_{a} and p_{b}.\n\nYou will construct a sequence with the following algorithm: Initially, start with the empty sequence. Each second, you do the following. With probability p_{a} / (p_{a} + p_{b}), add 'a' to the end of the sequence. Otherwise (with probability p_{b} / (p_{a} + p_{b})), add 'b' to the end of the sequence.\n\nYou stop once there are at least k subsequences that form 'ab'. Determine the expected number of times 'ab' is a subsequence in the resulting sequence. It can be shown that this can be represented by P / Q, where P and Q are coprime integers, and $Q \\neq 0 \\operatorname{mod}(10^{9} + 7)$. Print the value of $P \\cdot Q^{-1} \\operatorname{mod}(10^{9} + 7)$.\n\n\n-----Input-----\n\nThe first line will contain three integers integer k, p_{a}, p_{b} (1 ≤ k ≤ 1 000, 1 ≤ p_{a}, p_{b} ≤ 1 000 000).\n\n\n-----Output-----\n\nPrint a single integer, the answer to the problem.\n\n\n-----Examples-----\nInput\n1 1 1\n\nOutput\n2\n\nInput\n3 1 4\n\nOutput\n370000006\n\n\n\n-----Note-----\n\nThe first sample, we will keep appending to our sequence until we get the subsequence 'ab' at least once. For instance, we get the sequence 'ab' with probability 1/4, 'bbab' with probability 1/16, and 'aab' with probability 1/8. Note, it's impossible for us to end with a sequence like 'aabab', since we would have stopped our algorithm once we had the prefix 'aab'. \n\nThe expected amount of times that 'ab' will occur across all valid sequences is 2. \n\nFor the second sample, the answer is equal to $\\frac{341}{100}$.",
        "solutions": "[\"k, pa, pb = list(map(int, input().split()))\\n\\nMOD = 10**9 + 7\\nINF = ((pa + pb) * pow(pb, MOD-2, MOD)) % MOD\\nrAB = pow(pa+pb, MOD-2, MOD)\\nrB = pow(pb, MOD-2, MOD)\\n\\nmemo = {}\\n\\ndef dfs(a, ab):\\n    if ab >= k:\\n        return ab\\n    if a + ab >= k:\\n        #return INF\\n        #return (pa + pb) / pb\\n        return ((a + MOD-1) + (pa + pb) * rB + ab) % MOD\\n        return a - 1 + (pa + pb) / pb + ab\\n    if (a, ab) in memo:\\n        return memo[a, ab]\\n    #res = (((dfs(a+1, ab)+1) * pa * rAB) + ((dfs(a, ab+a)+1) * pb * rAB)) % MOD\\n    #res = (dfs(a+1, ab)) * pa / (pa + pb) + (dfs(a, ab+a)) * pb / (pa + pb)\\n    res = (dfs(a+1, ab) * pa * rAB) + (dfs(a, ab+a) * pb * rAB)\\n    #print(a, ab, res)\\n    memo[a, ab] = res = res % MOD\\n    return res\\n#print((dfs(1, 0) * pa * rAB + 1) % MOD)\\n#print((pb + dfs(1, 0)*pa) / pa)\\nprint(dfs(1, 0))\\n\", \"from collections import defaultdict as di\\nimport sys\\nsys.setreqursiondepth = 1000000\\nMOD = int(1e9+7)\\ndef modinvEuler(x,mod):\\n    # if mod is prime\\n    return pow(x, mod-2, mod)\\n    # otherwise exponent should be totient(mod)-1\\n\\nk,pa,pb = [int(x) for x in input().split()]\\nPa = (pa*modinvEuler(pa+pb,MOD))%MOD\\nPb = (1-Pa)%MOD\\n\\nEa = modinvEuler(Pa,MOD)\\nEb = modinvEuler(Pb,MOD)\\n\\nPbinv = modinvEuler(Pb,MOD)\\n\\nmem = di()\\ndef f(na,ns):\\n    #nonlocal k,Pa,Pb\\n    if ns>=k:\\n        return ns\\n    if na+ns>=k:\\n        total = ns\\n        total += na\\n        total += Pa*Pbinv\\n        total%= MOD\\n        return total\\n    if (na,ns) not in mem:\\n        mem[(na,ns)] = ( Pa*(f(na+1,ns))+Pb*(f(na,ns+na)) )%MOD\\n\\n    return mem[(na,ns)]\\nprint((f(1,0))%MOD)\\n\", \"import math\\nimport sys\\n\\nk, pa, pb = list( map( int, input().split() ) )\\n\\n\\nmemo = {}\\n\\nsys.setrecursionlimit(1500*1500*2)\\n\\nMOD = (10**9 + 7 )\\n\\ndef pow( a, b ):\\n\\n    ret = 1\\n\\n    while b > 0:\\n\\n        if b & 1:\\n            ret = (ret*a)%MOD\\n\\n        a = (a*a)%MOD\\n\\n        b //= 2\\n\\n    return ( ret )\\n\\ndef inv(a):\\n   return pow( a, MOD-2 )\\n\\nPa = pa * inv( pa + pb )\\nPb = pb * inv( pa + pb )\\npa_inv_pb = pa * inv(pb)\\n\\ndef f( total_a, total_ab ):\\n\\n    if total_a+total_ab >= k:\\n        return total_a+total_ab + pa_inv_pb\\n\\n    if (total_a,total_ab) in memo:\\n        return memo[ (total_a,total_ab) ]\\n\\n    Best = 0\\n\\n    Best += Pa * f( total_a+1, total_ab )\\n    Best %= MOD\\n\\n    Best += Pb * f( total_a, total_a+total_ab )\\n    Best %= MOD\\n\\n    memo[ (total_a,total_ab) ] = Best\\n\\n    return ( Best )\\n\\n#print( k, pa, pb )\\n\\nprint( ( f(1,0) ) % MOD )\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(N)] for y in range(N)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"mod = 1000000007\\nN   = 1005\\n#input start\\nk,pa,pb = list(map(int,input().split()))\\ndp = [[0 for x in range(k)] for y in range(k+1)]\\n#end of input\\n\\ndef fast_expo(a,b):\\n\\tret = 1\\n\\twhile b:\\n\\t\\tif b%2==1:\\n\\t\\t\\tret = ret*a%mod\\n\\t\\tb //= 2\\n\\t\\ta = a*a%mod\\n\\treturn ret\\ndef inv(x):\\n\\treturn fast_expo(x,mod-2)%mod\\n\\ndef dp_val(a,b):\\n\\tif b>=k:\\n\\t\\treturn b\\n\\treturn dp[a][b]\\n\\n\\nfor i in range(k):\\n\\tdp[k][i] = (i+k+pa*inv(pb))%mod\\n\\nden = inv(pa+pb)\\n\\nfor i in range(k-1,0,-1):\\n\\tfor j in range(k-1,-1,-1):\\n\\t\\tdp[i][j] = pa*dp_val(i+1,j)*den%mod+pb*dp_val(i,i+j)*den%mod\\n\\t\\tdp[i][j]%=mod\\n\\nprint(dp[1][0])\\n\", \"k, a, b = map(int, input().split())\\nm = 1000000007\\nd = a * pow(b, m - 2, m) % m\\nc = pow(a + b, m - 2, m)\\nu, v = [0] * k, [0] * k\\nfor s in range(k, 0, -1):\\n    v[s - 1] = s + d\\n    for i in range(s, k):\\n        j = max(i - s, s - 1)\\n        v[i] = c * (a * u[i] + b * (s + v[j])) % m\\n    u, v = v, u\\nprint(u[k - 1])\", \"import  sys\\n#input=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    p=int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\", \"import  sys\\ninput=sys.stdin.readline\\ndp={}\\nmod=int(1000000007)\\nsys.setrecursionlimit(100000)\\ndef bigmod(n,p):\\n    n,p=int(n),int(p)\\n    if p==0:\\n        return 1\\n    x=bigmod(n,p/2)\\n    x=(x*x)%mod\\n    if p%2==1:\\n        x=(x*n)%mod\\n    return x\\nk,pa,pb=list(map(int,input().split()))\\nr=bigmod(pa+pb,mod-2)\\nr1=bigmod(pb,mod-2)\\nr1=(pa*r1)%mod\\np=(pa*r)%mod\\nq=(pb*r)%mod\\n\\n\\n\\ndef cal(k1,a1):\\n    if k1+a1>=k:\\n        return (k1+a1+r1)%mod\\n    if (k1,a1) in dp:\\n        return dp[(k1,a1)]\\n    dp[(k1,a1)]=((cal(k1+a1,a1)*q)%mod+(cal(k1,a1+1)*p)%mod)%mod\\n    return dp[(k1,a1)]\\n\\nprint(cal(0,1))\\n\\n\"]",
        "difficulty": "interview",
        "input": "1000 1000000 1\n",
        "output": "478180868\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/908/D"
    },
    {
        "id": 228,
        "task_id": 1347,
        "test_case_id": 1,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n",
        "output": "2 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 229,
        "task_id": 1347,
        "test_case_id": 2,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "2\nRuruRu fedya\n1\nruruRU fedor\n",
        "output": "1 10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 230,
        "task_id": 1347,
        "test_case_id": 3,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "1\nffff\n1\nffff r\n",
        "output": "0 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 231,
        "task_id": 1347,
        "test_case_id": 4,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "2\nYURA YUrA\n1\nyura fedya\n",
        "output": "0 10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 232,
        "task_id": 1347,
        "test_case_id": 5,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "5\nhello my name is fedya\n2\nhello hi\nis i\n",
        "output": "0 14\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 233,
        "task_id": 1347,
        "test_case_id": 6,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "5\nawiuegjsrkjshegkjshegseg g soeigjseg www s\n3\nwww s\nawiuegjsrkjshegkjshegseg wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww\nwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww www\n",
        "output": "0 13\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 234,
        "task_id": 1347,
        "test_case_id": 7,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "5\naa bb cc ee ff\n5\naa a\nbb aa\ncc bb\nee cc\nff bb\n",
        "output": "0 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 235,
        "task_id": 1347,
        "test_case_id": 8,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "7\nraki vezde est awjgkawkgjn ttttt raki raks\n4\nraks rks\nrks raks\nraki raks\nvezde pss\n",
        "output": "3 31\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 236,
        "task_id": 1347,
        "test_case_id": 9,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "5\nfedor fedya www awwwwwww a\n5\nr a\nfedor fedr\nwww a\nawwwwwww www\na r\n",
        "output": "1 12\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 237,
        "task_id": 1347,
        "test_case_id": 10,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "1\nYURA\n1\nyura lesha\n",
        "output": "0 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 238,
        "task_id": 1347,
        "test_case_id": 11,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "2\nABBABAABBAABABBABAABABBAABBABAABBAABABBAABBABAABABBABAABBAABABBA ABBABAABBAABABBABAABABBAABBABAABBAABABBAABBABAABABBABAABBAABABA\n2\nABBABAABBAABABBABAABABBAABBABAABBAABABBAABBABAABABBABAABBAABABA neuzaiheshi\nABBABAABBAABABBABAABABBAABBABAABBAABABBAABBABAABABBABAABBAABABBA ABBABAABBAABABBABAABABBAABBABAABBAABABBAABBABAABABBABAABBAABABA\n",
        "output": "0 22\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 239,
        "task_id": 1347,
        "test_case_id": 12,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "10\nlalka lolka yura lesha fedya bredor tourist www qqq gruihdrkgjp\n11\nlalka lolka\nlolka lalka\nyura lolka\nlalka poka\nfedya bredor\nbredor yura\ntourist bredor\nwww qqq\nqqq w\nw g\ngruihdrkgjp bredor\n",
        "output": "0 35\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 240,
        "task_id": 1347,
        "test_case_id": 14,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "3\nreka greka rak\n11\nrek rak\nrak grek\nreka rak\ngreka reka\nrak reka\nrak greka\ngreka rak\nlol rek\nlol rak\nLO lol\nABA BA\n",
        "output": "3 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 241,
        "task_id": 1347,
        "test_case_id": 15,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "3\nreka greka rak\n13\nrek rak\nrak grek\nreka rak\ngreka reka\nrak reka\nrak greka\ngreka rak\nlol rek\nlol rak\nlol LO\nABA BA\nLOLKA rak\nrak lol\n",
        "output": "0 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 242,
        "task_id": 1347,
        "test_case_id": 17,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "5\nfEdOR Is A bAd BoY\n2\nboy boYy\nFeDor fedyaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\n",
        "output": "0 70\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 243,
        "task_id": 1347,
        "test_case_id": 18,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "1\nyrwlqadsfw\n2\nmnqdxczpyo a\na mnqdxczpyo\n",
        "output": "1 10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 244,
        "task_id": 1347,
        "test_case_id": 19,
        "question": "After you had helped Fedor to find friends in the «Call of Soldiers 3» game, he stopped studying completely. Today, the English teacher told him to prepare an essay. Fedor didn't want to prepare the essay, so he asked Alex for help. Alex came to help and wrote the essay for Fedor. But Fedor didn't like the essay at all. Now Fedor is going to change the essay using the synonym dictionary of the English language.\n\nFedor does not want to change the meaning of the essay. So the only change he would do: change a word from essay to one of its synonyms, basing on a replacement rule from the dictionary. Fedor may perform this operation any number of times.\n\nAs a result, Fedor wants to get an essay which contains as little letters «R» (the case doesn't matter) as possible. If there are multiple essays with minimum number of «R»s he wants to get the one with minimum length (length of essay is the sum of the lengths of all the words in it). Help Fedor get the required essay.\n\nPlease note that in this problem the case of letters doesn't matter. For example, if the synonym dictionary says that word cat can be replaced with word DOG, then it is allowed to replace the word Cat with the word doG.\n\n\n-----Input-----\n\nThe first line contains a single integer m (1 ≤ m ≤ 10^5) — the number of words in the initial essay. The second line contains words of the essay. The words are separated by a single space. It is guaranteed that the total length of the words won't exceed 10^5 characters.\n\nThe next line contains a single integer n (0 ≤ n ≤ 10^5) — the number of pairs of words in synonym dictionary. The i-th of the next n lines contains two space-separated non-empty words x_{i} and y_{i}. They mean that word x_{i} can be replaced with word y_{i} (but not vise versa). It is guaranteed that the total length of all pairs of synonyms doesn't exceed 5·10^5 characters.\n\nAll the words at input can only consist of uppercase and lowercase letters of the English alphabet.\n\n\n-----Output-----\n\nPrint two integers — the minimum number of letters «R» in an optimal essay and the minimum length of an optimal essay.\n\n\n-----Examples-----\nInput\n3\nAbRb r Zz\n4\nxR abRb\naA xr\nzz Z\nxr y\n\nOutput\n2 6\n\nInput\n2\nRuruRu fedya\n1\nruruRU fedor\n\nOutput\n1 10",
        "solutions": "[\"from sys import stdin\\nfrom collections import defaultdict\\n\\ndef main():\\n  stdin.readline()\\n  num = {}\\n  stat = lambda word: (word.count('r'), \\n      len(word), num.setdefault(word, len(num)))\\n  essay = list(map(stat, stdin.readline().lower().split()))\\n  queue = []\\n  for word in essay:\\n    queue.append(word)\\n  n_synonym = int(stdin.readline())\\n  synonym = defaultdict(list)\\n  for i in range(n_synonym):\\n    word, rep = map(stat, stdin.readline().lower().split())\\n    synonym[rep[2]].append(word[2])\\n    queue.append(rep)\\n  queue.sort(reverse=True)\\n  best = {}\\n  while queue:\\n    n_r, length, word = queue.pop()\\n    if word in best:\\n      continue\\n    best[word] = n_r, length\\n    for rep in synonym[word]:\\n      if rep not in best:\\n        queue.append((n_r, length, rep))\\n\\n  sum_n_r, sum_len = 0, 0\\n  for n_r, length, word in essay:\\n    n_r, length = best[word]\\n    sum_n_r += n_r\\n    sum_len += length\\n  print(sum_n_r, sum_len)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = map(stat, input().lower().split())\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in map(lambda w: best[w[2]], essay):\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\", \"import collections\\n\\ndef solve():\\n  m = int(input())\\n  essay = [s for s in input().lower().split()]\\n  n = int(input()) \\n  sti = dict()\\n  pack = lambda word: (word.count('r'), len(word), sti.setdefault(word, len(sti)))\\n  edge = collections.defaultdict(list)\\n  nodes = list()\\n\\n  for _ in range(n):\\n    word, synon = list(map(pack, input().lower().split()))\\n    edge[synon[-1]].append(word[-1])\\n    nodes.append(word)\\n    nodes.append(synon)\\n\\n  nodes.sort()\\n\\n  best = dict()\\n  for node in nodes:\\n    if node[2] not in best:\\n      stack = [node[2]]\\n      while stack:\\n        top = stack.pop()\\n        if top not in best:\\n          best[top] = node[:2]\\n          for n in edge[top]:\\n            if n is not best:\\n              stack.append(n)\\n\\n  tr = 0\\n  tl = 0\\n  for word in essay:\\n    if word in sti:\\n      wid = sti[word]\\n      tr += best[wid][0]\\n      tl += best[wid][1]\\n    else:\\n      tr += word.count('r')\\n      tl += len(word)\\n  print(tr, ' ', tl)\\n\\ndef __starting_point():\\n  solve()\\n\\n__starting_point()\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\", \"from collections import defaultdict\\n\\ninput()\\nindex = {}\\nstat = lambda word: (word.count('r'), \\n    len(word), index.setdefault(word, len(index)))\\nessay = list(map(stat, input().lower().split()))\\nqueue = essay[:]\\n\\nsyn = defaultdict(list)\\nfor i in range(int(input())):\\n  word, rep = list(map(stat, input().lower().split()))\\n  syn[rep[2]].append(word[2])\\n  queue.append(rep)\\nqueue.sort(reverse=True)\\nbest = {}\\nwhile queue:\\n  n_r, length, word = queue.pop()\\n  if word in best:\\n    continue\\n  best[word] = n_r, length\\n  for rep in syn[word]:\\n    if rep not in best:\\n      queue.append((n_r, length, rep))\\nsum_n_r, sum_len = 0, 0\\nfor n_r, length in [best[w[2]] for w in essay]:\\n  sum_n_r += n_r\\n  sum_len += length\\nprint(sum_n_r, sum_len)\\n\"]",
        "difficulty": "interview",
        "input": "4\nr rr rrr rrrr\n9\nrr rrr\nrrrr rr\nr rr\nr rrrr\nrrr rr\nrrr rrr\nrr rrr\nrr r\nr r\n",
        "output": "4 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/467/D"
    },
    {
        "id": 245,
        "task_id": 1813,
        "test_case_id": 1,
        "question": "Alexandra has a paper strip with n numbers on it. Let's call them a_{i} from left to right.\n\nNow Alexandra wants to split it into some pieces (possibly 1). For each piece of strip, it must satisfy:\n\n\n\n Each piece should contain at least l numbers.\n\n The difference between the maximal and the minimal number on the piece should be at most s.\n\nPlease help Alexandra to find the minimal number of pieces meeting the condition above.\n\n\n-----Input-----\n\nThe first line contains three space-separated integers n, s, l (1 ≤ n ≤ 10^5, 0 ≤ s ≤ 10^9, 1 ≤ l ≤ 10^5).\n\nThe second line contains n integers a_{i} separated by spaces ( - 10^9 ≤ a_{i} ≤ 10^9).\n\n\n-----Output-----\n\nOutput the minimal number of strip pieces.\n\nIf there are no ways to split the strip, output -1.\n\n\n-----Examples-----\nInput\n7 2 2\n1 3 1 2 4 1 2\n\nOutput\n3\n\nInput\n7 2 2\n1 100 1 100 1 100 1\n\nOutput\n-1\n\n\n\n-----Note-----\n\nFor the first sample, we can split the strip into 3 pieces: [1, 3, 1], [2, 4], [1, 2].\n\nFor the second sample, we can't let 1 and 100 be on the same piece, so no solution exists.",
        "solutions": "[\"def split(a,n,s,l):\\n    pieces = []\\n\\n    i = 1\\n    tmpmin = a[0]\\n    tmpmax = a[0]\\n    tmppc  = [a[0]]\\n    while i<n:\\n        if abs(a[i]-tmpmin)<=s and abs(a[i]-tmpmax)<=s:\\n            tmppc.append(a[i])\\n            if a[i]<tmpmin: tmpmin=a[i]\\n            elif a[i]>tmpmax: tmpmax = a[i]\\n        else:\\n            pieces.append(tmppc)\\n            tmppc = [a[i]]\\n            tmpmin = a[i]\\n            tmpmax = a[i]\\n        i += 1\\n    pieces.append(tmppc)\\n\\n    fail = False        \\n    for j in range(len(pieces)):\\n        if len(pieces[j])<l:\\n            if j>0:\\n                prevpc = pieces[j-1]\\n                minj = min(pieces[j])\\n                maxj = max(pieces[j])\\n                \\n                while len(pieces[j])<l:\\n                    tmp = prevpc.pop()\\n                    if abs(tmp-minj)<=s and abs(tmp-maxj)<=s:\\n                        pieces[j].insert(0,tmp)\\n                        if tmp<minj: minj=tmp\\n                        elif tmp>maxj: maxj=tmp\\n                    else:\\n                        return -1\\n                    if len(prevpc)<l:\\n                        return -1\\n            else:\\n                return -1\\n    return len(pieces)\\n\\nn,s,l = [int(s) for s in input().split()]\\na = [int(s) for s in input().split()]\\n\\n\\nres = split(a,n,s,l)\\nif res<0:\\n    a.reverse()\\n    res = split(a,n,s,l)\\nprint(res)\\n    \\n\\n\\n\\n\"]",
        "difficulty": "interview",
        "input": "7 2 2\n1 3 1 2 4 1 2\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/488/D"
    },
    {
        "id": 246,
        "task_id": 1813,
        "test_case_id": 2,
        "question": "Alexandra has a paper strip with n numbers on it. Let's call them a_{i} from left to right.\n\nNow Alexandra wants to split it into some pieces (possibly 1). For each piece of strip, it must satisfy:\n\n\n\n Each piece should contain at least l numbers.\n\n The difference between the maximal and the minimal number on the piece should be at most s.\n\nPlease help Alexandra to find the minimal number of pieces meeting the condition above.\n\n\n-----Input-----\n\nThe first line contains three space-separated integers n, s, l (1 ≤ n ≤ 10^5, 0 ≤ s ≤ 10^9, 1 ≤ l ≤ 10^5).\n\nThe second line contains n integers a_{i} separated by spaces ( - 10^9 ≤ a_{i} ≤ 10^9).\n\n\n-----Output-----\n\nOutput the minimal number of strip pieces.\n\nIf there are no ways to split the strip, output -1.\n\n\n-----Examples-----\nInput\n7 2 2\n1 3 1 2 4 1 2\n\nOutput\n3\n\nInput\n7 2 2\n1 100 1 100 1 100 1\n\nOutput\n-1\n\n\n\n-----Note-----\n\nFor the first sample, we can split the strip into 3 pieces: [1, 3, 1], [2, 4], [1, 2].\n\nFor the second sample, we can't let 1 and 100 be on the same piece, so no solution exists.",
        "solutions": "[\"def split(a,n,s,l):\\n    pieces = []\\n\\n    i = 1\\n    tmpmin = a[0]\\n    tmpmax = a[0]\\n    tmppc  = [a[0]]\\n    while i<n:\\n        if abs(a[i]-tmpmin)<=s and abs(a[i]-tmpmax)<=s:\\n            tmppc.append(a[i])\\n            if a[i]<tmpmin: tmpmin=a[i]\\n            elif a[i]>tmpmax: tmpmax = a[i]\\n        else:\\n            pieces.append(tmppc)\\n            tmppc = [a[i]]\\n            tmpmin = a[i]\\n            tmpmax = a[i]\\n        i += 1\\n    pieces.append(tmppc)\\n\\n    fail = False        \\n    for j in range(len(pieces)):\\n        if len(pieces[j])<l:\\n            if j>0:\\n                prevpc = pieces[j-1]\\n                minj = min(pieces[j])\\n                maxj = max(pieces[j])\\n                \\n                while len(pieces[j])<l:\\n                    tmp = prevpc.pop()\\n                    if abs(tmp-minj)<=s and abs(tmp-maxj)<=s:\\n                        pieces[j].insert(0,tmp)\\n                        if tmp<minj: minj=tmp\\n                        elif tmp>maxj: maxj=tmp\\n                    else:\\n                        return -1\\n                    if len(prevpc)<l:\\n                        return -1\\n            else:\\n                return -1\\n    return len(pieces)\\n\\nn,s,l = [int(s) for s in input().split()]\\na = [int(s) for s in input().split()]\\n\\n\\nres = split(a,n,s,l)\\nif res<0:\\n    a.reverse()\\n    res = split(a,n,s,l)\\nprint(res)\\n    \\n\\n\\n\\n\"]",
        "difficulty": "interview",
        "input": "7 2 2\n1 100 1 100 1 100 1\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/488/D"
    },
    {
        "id": 247,
        "task_id": 1813,
        "test_case_id": 8,
        "question": "Alexandra has a paper strip with n numbers on it. Let's call them a_{i} from left to right.\n\nNow Alexandra wants to split it into some pieces (possibly 1). For each piece of strip, it must satisfy:\n\n\n\n Each piece should contain at least l numbers.\n\n The difference between the maximal and the minimal number on the piece should be at most s.\n\nPlease help Alexandra to find the minimal number of pieces meeting the condition above.\n\n\n-----Input-----\n\nThe first line contains three space-separated integers n, s, l (1 ≤ n ≤ 10^5, 0 ≤ s ≤ 10^9, 1 ≤ l ≤ 10^5).\n\nThe second line contains n integers a_{i} separated by spaces ( - 10^9 ≤ a_{i} ≤ 10^9).\n\n\n-----Output-----\n\nOutput the minimal number of strip pieces.\n\nIf there are no ways to split the strip, output -1.\n\n\n-----Examples-----\nInput\n7 2 2\n1 3 1 2 4 1 2\n\nOutput\n3\n\nInput\n7 2 2\n1 100 1 100 1 100 1\n\nOutput\n-1\n\n\n\n-----Note-----\n\nFor the first sample, we can split the strip into 3 pieces: [1, 3, 1], [2, 4], [1, 2].\n\nFor the second sample, we can't let 1 and 100 be on the same piece, so no solution exists.",
        "solutions": "[\"def split(a,n,s,l):\\n    pieces = []\\n\\n    i = 1\\n    tmpmin = a[0]\\n    tmpmax = a[0]\\n    tmppc  = [a[0]]\\n    while i<n:\\n        if abs(a[i]-tmpmin)<=s and abs(a[i]-tmpmax)<=s:\\n            tmppc.append(a[i])\\n            if a[i]<tmpmin: tmpmin=a[i]\\n            elif a[i]>tmpmax: tmpmax = a[i]\\n        else:\\n            pieces.append(tmppc)\\n            tmppc = [a[i]]\\n            tmpmin = a[i]\\n            tmpmax = a[i]\\n        i += 1\\n    pieces.append(tmppc)\\n\\n    fail = False        \\n    for j in range(len(pieces)):\\n        if len(pieces[j])<l:\\n            if j>0:\\n                prevpc = pieces[j-1]\\n                minj = min(pieces[j])\\n                maxj = max(pieces[j])\\n                \\n                while len(pieces[j])<l:\\n                    tmp = prevpc.pop()\\n                    if abs(tmp-minj)<=s and abs(tmp-maxj)<=s:\\n                        pieces[j].insert(0,tmp)\\n                        if tmp<minj: minj=tmp\\n                        elif tmp>maxj: maxj=tmp\\n                    else:\\n                        return -1\\n                    if len(prevpc)<l:\\n                        return -1\\n            else:\\n                return -1\\n    return len(pieces)\\n\\nn,s,l = [int(s) for s in input().split()]\\na = [int(s) for s in input().split()]\\n\\n\\nres = split(a,n,s,l)\\nif res<0:\\n    a.reverse()\\n    res = split(a,n,s,l)\\nprint(res)\\n    \\n\\n\\n\\n\"]",
        "difficulty": "interview",
        "input": "2 1000000000 1\n-1000000000 1000000000\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/488/D"
    },
    {
        "id": 248,
        "task_id": 1813,
        "test_case_id": 9,
        "question": "Alexandra has a paper strip with n numbers on it. Let's call them a_{i} from left to right.\n\nNow Alexandra wants to split it into some pieces (possibly 1). For each piece of strip, it must satisfy:\n\n\n\n Each piece should contain at least l numbers.\n\n The difference between the maximal and the minimal number on the piece should be at most s.\n\nPlease help Alexandra to find the minimal number of pieces meeting the condition above.\n\n\n-----Input-----\n\nThe first line contains three space-separated integers n, s, l (1 ≤ n ≤ 10^5, 0 ≤ s ≤ 10^9, 1 ≤ l ≤ 10^5).\n\nThe second line contains n integers a_{i} separated by spaces ( - 10^9 ≤ a_{i} ≤ 10^9).\n\n\n-----Output-----\n\nOutput the minimal number of strip pieces.\n\nIf there are no ways to split the strip, output -1.\n\n\n-----Examples-----\nInput\n7 2 2\n1 3 1 2 4 1 2\n\nOutput\n3\n\nInput\n7 2 2\n1 100 1 100 1 100 1\n\nOutput\n-1\n\n\n\n-----Note-----\n\nFor the first sample, we can split the strip into 3 pieces: [1, 3, 1], [2, 4], [1, 2].\n\nFor the second sample, we can't let 1 and 100 be on the same piece, so no solution exists.",
        "solutions": "[\"def split(a,n,s,l):\\n    pieces = []\\n\\n    i = 1\\n    tmpmin = a[0]\\n    tmpmax = a[0]\\n    tmppc  = [a[0]]\\n    while i<n:\\n        if abs(a[i]-tmpmin)<=s and abs(a[i]-tmpmax)<=s:\\n            tmppc.append(a[i])\\n            if a[i]<tmpmin: tmpmin=a[i]\\n            elif a[i]>tmpmax: tmpmax = a[i]\\n        else:\\n            pieces.append(tmppc)\\n            tmppc = [a[i]]\\n            tmpmin = a[i]\\n            tmpmax = a[i]\\n        i += 1\\n    pieces.append(tmppc)\\n\\n    fail = False        \\n    for j in range(len(pieces)):\\n        if len(pieces[j])<l:\\n            if j>0:\\n                prevpc = pieces[j-1]\\n                minj = min(pieces[j])\\n                maxj = max(pieces[j])\\n                \\n                while len(pieces[j])<l:\\n                    tmp = prevpc.pop()\\n                    if abs(tmp-minj)<=s and abs(tmp-maxj)<=s:\\n                        pieces[j].insert(0,tmp)\\n                        if tmp<minj: minj=tmp\\n                        elif tmp>maxj: maxj=tmp\\n                    else:\\n                        return -1\\n                    if len(prevpc)<l:\\n                        return -1\\n            else:\\n                return -1\\n    return len(pieces)\\n\\nn,s,l = [int(s) for s in input().split()]\\na = [int(s) for s in input().split()]\\n\\n\\nres = split(a,n,s,l)\\nif res<0:\\n    a.reverse()\\n    res = split(a,n,s,l)\\nprint(res)\\n    \\n\\n\\n\\n\"]",
        "difficulty": "interview",
        "input": "2 1000000000 2\n-1000000000 1000000000\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/488/D"
    },
    {
        "id": 249,
        "task_id": 1813,
        "test_case_id": 10,
        "question": "Alexandra has a paper strip with n numbers on it. Let's call them a_{i} from left to right.\n\nNow Alexandra wants to split it into some pieces (possibly 1). For each piece of strip, it must satisfy:\n\n\n\n Each piece should contain at least l numbers.\n\n The difference between the maximal and the minimal number on the piece should be at most s.\n\nPlease help Alexandra to find the minimal number of pieces meeting the condition above.\n\n\n-----Input-----\n\nThe first line contains three space-separated integers n, s, l (1 ≤ n ≤ 10^5, 0 ≤ s ≤ 10^9, 1 ≤ l ≤ 10^5).\n\nThe second line contains n integers a_{i} separated by spaces ( - 10^9 ≤ a_{i} ≤ 10^9).\n\n\n-----Output-----\n\nOutput the minimal number of strip pieces.\n\nIf there are no ways to split the strip, output -1.\n\n\n-----Examples-----\nInput\n7 2 2\n1 3 1 2 4 1 2\n\nOutput\n3\n\nInput\n7 2 2\n1 100 1 100 1 100 1\n\nOutput\n-1\n\n\n\n-----Note-----\n\nFor the first sample, we can split the strip into 3 pieces: [1, 3, 1], [2, 4], [1, 2].\n\nFor the second sample, we can't let 1 and 100 be on the same piece, so no solution exists.",
        "solutions": "[\"def split(a,n,s,l):\\n    pieces = []\\n\\n    i = 1\\n    tmpmin = a[0]\\n    tmpmax = a[0]\\n    tmppc  = [a[0]]\\n    while i<n:\\n        if abs(a[i]-tmpmin)<=s and abs(a[i]-tmpmax)<=s:\\n            tmppc.append(a[i])\\n            if a[i]<tmpmin: tmpmin=a[i]\\n            elif a[i]>tmpmax: tmpmax = a[i]\\n        else:\\n            pieces.append(tmppc)\\n            tmppc = [a[i]]\\n            tmpmin = a[i]\\n            tmpmax = a[i]\\n        i += 1\\n    pieces.append(tmppc)\\n\\n    fail = False        \\n    for j in range(len(pieces)):\\n        if len(pieces[j])<l:\\n            if j>0:\\n                prevpc = pieces[j-1]\\n                minj = min(pieces[j])\\n                maxj = max(pieces[j])\\n                \\n                while len(pieces[j])<l:\\n                    tmp = prevpc.pop()\\n                    if abs(tmp-minj)<=s and abs(tmp-maxj)<=s:\\n                        pieces[j].insert(0,tmp)\\n                        if tmp<minj: minj=tmp\\n                        elif tmp>maxj: maxj=tmp\\n                    else:\\n                        return -1\\n                    if len(prevpc)<l:\\n                        return -1\\n            else:\\n                return -1\\n    return len(pieces)\\n\\nn,s,l = [int(s) for s in input().split()]\\na = [int(s) for s in input().split()]\\n\\n\\nres = split(a,n,s,l)\\nif res<0:\\n    a.reverse()\\n    res = split(a,n,s,l)\\nprint(res)\\n    \\n\\n\\n\\n\"]",
        "difficulty": "interview",
        "input": "10 3 3\n1 1 1 1 1 5 6 7 8 9\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/488/D"
    },
    {
        "id": 250,
        "task_id": 1813,
        "test_case_id": 11,
        "question": "Alexandra has a paper strip with n numbers on it. Let's call them a_{i} from left to right.\n\nNow Alexandra wants to split it into some pieces (possibly 1). For each piece of strip, it must satisfy:\n\n\n\n Each piece should contain at least l numbers.\n\n The difference between the maximal and the minimal number on the piece should be at most s.\n\nPlease help Alexandra to find the minimal number of pieces meeting the condition above.\n\n\n-----Input-----\n\nThe first line contains three space-separated integers n, s, l (1 ≤ n ≤ 10^5, 0 ≤ s ≤ 10^9, 1 ≤ l ≤ 10^5).\n\nThe second line contains n integers a_{i} separated by spaces ( - 10^9 ≤ a_{i} ≤ 10^9).\n\n\n-----Output-----\n\nOutput the minimal number of strip pieces.\n\nIf there are no ways to split the strip, output -1.\n\n\n-----Examples-----\nInput\n7 2 2\n1 3 1 2 4 1 2\n\nOutput\n3\n\nInput\n7 2 2\n1 100 1 100 1 100 1\n\nOutput\n-1\n\n\n\n-----Note-----\n\nFor the first sample, we can split the strip into 3 pieces: [1, 3, 1], [2, 4], [1, 2].\n\nFor the second sample, we can't let 1 and 100 be on the same piece, so no solution exists.",
        "solutions": "[\"def split(a,n,s,l):\\n    pieces = []\\n\\n    i = 1\\n    tmpmin = a[0]\\n    tmpmax = a[0]\\n    tmppc  = [a[0]]\\n    while i<n:\\n        if abs(a[i]-tmpmin)<=s and abs(a[i]-tmpmax)<=s:\\n            tmppc.append(a[i])\\n            if a[i]<tmpmin: tmpmin=a[i]\\n            elif a[i]>tmpmax: tmpmax = a[i]\\n        else:\\n            pieces.append(tmppc)\\n            tmppc = [a[i]]\\n            tmpmin = a[i]\\n            tmpmax = a[i]\\n        i += 1\\n    pieces.append(tmppc)\\n\\n    fail = False        \\n    for j in range(len(pieces)):\\n        if len(pieces[j])<l:\\n            if j>0:\\n                prevpc = pieces[j-1]\\n                minj = min(pieces[j])\\n                maxj = max(pieces[j])\\n                \\n                while len(pieces[j])<l:\\n                    tmp = prevpc.pop()\\n                    if abs(tmp-minj)<=s and abs(tmp-maxj)<=s:\\n                        pieces[j].insert(0,tmp)\\n                        if tmp<minj: minj=tmp\\n                        elif tmp>maxj: maxj=tmp\\n                    else:\\n                        return -1\\n                    if len(prevpc)<l:\\n                        return -1\\n            else:\\n                return -1\\n    return len(pieces)\\n\\nn,s,l = [int(s) for s in input().split()]\\na = [int(s) for s in input().split()]\\n\\n\\nres = split(a,n,s,l)\\nif res<0:\\n    a.reverse()\\n    res = split(a,n,s,l)\\nprint(res)\\n    \\n\\n\\n\\n\"]",
        "difficulty": "interview",
        "input": "10 3 3\n1 1 1 2 2 5 6 7 8 9\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/488/D"
    },
    {
        "id": 251,
        "task_id": 1855,
        "test_case_id": 1,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "3 2\n2 1 3\n",
        "output": "5 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 252,
        "task_id": 1855,
        "test_case_id": 2,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "5 5\n2 1 5 3 4\n",
        "output": "15 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 253,
        "task_id": 1855,
        "test_case_id": 3,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "7 3\n2 7 3 1 5 4 6\n",
        "output": "18 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 254,
        "task_id": 1855,
        "test_case_id": 4,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "1 1\n1\n",
        "output": "1 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 255,
        "task_id": 1855,
        "test_case_id": 5,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "2 1\n1 2\n",
        "output": "2 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 256,
        "task_id": 1855,
        "test_case_id": 6,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "2 2\n2 1\n",
        "output": "3 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 257,
        "task_id": 1855,
        "test_case_id": 7,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "3 2\n3 2 1\n",
        "output": "5 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 258,
        "task_id": 1855,
        "test_case_id": 8,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "5 4\n2 1 3 5 4\n",
        "output": "14 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 259,
        "task_id": 1855,
        "test_case_id": 9,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "10 3\n4 6 7 8 9 1 10 3 5 2\n",
        "output": "27 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 260,
        "task_id": 1855,
        "test_case_id": 10,
        "question": "You are given a permutation $p_1, p_2, \\ldots, p_n$ of integers from $1$ to $n$ and an integer $k$, such that $1 \\leq k \\leq n$. A permutation means that every number from $1$ to $n$ is contained in $p$ exactly once.\n\nLet's consider all partitions of this permutation into $k$ disjoint segments. Formally, a partition is a set of segments $\\{[l_1, r_1], [l_2, r_2], \\ldots, [l_k, r_k]\\}$, such that:\n\n  $1 \\leq l_i \\leq r_i \\leq n$ for all $1 \\leq i \\leq k$;  For all $1 \\leq j \\leq n$ there exists exactly one segment $[l_i, r_i]$, such that $l_i \\leq j \\leq r_i$. \n\nTwo partitions are different if there exists a segment that lies in one partition but not the other.\n\nLet's calculate the partition value, defined as $\\sum\\limits_{i=1}^{k} {\\max\\limits_{l_i \\leq j \\leq r_i} {p_j}}$, for all possible partitions of the permutation into $k$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $998\\,244\\,353$.\n\n\n-----Input-----\n\nThe first line contains two integers, $n$ and $k$ ($1 \\leq k \\leq n \\leq 200\\,000$) — the size of the given permutation and the number of segments in a partition.\n\nThe second line contains $n$ different integers $p_1, p_2, \\ldots, p_n$ ($1 \\leq p_i \\leq n$) — the given permutation.\n\n\n-----Output-----\n\nPrint two integers — the maximum possible partition value over all partitions of the permutation into $k$ disjoint segments and the number of such partitions for which the partition value is equal to the maximum possible value, modulo $998\\,244\\,353$.\n\nPlease note that you should only find the second value modulo $998\\,244\\,353$.\n\n\n-----Examples-----\nInput\n3 2\n2 1 3\n\nOutput\n5 2\n\nInput\n5 5\n2 1 5 3 4\n\nOutput\n15 1\n\nInput\n7 3\n2 7 3 1 5 4 6\n\nOutput\n18 6\n\n\n\n-----Note-----\n\nIn the first test, for $k = 2$, there exists only two valid partitions: $\\{[1, 1], [2, 3]\\}$ and $\\{[1, 2], [3, 3]\\}$. For each partition, the partition value is equal to $2 + 3 = 5$. So, the maximum possible value is $5$ and the number of partitions is $2$.\n\nIn the third test, for $k = 3$, the partitions with the maximum possible partition value are $\\{[1, 2], [3, 5], [6, 7]\\}$, $\\{[1, 3], [4, 5], [6, 7]\\}$, $\\{[1, 4], [5, 5], [6, 7]\\}$, $\\{[1, 2], [3, 6], [7, 7]\\}$, $\\{[1, 3], [4, 6], [7, 7]\\}$, $\\{[1, 4], [5, 6], [7, 7]\\}$. For all of them, the partition value is equal to $7 + 5 + 6 = 18$. \n\nThe partition $\\{[1, 2], [3, 4], [5, 7]\\}$, however, has the partition value $7 + 3 + 6 = 16$. This is not the maximum possible value, so we don't count it.",
        "solutions": "[\"n,k=map(int,input().split())\\nL=list(map(int,input().split()))\\nind=[]\\nfor i in range(n):\\n    if L[i]>n-k:ind.append(i)\\nm=1\\nfor i in range(len(ind)-1):\\n    m*=(ind[i+1]-ind[i])\\n    m%=998244353\\nprint(((n*(n+1)//2)-((n-k)*((n-k)+1))//2),m)\"]",
        "difficulty": "interview",
        "input": "100 77\n59 92 18 16 45 82 63 43 50 68 19 13 53 79 48 28 94 49 25 77 54 8 61 66 40 100 99 20 35 14 52 56 22 17 57 36 23 90 4 65 84 42 30 27 3 15 87 32 93 74 46 91 41 9 34 12 11 7 10 86 78 72 81 73 51 55 58 97 39 31 5 24 29 88 95 6 44 37 60 62 83 33 47 21 2 38 26 98 71 75 96 70 76 69 64 89 67 1 80 85\n",
        "output": "4774 622080\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1326/C"
    },
    {
        "id": 261,
        "task_id": 2041,
        "test_case_id": 5,
        "question": "This is the harder version of the problem. In this version, $1 \\le n, m \\le 2\\cdot10^5$. You can hack this problem if you locked it. But you can hack the previous problem only if you locked both problems.\n\nYou are given a sequence of integers $a=[a_1,a_2,\\dots,a_n]$ of length $n$. Its subsequence is obtained by removing zero or more elements from the sequence $a$ (they do not necessarily go consecutively). For example, for the sequence $a=[11,20,11,33,11,20,11]$:  $[11,20,11,33,11,20,11]$, $[11,20,11,33,11,20]$, $[11,11,11,11]$, $[20]$, $[33,20]$ are subsequences (these are just some of the long list);  $[40]$, $[33,33]$, $[33,20,20]$, $[20,20,11,11]$ are not subsequences. \n\nSuppose that an additional non-negative integer $k$ ($1 \\le k \\le n$) is given, then the subsequence is called optimal if:  it has a length of $k$ and the sum of its elements is the maximum possible among all subsequences of length $k$;  and among all subsequences of length $k$ that satisfy the previous item, it is lexicographically minimal. \n\nRecall that the sequence $b=[b_1, b_2, \\dots, b_k]$ is lexicographically smaller than the sequence $c=[c_1, c_2, \\dots, c_k]$ if the first element (from the left) in which they differ less in the sequence $b$ than in $c$. Formally: there exists $t$ ($1 \\le t \\le k$) such that $b_1=c_1$, $b_2=c_2$, ..., $b_{t-1}=c_{t-1}$ and at the same time $b_t<c_t$. For example:  $[10, 20, 20]$ lexicographically less than $[10, 21, 1]$,  $[7, 99, 99]$ is lexicographically less than $[10, 21, 1]$,  $[10, 21, 0]$ is lexicographically less than $[10, 21, 1]$. \n\nYou are given a sequence of $a=[a_1,a_2,\\dots,a_n]$ and $m$ requests, each consisting of two numbers $k_j$ and $pos_j$ ($1 \\le k \\le n$, $1 \\le pos_j \\le k_j$). For each query, print the value that is in the index $pos_j$ of the optimal subsequence of the given sequence $a$ for $k=k_j$.\n\nFor example, if $n=4$, $a=[10,20,30,20]$, $k_j=2$, then the optimal subsequence is $[20,30]$ — it is the minimum lexicographically among all subsequences of length $2$ with the maximum total sum of items. Thus, the answer to the request $k_j=2$, $pos_j=1$ is the number $20$, and the answer to the request $k_j=2$, $pos_j=2$ is the number $30$.\n\n\n-----Input-----\n\nThe first line contains an integer $n$ ($1 \\le n \\le 2\\cdot10^5$) — the length of the sequence $a$.\n\nThe second line contains elements of the sequence $a$: integer numbers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^9$).\n\nThe third line contains an integer $m$ ($1 \\le m \\le 2\\cdot10^5$) — the number of requests.\n\nThe following $m$ lines contain pairs of integers $k_j$ and $pos_j$ ($1 \\le k \\le n$, $1 \\le pos_j \\le k_j$) — the requests.\n\n\n-----Output-----\n\nPrint $m$ integers $r_1, r_2, \\dots, r_m$ ($1 \\le r_j \\le 10^9$) one per line: answers to the requests in the order they appear in the input. The value of $r_j$ should be equal to the value contained in the position $pos_j$ of the optimal subsequence for $k=k_j$.\n\n\n-----Examples-----\nInput\n3\n10 20 10\n6\n1 1\n2 1\n2 2\n3 1\n3 2\n3 3\n\nOutput\n20\n10\n20\n10\n20\n10\n\nInput\n7\n1 2 1 3 1 2 1\n9\n2 1\n2 2\n3 1\n3 2\n3 3\n1 1\n7 1\n7 7\n7 4\n\nOutput\n2\n3\n2\n3\n2\n3\n1\n1\n3\n\n\n\n-----Note-----\n\nIn the first example, for $a=[10,20,10]$ the optimal subsequences are:   for $k=1$: $[20]$,  for $k=2$: $[10,20]$,  for $k=3$: $[10,20,10]$.",
        "solutions": "[\"import sys\\n\\nclass TreeNode:\\n    def __init__(self, k, v):\\n        self.key = k\\n        self.value = v\\n        self.left = None\\n        self.right = None\\n        self.parent = None\\n        self.height = 1\\n        self.num_left = 1\\n        self.num_total = 1\\n\\n\\nclass AvlTree:\\n\\n    def __init__(self):\\n        self._tree = None\\n\\n    def add(self, k, v):\\n        if not self._tree:\\n            self._tree = TreeNode(k, v)\\n            return\\n        node = self._add(k, v)\\n        if node:\\n            self._rebalance(node)\\n\\n    def _add(self, k, v):\\n        node = self._tree\\n        while node:\\n            if k < node.key:\\n                if node.left:\\n                    node = node.left\\n                else:\\n                    node.left = TreeNode(k, v)\\n                    node.left.parent = node\\n                    return node.left\\n            elif node.key < k:\\n                if node.right:\\n                    node = node.right\\n                else:\\n                    node.right = TreeNode(k, v)\\n                    node.right.parent = node\\n                    return node.right\\n            else:\\n                node.value = v\\n                return\\n\\n    @staticmethod\\n    def get_height(x):\\n        return x.height if x else 0\\n\\n    @staticmethod\\n    def get_num_total(x):\\n        return x.num_total if x else 0\\n\\n    def _rebalance(self, node):\\n\\n        n = node\\n        while n:\\n            lh = self.get_height(n.left)\\n            rh = self.get_height(n.right)\\n            n.height = max(lh, rh) + 1\\n            balance_factor = lh - rh\\n            n.num_total = 1 + self.get_num_total(n.left) + self.get_num_total(n.right)\\n            n.num_left = 1 + self.get_num_total(n.left)\\n\\n            if balance_factor > 1:\\n                if self.get_height(n.left.left) < self.get_height(n.left.right):\\n                    self._rotate_left(n.left)\\n                self._rotate_right(n)\\n            elif balance_factor < -1:\\n                if self.get_height(n.right.right) < self.get_height(n.right.left):\\n                    self._rotate_right(n.right)\\n                self._rotate_left(n)\\n            else:\\n                n = n.parent\\n\\n    def _remove_one(self, node):\\n        \\\"\\\"\\\"\\n        Side effect!!! Changes node. Node should have exactly one child\\n        \\\"\\\"\\\"\\n        replacement = node.left or node.right\\n        if node.parent:\\n            if AvlTree._is_left(node):\\n                node.parent.left = replacement\\n            else:\\n                node.parent.right = replacement\\n            replacement.parent = node.parent\\n            node.parent = None\\n        else:\\n            self._tree = replacement\\n            replacement.parent = None\\n        node.left = None\\n        node.right = None\\n        node.parent = None\\n        self._rebalance(replacement)\\n\\n    def _remove_leaf(self, node):\\n        if node.parent:\\n            if AvlTree._is_left(node):\\n                node.parent.left = None\\n            else:\\n                node.parent.right = None\\n            self._rebalance(node.parent)\\n        else:\\n            self._tree = None\\n        node.parent = None\\n        node.left = None\\n        node.right = None\\n\\n    def remove(self, k):\\n        node = self._get_node(k)\\n        if not node:\\n            return\\n        if AvlTree._is_leaf(node):\\n            self._remove_leaf(node)\\n            return\\n        if node.left and node.right:\\n            nxt = AvlTree._get_next(node)\\n            node.key = nxt.key\\n            node.value = nxt.value\\n            if self._is_leaf(nxt):\\n                self._remove_leaf(nxt)\\n            else:\\n                self._remove_one(nxt)\\n            self._rebalance(node)\\n        else:\\n            self._remove_one(node)\\n\\n    def get(self, k):\\n        node = self._get_node(k)\\n        return node.value if node else -1\\n\\n    def _get_node(self, k):\\n        if not self._tree:\\n            return None\\n        node = self._tree\\n        while node:\\n            if k < node.key:\\n                node = node.left\\n            elif node.key < k:\\n                node = node.right\\n            else:\\n                return node\\n        return None\\n\\n    def get_at(self, pos):\\n        x = pos + 1\\n        node = self._tree\\n        while node:\\n            if x < node.num_left:\\n                node = node.left\\n            elif node.num_left < x:\\n                x -= node.num_left\\n                node = node.right\\n            else:\\n                return (node.key, node.value)\\n        raise IndexError(\\\"Out of ranges\\\")\\n\\n    @staticmethod\\n    def _is_left(node):\\n        return node.parent.left and node.parent.left == node\\n\\n    @staticmethod\\n    def _is_leaf(node):\\n        return node.left is None and node.right is None\\n\\n    def _rotate_right(self, node):\\n        if not node.parent:\\n            self._tree = node.left\\n            node.left.parent = None\\n        elif AvlTree._is_left(node):\\n            node.parent.left = node.left\\n            node.left.parent = node.parent\\n        else:\\n            node.parent.right = node.left\\n            node.left.parent = node.parent\\n        bk = node.left.right\\n        node.left.right = node\\n        node.parent = node.left\\n        node.left = bk\\n        if bk:\\n            bk.parent = node\\n        node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1\\n        node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)\\n        node.num_left = 1 + self.get_num_total(node.left)\\n\\n    def _rotate_left(self, node):\\n        if not node.parent:\\n            self._tree = node.right\\n            node.right.parent = None\\n        elif AvlTree._is_left(node):\\n            node.parent.left = node.right\\n            node.right.parent = node.parent\\n        else:\\n            node.parent.right = node.right\\n            node.right.parent = node.parent\\n        bk = node.right.left\\n        node.right.left = node\\n        node.parent = node.right\\n        node.right = bk\\n        if bk:\\n            bk.parent = node\\n        node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1\\n        node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)\\n        node.num_left = 1 + self.get_num_total(node.left)\\n\\n    @staticmethod\\n    def _get_next(node):\\n        if not node.right:\\n            return node.parent\\n        n = node.right\\n        while n.left:\\n            n = n.left\\n        return n\\n\\n\\ndef __starting_point():\\n    lines = sys.stdin.readlines()\\n    n = int(lines[0])\\n    aa = [(a, i) for i, a in enumerate(map(int, lines[1].split()))]\\n    m = int(lines[2])\\n    qs = [None]*m\\n    ans = [None]*m\\n    for i in range(m):\\n        k, pos = list(map(int, lines[i+3].split()))\\n        qs[i] = (pos, k, i)\\n    qs.sort(key=lambda x: x[1])\\n    aa.sort(key=lambda x: x[1])\\n    aa.sort(key=lambda x: x[0], reverse=True)\\n    avl = AvlTree()\\n    s = 0\\n    for pos, k, i in qs:\\n        for a, j in aa[s: k]:\\n            avl.add(j, a)\\n        ans[i] = str(avl.get_at(pos - 1)[1])\\n        s = k\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n__starting_point()\", \"WIDTH = 10\\n\\ndef index_tree(n):\\n    levels = [ [1]*n ]\\n\\n    size = WIDTH\\n    while size < n:\\n        m, r = n // size, n % size\\n        levels.append( [size]*m + ([r] if r > 0 else []) )\\n        size *= WIDTH\\n\\n    return levels\\n\\ndef dec_index(levels, i):\\n    for level in levels:\\n        level[i] -= 1\\n        i //= WIDTH\\n\\ndef find_pos(levels, pos):\\n    i, l = 0, len(levels) - 1\\n    \\n    total = 0\\n    while True:\\n        level = levels[l]\\n        while total + level[i] < pos:\\n            total += level[i]\\n            i += 1\\n        \\n        if l == 0: return i\\n        i *= WIDTH\\n        l -= 1\\n\\n\\nimport sys\\n\\ndef main():\\n    ## INPUT\\n    numbers = [int(x) for x in sys.stdin.read().split()]\\n    n = numbers[0]\\n    sequence = numbers[1:n+1]\\n    m = numbers[n+1]\\n\\n    queries = {}\\n    for i in range(n+2, n+2 + 2*m, 2):\\n        k, pos = numbers[i], numbers[i+1]\\n        if k in queries: queries[k][pos] = None\\n        else: queries[k] = { pos: None }\\n    \\n    ## WORK\\n    sequence1 = sorted([ (s,-i) for i,s in enumerate(sequence) ])\\n    tree = index_tree(n)\\n    size = n\\n\\n    for _, neg_i in sequence1: \\n        if size in queries:\\n            for pos in queries[size]:\\n                queries[size][pos] = find_pos(tree, pos)\\n\\n        dec_index(tree, -neg_i)\\n        size -= 1\\n\\n    ## PRINT \\n    for i in range(n+2, n+2 + 2*m, 2):\\n        k, pos = numbers[i], numbers[i+1]\\n        print(sequence[ queries[k][pos] ])\\n\\n\\nmain()\\n\\n\", \"from collections import defaultdict\\nimport sys as _sys\\n\\n\\ndef main():\\n    n, = _read_ints()\\n    a = tuple(_read_ints())\\n    m, = _read_ints()\\n    queries = (tuple(_read_ints()) for i_query in range(m))\\n    result = process_queries(a, queries)\\n    print(*result, sep='\\\\n')\\n\\n\\ndef _read_line():\\n    result = _sys.stdin.readline()\\n    assert result[-1] == \\\"\\\\n\\\"\\n    return result[:-1]\\n\\n\\ndef _read_ints():\\n    return map(int, _read_line().split())\\n\\n\\ndef process_queries(sequence, queries):\\n    sequence = tuple(sequence)\\n    \\n    indices_by_values = defaultdict(list)\\n    for i, x in enumerate(sequence):\\n        indices_by_values[x].append(i)\\n    \\n    enumerated_queries = sorted(enumerate(queries), key=lambda iv: iv[1][0])[::-1]\\n    queries_responses = [None] * len(enumerated_queries)\\n    \\n    selections_tree = [0] * (len(sequence) + 1)\\n    \\n    k = 0\\n    for value, indices in sorted(indices_by_values.items(), reverse=True):\\n        for index_to_select in indices:\\n            _fenwick_tree_add(selections_tree, index_to_select, 1)\\n            k += 1\\n            while enumerated_queries and enumerated_queries[-1][1][0] == k:\\n                query_index, (_k, subseq_index) = enumerated_queries.pop()\\n                seq_index = _find_seq_index_by_subseq_index(selections_tree, subseq_index)\\n                queries_responses[query_index] = sequence[seq_index]\\n\\n    return queries_responses\\n\\n\\ndef _find_seq_index_by_subseq_index(tree, subseq_i):\\n    min_i = 0\\n    max_i = len(tree) - 1\\n    while min_i != max_i:\\n        mid_i = (min_i + max_i) // 2\\n        if _fenwick_tree_prefix_sum(tree, mid_i) < subseq_i:\\n            min_i = mid_i + 1\\n        else:\\n            max_i = mid_i\\n    return min_i\\n\\n\\ndef _fenwick_tree_prefix_sum(tree, i):\\n    i += 1\\n    result = 0\\n    while i != 0:\\n        result += tree[i]\\n        i -= i & (-i)\\n    return result\\n\\n\\ndef _fenwick_tree_add(tree, i, x):\\n    i += 1\\n    while i < len(tree):\\n        tree[i] += x\\n        i += i & (-i)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys as _sys\\n\\n\\ndef main():\\n    n, = _read_ints()\\n    a = tuple(_read_ints())\\n    m, = _read_ints()\\n    queries = (tuple(_read_ints()) for i_query in range(m))\\n    result = process_queries(a, queries)\\n    print(*result, sep='\\\\n')\\n\\n\\ndef _read_line():\\n    result = _sys.stdin.readline()\\n    assert result[-1] == \\\"\\\\n\\\"\\n    return result[:-1]\\n\\n\\ndef _read_ints():\\n    return map(int, _read_line().split())\\n\\n\\ndef process_queries(sequence, queries):\\n    sequence = tuple(sequence)\\n    \\n    indices_to_select = sorted(\\n        range(len(sequence)),\\n        key=lambda index: (-sequence[index], index)\\n    )\\n    \\n    enumerated_queries = sorted(enumerate(queries), key=lambda iv: iv[1][0])[::-1]\\n    queries_responses = [None] * len(enumerated_queries)\\n    \\n    selections_tree = [0] * (len(sequence) + 1)\\n    \\n    selected_n = 0\\n    for index_to_select in indices_to_select:\\n        _fenwick_tree_add(selections_tree, index_to_select, 1)\\n        selected_n += 1\\n        while enumerated_queries and enumerated_queries[-1][1][0] == selected_n:\\n            query_index, (_k, subseq_index) = enumerated_queries.pop()\\n            seq_index = _find_seq_index_by_subseq_index(selections_tree, subseq_index)\\n            queries_responses[query_index] = sequence[seq_index]\\n    \\n    return queries_responses\\n\\n\\ndef _find_seq_index_by_subseq_index(tree, subseq_i):\\n    seq_length = len(tree) - 1\\n    min_i = 0\\n    max_i = seq_length - 1\\n    while min_i != max_i:\\n        mid_i = (min_i + max_i) // 2\\n        if _fenwick_tree_prefix_sum(tree, mid_i) < subseq_i:\\n            min_i = mid_i + 1\\n        else:\\n            max_i = mid_i\\n    return min_i\\n\\n\\ndef _fenwick_tree_prefix_sum(tree, i):\\n    i += 1\\n    result = 0\\n    while i != 0:\\n        result += tree[i]\\n        i -= i & (-i)\\n    return result\\n\\n\\ndef _fenwick_tree_add(tree, i, x):\\n    i += 1\\n    while i < len(tree):\\n        tree[i] += x\\n        i += i & (-i)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "2\n1 10\n3\n2 2\n2 1\n1 1\n",
        "output": "10\n1\n10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1227/D2"
    },
    {
        "id": 262,
        "task_id": 2041,
        "test_case_id": 6,
        "question": "This is the harder version of the problem. In this version, $1 \\le n, m \\le 2\\cdot10^5$. You can hack this problem if you locked it. But you can hack the previous problem only if you locked both problems.\n\nYou are given a sequence of integers $a=[a_1,a_2,\\dots,a_n]$ of length $n$. Its subsequence is obtained by removing zero or more elements from the sequence $a$ (they do not necessarily go consecutively). For example, for the sequence $a=[11,20,11,33,11,20,11]$:  $[11,20,11,33,11,20,11]$, $[11,20,11,33,11,20]$, $[11,11,11,11]$, $[20]$, $[33,20]$ are subsequences (these are just some of the long list);  $[40]$, $[33,33]$, $[33,20,20]$, $[20,20,11,11]$ are not subsequences. \n\nSuppose that an additional non-negative integer $k$ ($1 \\le k \\le n$) is given, then the subsequence is called optimal if:  it has a length of $k$ and the sum of its elements is the maximum possible among all subsequences of length $k$;  and among all subsequences of length $k$ that satisfy the previous item, it is lexicographically minimal. \n\nRecall that the sequence $b=[b_1, b_2, \\dots, b_k]$ is lexicographically smaller than the sequence $c=[c_1, c_2, \\dots, c_k]$ if the first element (from the left) in which they differ less in the sequence $b$ than in $c$. Formally: there exists $t$ ($1 \\le t \\le k$) such that $b_1=c_1$, $b_2=c_2$, ..., $b_{t-1}=c_{t-1}$ and at the same time $b_t<c_t$. For example:  $[10, 20, 20]$ lexicographically less than $[10, 21, 1]$,  $[7, 99, 99]$ is lexicographically less than $[10, 21, 1]$,  $[10, 21, 0]$ is lexicographically less than $[10, 21, 1]$. \n\nYou are given a sequence of $a=[a_1,a_2,\\dots,a_n]$ and $m$ requests, each consisting of two numbers $k_j$ and $pos_j$ ($1 \\le k \\le n$, $1 \\le pos_j \\le k_j$). For each query, print the value that is in the index $pos_j$ of the optimal subsequence of the given sequence $a$ for $k=k_j$.\n\nFor example, if $n=4$, $a=[10,20,30,20]$, $k_j=2$, then the optimal subsequence is $[20,30]$ — it is the minimum lexicographically among all subsequences of length $2$ with the maximum total sum of items. Thus, the answer to the request $k_j=2$, $pos_j=1$ is the number $20$, and the answer to the request $k_j=2$, $pos_j=2$ is the number $30$.\n\n\n-----Input-----\n\nThe first line contains an integer $n$ ($1 \\le n \\le 2\\cdot10^5$) — the length of the sequence $a$.\n\nThe second line contains elements of the sequence $a$: integer numbers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^9$).\n\nThe third line contains an integer $m$ ($1 \\le m \\le 2\\cdot10^5$) — the number of requests.\n\nThe following $m$ lines contain pairs of integers $k_j$ and $pos_j$ ($1 \\le k \\le n$, $1 \\le pos_j \\le k_j$) — the requests.\n\n\n-----Output-----\n\nPrint $m$ integers $r_1, r_2, \\dots, r_m$ ($1 \\le r_j \\le 10^9$) one per line: answers to the requests in the order they appear in the input. The value of $r_j$ should be equal to the value contained in the position $pos_j$ of the optimal subsequence for $k=k_j$.\n\n\n-----Examples-----\nInput\n3\n10 20 10\n6\n1 1\n2 1\n2 2\n3 1\n3 2\n3 3\n\nOutput\n20\n10\n20\n10\n20\n10\n\nInput\n7\n1 2 1 3 1 2 1\n9\n2 1\n2 2\n3 1\n3 2\n3 3\n1 1\n7 1\n7 7\n7 4\n\nOutput\n2\n3\n2\n3\n2\n3\n1\n1\n3\n\n\n\n-----Note-----\n\nIn the first example, for $a=[10,20,10]$ the optimal subsequences are:   for $k=1$: $[20]$,  for $k=2$: $[10,20]$,  for $k=3$: $[10,20,10]$.",
        "solutions": "[\"import sys\\n\\nclass TreeNode:\\n    def __init__(self, k, v):\\n        self.key = k\\n        self.value = v\\n        self.left = None\\n        self.right = None\\n        self.parent = None\\n        self.height = 1\\n        self.num_left = 1\\n        self.num_total = 1\\n\\n\\nclass AvlTree:\\n\\n    def __init__(self):\\n        self._tree = None\\n\\n    def add(self, k, v):\\n        if not self._tree:\\n            self._tree = TreeNode(k, v)\\n            return\\n        node = self._add(k, v)\\n        if node:\\n            self._rebalance(node)\\n\\n    def _add(self, k, v):\\n        node = self._tree\\n        while node:\\n            if k < node.key:\\n                if node.left:\\n                    node = node.left\\n                else:\\n                    node.left = TreeNode(k, v)\\n                    node.left.parent = node\\n                    return node.left\\n            elif node.key < k:\\n                if node.right:\\n                    node = node.right\\n                else:\\n                    node.right = TreeNode(k, v)\\n                    node.right.parent = node\\n                    return node.right\\n            else:\\n                node.value = v\\n                return\\n\\n    @staticmethod\\n    def get_height(x):\\n        return x.height if x else 0\\n\\n    @staticmethod\\n    def get_num_total(x):\\n        return x.num_total if x else 0\\n\\n    def _rebalance(self, node):\\n\\n        n = node\\n        while n:\\n            lh = self.get_height(n.left)\\n            rh = self.get_height(n.right)\\n            n.height = max(lh, rh) + 1\\n            balance_factor = lh - rh\\n            n.num_total = 1 + self.get_num_total(n.left) + self.get_num_total(n.right)\\n            n.num_left = 1 + self.get_num_total(n.left)\\n\\n            if balance_factor > 1:\\n                if self.get_height(n.left.left) < self.get_height(n.left.right):\\n                    self._rotate_left(n.left)\\n                self._rotate_right(n)\\n            elif balance_factor < -1:\\n                if self.get_height(n.right.right) < self.get_height(n.right.left):\\n                    self._rotate_right(n.right)\\n                self._rotate_left(n)\\n            else:\\n                n = n.parent\\n\\n    def _remove_one(self, node):\\n        \\\"\\\"\\\"\\n        Side effect!!! Changes node. Node should have exactly one child\\n        \\\"\\\"\\\"\\n        replacement = node.left or node.right\\n        if node.parent:\\n            if AvlTree._is_left(node):\\n                node.parent.left = replacement\\n            else:\\n                node.parent.right = replacement\\n            replacement.parent = node.parent\\n            node.parent = None\\n        else:\\n            self._tree = replacement\\n            replacement.parent = None\\n        node.left = None\\n        node.right = None\\n        node.parent = None\\n        self._rebalance(replacement)\\n\\n    def _remove_leaf(self, node):\\n        if node.parent:\\n            if AvlTree._is_left(node):\\n                node.parent.left = None\\n            else:\\n                node.parent.right = None\\n            self._rebalance(node.parent)\\n        else:\\n            self._tree = None\\n        node.parent = None\\n        node.left = None\\n        node.right = None\\n\\n    def remove(self, k):\\n        node = self._get_node(k)\\n        if not node:\\n            return\\n        if AvlTree._is_leaf(node):\\n            self._remove_leaf(node)\\n            return\\n        if node.left and node.right:\\n            nxt = AvlTree._get_next(node)\\n            node.key = nxt.key\\n            node.value = nxt.value\\n            if self._is_leaf(nxt):\\n                self._remove_leaf(nxt)\\n            else:\\n                self._remove_one(nxt)\\n            self._rebalance(node)\\n        else:\\n            self._remove_one(node)\\n\\n    def get(self, k):\\n        node = self._get_node(k)\\n        return node.value if node else -1\\n\\n    def _get_node(self, k):\\n        if not self._tree:\\n            return None\\n        node = self._tree\\n        while node:\\n            if k < node.key:\\n                node = node.left\\n            elif node.key < k:\\n                node = node.right\\n            else:\\n                return node\\n        return None\\n\\n    def get_at(self, pos):\\n        x = pos + 1\\n        node = self._tree\\n        while node:\\n            if x < node.num_left:\\n                node = node.left\\n            elif node.num_left < x:\\n                x -= node.num_left\\n                node = node.right\\n            else:\\n                return (node.key, node.value)\\n        raise IndexError(\\\"Out of ranges\\\")\\n\\n    @staticmethod\\n    def _is_left(node):\\n        return node.parent.left and node.parent.left == node\\n\\n    @staticmethod\\n    def _is_leaf(node):\\n        return node.left is None and node.right is None\\n\\n    def _rotate_right(self, node):\\n        if not node.parent:\\n            self._tree = node.left\\n            node.left.parent = None\\n        elif AvlTree._is_left(node):\\n            node.parent.left = node.left\\n            node.left.parent = node.parent\\n        else:\\n            node.parent.right = node.left\\n            node.left.parent = node.parent\\n        bk = node.left.right\\n        node.left.right = node\\n        node.parent = node.left\\n        node.left = bk\\n        if bk:\\n            bk.parent = node\\n        node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1\\n        node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)\\n        node.num_left = 1 + self.get_num_total(node.left)\\n\\n    def _rotate_left(self, node):\\n        if not node.parent:\\n            self._tree = node.right\\n            node.right.parent = None\\n        elif AvlTree._is_left(node):\\n            node.parent.left = node.right\\n            node.right.parent = node.parent\\n        else:\\n            node.parent.right = node.right\\n            node.right.parent = node.parent\\n        bk = node.right.left\\n        node.right.left = node\\n        node.parent = node.right\\n        node.right = bk\\n        if bk:\\n            bk.parent = node\\n        node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1\\n        node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)\\n        node.num_left = 1 + self.get_num_total(node.left)\\n\\n    @staticmethod\\n    def _get_next(node):\\n        if not node.right:\\n            return node.parent\\n        n = node.right\\n        while n.left:\\n            n = n.left\\n        return n\\n\\n\\ndef __starting_point():\\n    lines = sys.stdin.readlines()\\n    n = int(lines[0])\\n    aa = [(a, i) for i, a in enumerate(map(int, lines[1].split()))]\\n    m = int(lines[2])\\n    qs = [None]*m\\n    ans = [None]*m\\n    for i in range(m):\\n        k, pos = list(map(int, lines[i+3].split()))\\n        qs[i] = (pos, k, i)\\n    qs.sort(key=lambda x: x[1])\\n    aa.sort(key=lambda x: x[1])\\n    aa.sort(key=lambda x: x[0], reverse=True)\\n    avl = AvlTree()\\n    s = 0\\n    for pos, k, i in qs:\\n        for a, j in aa[s: k]:\\n            avl.add(j, a)\\n        ans[i] = str(avl.get_at(pos - 1)[1])\\n        s = k\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n__starting_point()\", \"WIDTH = 10\\n\\ndef index_tree(n):\\n    levels = [ [1]*n ]\\n\\n    size = WIDTH\\n    while size < n:\\n        m, r = n // size, n % size\\n        levels.append( [size]*m + ([r] if r > 0 else []) )\\n        size *= WIDTH\\n\\n    return levels\\n\\ndef dec_index(levels, i):\\n    for level in levels:\\n        level[i] -= 1\\n        i //= WIDTH\\n\\ndef find_pos(levels, pos):\\n    i, l = 0, len(levels) - 1\\n    \\n    total = 0\\n    while True:\\n        level = levels[l]\\n        while total + level[i] < pos:\\n            total += level[i]\\n            i += 1\\n        \\n        if l == 0: return i\\n        i *= WIDTH\\n        l -= 1\\n\\n\\nimport sys\\n\\ndef main():\\n    ## INPUT\\n    numbers = [int(x) for x in sys.stdin.read().split()]\\n    n = numbers[0]\\n    sequence = numbers[1:n+1]\\n    m = numbers[n+1]\\n\\n    queries = {}\\n    for i in range(n+2, n+2 + 2*m, 2):\\n        k, pos = numbers[i], numbers[i+1]\\n        if k in queries: queries[k][pos] = None\\n        else: queries[k] = { pos: None }\\n    \\n    ## WORK\\n    sequence1 = sorted([ (s,-i) for i,s in enumerate(sequence) ])\\n    tree = index_tree(n)\\n    size = n\\n\\n    for _, neg_i in sequence1: \\n        if size in queries:\\n            for pos in queries[size]:\\n                queries[size][pos] = find_pos(tree, pos)\\n\\n        dec_index(tree, -neg_i)\\n        size -= 1\\n\\n    ## PRINT \\n    for i in range(n+2, n+2 + 2*m, 2):\\n        k, pos = numbers[i], numbers[i+1]\\n        print(sequence[ queries[k][pos] ])\\n\\n\\nmain()\\n\\n\", \"from collections import defaultdict\\nimport sys as _sys\\n\\n\\ndef main():\\n    n, = _read_ints()\\n    a = tuple(_read_ints())\\n    m, = _read_ints()\\n    queries = (tuple(_read_ints()) for i_query in range(m))\\n    result = process_queries(a, queries)\\n    print(*result, sep='\\\\n')\\n\\n\\ndef _read_line():\\n    result = _sys.stdin.readline()\\n    assert result[-1] == \\\"\\\\n\\\"\\n    return result[:-1]\\n\\n\\ndef _read_ints():\\n    return map(int, _read_line().split())\\n\\n\\ndef process_queries(sequence, queries):\\n    sequence = tuple(sequence)\\n    \\n    indices_by_values = defaultdict(list)\\n    for i, x in enumerate(sequence):\\n        indices_by_values[x].append(i)\\n    \\n    enumerated_queries = sorted(enumerate(queries), key=lambda iv: iv[1][0])[::-1]\\n    queries_responses = [None] * len(enumerated_queries)\\n    \\n    selections_tree = [0] * (len(sequence) + 1)\\n    \\n    k = 0\\n    for value, indices in sorted(indices_by_values.items(), reverse=True):\\n        for index_to_select in indices:\\n            _fenwick_tree_add(selections_tree, index_to_select, 1)\\n            k += 1\\n            while enumerated_queries and enumerated_queries[-1][1][0] == k:\\n                query_index, (_k, subseq_index) = enumerated_queries.pop()\\n                seq_index = _find_seq_index_by_subseq_index(selections_tree, subseq_index)\\n                queries_responses[query_index] = sequence[seq_index]\\n\\n    return queries_responses\\n\\n\\ndef _find_seq_index_by_subseq_index(tree, subseq_i):\\n    min_i = 0\\n    max_i = len(tree) - 1\\n    while min_i != max_i:\\n        mid_i = (min_i + max_i) // 2\\n        if _fenwick_tree_prefix_sum(tree, mid_i) < subseq_i:\\n            min_i = mid_i + 1\\n        else:\\n            max_i = mid_i\\n    return min_i\\n\\n\\ndef _fenwick_tree_prefix_sum(tree, i):\\n    i += 1\\n    result = 0\\n    while i != 0:\\n        result += tree[i]\\n        i -= i & (-i)\\n    return result\\n\\n\\ndef _fenwick_tree_add(tree, i, x):\\n    i += 1\\n    while i < len(tree):\\n        tree[i] += x\\n        i += i & (-i)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys as _sys\\n\\n\\ndef main():\\n    n, = _read_ints()\\n    a = tuple(_read_ints())\\n    m, = _read_ints()\\n    queries = (tuple(_read_ints()) for i_query in range(m))\\n    result = process_queries(a, queries)\\n    print(*result, sep='\\\\n')\\n\\n\\ndef _read_line():\\n    result = _sys.stdin.readline()\\n    assert result[-1] == \\\"\\\\n\\\"\\n    return result[:-1]\\n\\n\\ndef _read_ints():\\n    return map(int, _read_line().split())\\n\\n\\ndef process_queries(sequence, queries):\\n    sequence = tuple(sequence)\\n    \\n    indices_to_select = sorted(\\n        range(len(sequence)),\\n        key=lambda index: (-sequence[index], index)\\n    )\\n    \\n    enumerated_queries = sorted(enumerate(queries), key=lambda iv: iv[1][0])[::-1]\\n    queries_responses = [None] * len(enumerated_queries)\\n    \\n    selections_tree = [0] * (len(sequence) + 1)\\n    \\n    selected_n = 0\\n    for index_to_select in indices_to_select:\\n        _fenwick_tree_add(selections_tree, index_to_select, 1)\\n        selected_n += 1\\n        while enumerated_queries and enumerated_queries[-1][1][0] == selected_n:\\n            query_index, (_k, subseq_index) = enumerated_queries.pop()\\n            seq_index = _find_seq_index_by_subseq_index(selections_tree, subseq_index)\\n            queries_responses[query_index] = sequence[seq_index]\\n    \\n    return queries_responses\\n\\n\\ndef _find_seq_index_by_subseq_index(tree, subseq_i):\\n    seq_length = len(tree) - 1\\n    min_i = 0\\n    max_i = seq_length - 1\\n    while min_i != max_i:\\n        mid_i = (min_i + max_i) // 2\\n        if _fenwick_tree_prefix_sum(tree, mid_i) < subseq_i:\\n            min_i = mid_i + 1\\n        else:\\n            max_i = mid_i\\n    return min_i\\n\\n\\ndef _fenwick_tree_prefix_sum(tree, i):\\n    i += 1\\n    result = 0\\n    while i != 0:\\n        result += tree[i]\\n        i -= i & (-i)\\n    return result\\n\\n\\ndef _fenwick_tree_add(tree, i, x):\\n    i += 1\\n    while i < len(tree):\\n        tree[i] += x\\n        i += i & (-i)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "2\n3922 3922\n3\n2 2\n2 1\n1 1\n",
        "output": "3922\n3922\n3922\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1227/D2"
    },
    {
        "id": 263,
        "task_id": 2041,
        "test_case_id": 7,
        "question": "This is the harder version of the problem. In this version, $1 \\le n, m \\le 2\\cdot10^5$. You can hack this problem if you locked it. But you can hack the previous problem only if you locked both problems.\n\nYou are given a sequence of integers $a=[a_1,a_2,\\dots,a_n]$ of length $n$. Its subsequence is obtained by removing zero or more elements from the sequence $a$ (they do not necessarily go consecutively). For example, for the sequence $a=[11,20,11,33,11,20,11]$:  $[11,20,11,33,11,20,11]$, $[11,20,11,33,11,20]$, $[11,11,11,11]$, $[20]$, $[33,20]$ are subsequences (these are just some of the long list);  $[40]$, $[33,33]$, $[33,20,20]$, $[20,20,11,11]$ are not subsequences. \n\nSuppose that an additional non-negative integer $k$ ($1 \\le k \\le n$) is given, then the subsequence is called optimal if:  it has a length of $k$ and the sum of its elements is the maximum possible among all subsequences of length $k$;  and among all subsequences of length $k$ that satisfy the previous item, it is lexicographically minimal. \n\nRecall that the sequence $b=[b_1, b_2, \\dots, b_k]$ is lexicographically smaller than the sequence $c=[c_1, c_2, \\dots, c_k]$ if the first element (from the left) in which they differ less in the sequence $b$ than in $c$. Formally: there exists $t$ ($1 \\le t \\le k$) such that $b_1=c_1$, $b_2=c_2$, ..., $b_{t-1}=c_{t-1}$ and at the same time $b_t<c_t$. For example:  $[10, 20, 20]$ lexicographically less than $[10, 21, 1]$,  $[7, 99, 99]$ is lexicographically less than $[10, 21, 1]$,  $[10, 21, 0]$ is lexicographically less than $[10, 21, 1]$. \n\nYou are given a sequence of $a=[a_1,a_2,\\dots,a_n]$ and $m$ requests, each consisting of two numbers $k_j$ and $pos_j$ ($1 \\le k \\le n$, $1 \\le pos_j \\le k_j$). For each query, print the value that is in the index $pos_j$ of the optimal subsequence of the given sequence $a$ for $k=k_j$.\n\nFor example, if $n=4$, $a=[10,20,30,20]$, $k_j=2$, then the optimal subsequence is $[20,30]$ — it is the minimum lexicographically among all subsequences of length $2$ with the maximum total sum of items. Thus, the answer to the request $k_j=2$, $pos_j=1$ is the number $20$, and the answer to the request $k_j=2$, $pos_j=2$ is the number $30$.\n\n\n-----Input-----\n\nThe first line contains an integer $n$ ($1 \\le n \\le 2\\cdot10^5$) — the length of the sequence $a$.\n\nThe second line contains elements of the sequence $a$: integer numbers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^9$).\n\nThe third line contains an integer $m$ ($1 \\le m \\le 2\\cdot10^5$) — the number of requests.\n\nThe following $m$ lines contain pairs of integers $k_j$ and $pos_j$ ($1 \\le k \\le n$, $1 \\le pos_j \\le k_j$) — the requests.\n\n\n-----Output-----\n\nPrint $m$ integers $r_1, r_2, \\dots, r_m$ ($1 \\le r_j \\le 10^9$) one per line: answers to the requests in the order they appear in the input. The value of $r_j$ should be equal to the value contained in the position $pos_j$ of the optimal subsequence for $k=k_j$.\n\n\n-----Examples-----\nInput\n3\n10 20 10\n6\n1 1\n2 1\n2 2\n3 1\n3 2\n3 3\n\nOutput\n20\n10\n20\n10\n20\n10\n\nInput\n7\n1 2 1 3 1 2 1\n9\n2 1\n2 2\n3 1\n3 2\n3 3\n1 1\n7 1\n7 7\n7 4\n\nOutput\n2\n3\n2\n3\n2\n3\n1\n1\n3\n\n\n\n-----Note-----\n\nIn the first example, for $a=[10,20,10]$ the optimal subsequences are:   for $k=1$: $[20]$,  for $k=2$: $[10,20]$,  for $k=3$: $[10,20,10]$.",
        "solutions": "[\"import sys\\n\\nclass TreeNode:\\n    def __init__(self, k, v):\\n        self.key = k\\n        self.value = v\\n        self.left = None\\n        self.right = None\\n        self.parent = None\\n        self.height = 1\\n        self.num_left = 1\\n        self.num_total = 1\\n\\n\\nclass AvlTree:\\n\\n    def __init__(self):\\n        self._tree = None\\n\\n    def add(self, k, v):\\n        if not self._tree:\\n            self._tree = TreeNode(k, v)\\n            return\\n        node = self._add(k, v)\\n        if node:\\n            self._rebalance(node)\\n\\n    def _add(self, k, v):\\n        node = self._tree\\n        while node:\\n            if k < node.key:\\n                if node.left:\\n                    node = node.left\\n                else:\\n                    node.left = TreeNode(k, v)\\n                    node.left.parent = node\\n                    return node.left\\n            elif node.key < k:\\n                if node.right:\\n                    node = node.right\\n                else:\\n                    node.right = TreeNode(k, v)\\n                    node.right.parent = node\\n                    return node.right\\n            else:\\n                node.value = v\\n                return\\n\\n    @staticmethod\\n    def get_height(x):\\n        return x.height if x else 0\\n\\n    @staticmethod\\n    def get_num_total(x):\\n        return x.num_total if x else 0\\n\\n    def _rebalance(self, node):\\n\\n        n = node\\n        while n:\\n            lh = self.get_height(n.left)\\n            rh = self.get_height(n.right)\\n            n.height = max(lh, rh) + 1\\n            balance_factor = lh - rh\\n            n.num_total = 1 + self.get_num_total(n.left) + self.get_num_total(n.right)\\n            n.num_left = 1 + self.get_num_total(n.left)\\n\\n            if balance_factor > 1:\\n                if self.get_height(n.left.left) < self.get_height(n.left.right):\\n                    self._rotate_left(n.left)\\n                self._rotate_right(n)\\n            elif balance_factor < -1:\\n                if self.get_height(n.right.right) < self.get_height(n.right.left):\\n                    self._rotate_right(n.right)\\n                self._rotate_left(n)\\n            else:\\n                n = n.parent\\n\\n    def _remove_one(self, node):\\n        \\\"\\\"\\\"\\n        Side effect!!! Changes node. Node should have exactly one child\\n        \\\"\\\"\\\"\\n        replacement = node.left or node.right\\n        if node.parent:\\n            if AvlTree._is_left(node):\\n                node.parent.left = replacement\\n            else:\\n                node.parent.right = replacement\\n            replacement.parent = node.parent\\n            node.parent = None\\n        else:\\n            self._tree = replacement\\n            replacement.parent = None\\n        node.left = None\\n        node.right = None\\n        node.parent = None\\n        self._rebalance(replacement)\\n\\n    def _remove_leaf(self, node):\\n        if node.parent:\\n            if AvlTree._is_left(node):\\n                node.parent.left = None\\n            else:\\n                node.parent.right = None\\n            self._rebalance(node.parent)\\n        else:\\n            self._tree = None\\n        node.parent = None\\n        node.left = None\\n        node.right = None\\n\\n    def remove(self, k):\\n        node = self._get_node(k)\\n        if not node:\\n            return\\n        if AvlTree._is_leaf(node):\\n            self._remove_leaf(node)\\n            return\\n        if node.left and node.right:\\n            nxt = AvlTree._get_next(node)\\n            node.key = nxt.key\\n            node.value = nxt.value\\n            if self._is_leaf(nxt):\\n                self._remove_leaf(nxt)\\n            else:\\n                self._remove_one(nxt)\\n            self._rebalance(node)\\n        else:\\n            self._remove_one(node)\\n\\n    def get(self, k):\\n        node = self._get_node(k)\\n        return node.value if node else -1\\n\\n    def _get_node(self, k):\\n        if not self._tree:\\n            return None\\n        node = self._tree\\n        while node:\\n            if k < node.key:\\n                node = node.left\\n            elif node.key < k:\\n                node = node.right\\n            else:\\n                return node\\n        return None\\n\\n    def get_at(self, pos):\\n        x = pos + 1\\n        node = self._tree\\n        while node:\\n            if x < node.num_left:\\n                node = node.left\\n            elif node.num_left < x:\\n                x -= node.num_left\\n                node = node.right\\n            else:\\n                return (node.key, node.value)\\n        raise IndexError(\\\"Out of ranges\\\")\\n\\n    @staticmethod\\n    def _is_left(node):\\n        return node.parent.left and node.parent.left == node\\n\\n    @staticmethod\\n    def _is_leaf(node):\\n        return node.left is None and node.right is None\\n\\n    def _rotate_right(self, node):\\n        if not node.parent:\\n            self._tree = node.left\\n            node.left.parent = None\\n        elif AvlTree._is_left(node):\\n            node.parent.left = node.left\\n            node.left.parent = node.parent\\n        else:\\n            node.parent.right = node.left\\n            node.left.parent = node.parent\\n        bk = node.left.right\\n        node.left.right = node\\n        node.parent = node.left\\n        node.left = bk\\n        if bk:\\n            bk.parent = node\\n        node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1\\n        node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)\\n        node.num_left = 1 + self.get_num_total(node.left)\\n\\n    def _rotate_left(self, node):\\n        if not node.parent:\\n            self._tree = node.right\\n            node.right.parent = None\\n        elif AvlTree._is_left(node):\\n            node.parent.left = node.right\\n            node.right.parent = node.parent\\n        else:\\n            node.parent.right = node.right\\n            node.right.parent = node.parent\\n        bk = node.right.left\\n        node.right.left = node\\n        node.parent = node.right\\n        node.right = bk\\n        if bk:\\n            bk.parent = node\\n        node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1\\n        node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)\\n        node.num_left = 1 + self.get_num_total(node.left)\\n\\n    @staticmethod\\n    def _get_next(node):\\n        if not node.right:\\n            return node.parent\\n        n = node.right\\n        while n.left:\\n            n = n.left\\n        return n\\n\\n\\ndef __starting_point():\\n    lines = sys.stdin.readlines()\\n    n = int(lines[0])\\n    aa = [(a, i) for i, a in enumerate(map(int, lines[1].split()))]\\n    m = int(lines[2])\\n    qs = [None]*m\\n    ans = [None]*m\\n    for i in range(m):\\n        k, pos = list(map(int, lines[i+3].split()))\\n        qs[i] = (pos, k, i)\\n    qs.sort(key=lambda x: x[1])\\n    aa.sort(key=lambda x: x[1])\\n    aa.sort(key=lambda x: x[0], reverse=True)\\n    avl = AvlTree()\\n    s = 0\\n    for pos, k, i in qs:\\n        for a, j in aa[s: k]:\\n            avl.add(j, a)\\n        ans[i] = str(avl.get_at(pos - 1)[1])\\n        s = k\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n__starting_point()\", \"WIDTH = 10\\n\\ndef index_tree(n):\\n    levels = [ [1]*n ]\\n\\n    size = WIDTH\\n    while size < n:\\n        m, r = n // size, n % size\\n        levels.append( [size]*m + ([r] if r > 0 else []) )\\n        size *= WIDTH\\n\\n    return levels\\n\\ndef dec_index(levels, i):\\n    for level in levels:\\n        level[i] -= 1\\n        i //= WIDTH\\n\\ndef find_pos(levels, pos):\\n    i, l = 0, len(levels) - 1\\n    \\n    total = 0\\n    while True:\\n        level = levels[l]\\n        while total + level[i] < pos:\\n            total += level[i]\\n            i += 1\\n        \\n        if l == 0: return i\\n        i *= WIDTH\\n        l -= 1\\n\\n\\nimport sys\\n\\ndef main():\\n    ## INPUT\\n    numbers = [int(x) for x in sys.stdin.read().split()]\\n    n = numbers[0]\\n    sequence = numbers[1:n+1]\\n    m = numbers[n+1]\\n\\n    queries = {}\\n    for i in range(n+2, n+2 + 2*m, 2):\\n        k, pos = numbers[i], numbers[i+1]\\n        if k in queries: queries[k][pos] = None\\n        else: queries[k] = { pos: None }\\n    \\n    ## WORK\\n    sequence1 = sorted([ (s,-i) for i,s in enumerate(sequence) ])\\n    tree = index_tree(n)\\n    size = n\\n\\n    for _, neg_i in sequence1: \\n        if size in queries:\\n            for pos in queries[size]:\\n                queries[size][pos] = find_pos(tree, pos)\\n\\n        dec_index(tree, -neg_i)\\n        size -= 1\\n\\n    ## PRINT \\n    for i in range(n+2, n+2 + 2*m, 2):\\n        k, pos = numbers[i], numbers[i+1]\\n        print(sequence[ queries[k][pos] ])\\n\\n\\nmain()\\n\\n\", \"from collections import defaultdict\\nimport sys as _sys\\n\\n\\ndef main():\\n    n, = _read_ints()\\n    a = tuple(_read_ints())\\n    m, = _read_ints()\\n    queries = (tuple(_read_ints()) for i_query in range(m))\\n    result = process_queries(a, queries)\\n    print(*result, sep='\\\\n')\\n\\n\\ndef _read_line():\\n    result = _sys.stdin.readline()\\n    assert result[-1] == \\\"\\\\n\\\"\\n    return result[:-1]\\n\\n\\ndef _read_ints():\\n    return map(int, _read_line().split())\\n\\n\\ndef process_queries(sequence, queries):\\n    sequence = tuple(sequence)\\n    \\n    indices_by_values = defaultdict(list)\\n    for i, x in enumerate(sequence):\\n        indices_by_values[x].append(i)\\n    \\n    enumerated_queries = sorted(enumerate(queries), key=lambda iv: iv[1][0])[::-1]\\n    queries_responses = [None] * len(enumerated_queries)\\n    \\n    selections_tree = [0] * (len(sequence) + 1)\\n    \\n    k = 0\\n    for value, indices in sorted(indices_by_values.items(), reverse=True):\\n        for index_to_select in indices:\\n            _fenwick_tree_add(selections_tree, index_to_select, 1)\\n            k += 1\\n            while enumerated_queries and enumerated_queries[-1][1][0] == k:\\n                query_index, (_k, subseq_index) = enumerated_queries.pop()\\n                seq_index = _find_seq_index_by_subseq_index(selections_tree, subseq_index)\\n                queries_responses[query_index] = sequence[seq_index]\\n\\n    return queries_responses\\n\\n\\ndef _find_seq_index_by_subseq_index(tree, subseq_i):\\n    min_i = 0\\n    max_i = len(tree) - 1\\n    while min_i != max_i:\\n        mid_i = (min_i + max_i) // 2\\n        if _fenwick_tree_prefix_sum(tree, mid_i) < subseq_i:\\n            min_i = mid_i + 1\\n        else:\\n            max_i = mid_i\\n    return min_i\\n\\n\\ndef _fenwick_tree_prefix_sum(tree, i):\\n    i += 1\\n    result = 0\\n    while i != 0:\\n        result += tree[i]\\n        i -= i & (-i)\\n    return result\\n\\n\\ndef _fenwick_tree_add(tree, i, x):\\n    i += 1\\n    while i < len(tree):\\n        tree[i] += x\\n        i += i & (-i)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys as _sys\\n\\n\\ndef main():\\n    n, = _read_ints()\\n    a = tuple(_read_ints())\\n    m, = _read_ints()\\n    queries = (tuple(_read_ints()) for i_query in range(m))\\n    result = process_queries(a, queries)\\n    print(*result, sep='\\\\n')\\n\\n\\ndef _read_line():\\n    result = _sys.stdin.readline()\\n    assert result[-1] == \\\"\\\\n\\\"\\n    return result[:-1]\\n\\n\\ndef _read_ints():\\n    return map(int, _read_line().split())\\n\\n\\ndef process_queries(sequence, queries):\\n    sequence = tuple(sequence)\\n    \\n    indices_to_select = sorted(\\n        range(len(sequence)),\\n        key=lambda index: (-sequence[index], index)\\n    )\\n    \\n    enumerated_queries = sorted(enumerate(queries), key=lambda iv: iv[1][0])[::-1]\\n    queries_responses = [None] * len(enumerated_queries)\\n    \\n    selections_tree = [0] * (len(sequence) + 1)\\n    \\n    selected_n = 0\\n    for index_to_select in indices_to_select:\\n        _fenwick_tree_add(selections_tree, index_to_select, 1)\\n        selected_n += 1\\n        while enumerated_queries and enumerated_queries[-1][1][0] == selected_n:\\n            query_index, (_k, subseq_index) = enumerated_queries.pop()\\n            seq_index = _find_seq_index_by_subseq_index(selections_tree, subseq_index)\\n            queries_responses[query_index] = sequence[seq_index]\\n    \\n    return queries_responses\\n\\n\\ndef _find_seq_index_by_subseq_index(tree, subseq_i):\\n    seq_length = len(tree) - 1\\n    min_i = 0\\n    max_i = seq_length - 1\\n    while min_i != max_i:\\n        mid_i = (min_i + max_i) // 2\\n        if _fenwick_tree_prefix_sum(tree, mid_i) < subseq_i:\\n            min_i = mid_i + 1\\n        else:\\n            max_i = mid_i\\n    return min_i\\n\\n\\ndef _fenwick_tree_prefix_sum(tree, i):\\n    i += 1\\n    result = 0\\n    while i != 0:\\n        result += tree[i]\\n        i -= i & (-i)\\n    return result\\n\\n\\ndef _fenwick_tree_add(tree, i, x):\\n    i += 1\\n    while i < len(tree):\\n        tree[i] += x\\n        i += i & (-i)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "2\n392222 322\n3\n2 2\n2 1\n1 1\n",
        "output": "322\n392222\n392222\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1227/D2"
    },
    {
        "id": 264,
        "task_id": 2082,
        "test_case_id": 1,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "4 3\n10 20 30 40\n1 3\n2 3\n4 3\n",
        "output": "16.666667\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 265,
        "task_id": 2082,
        "test_case_id": 2,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "3 3\n10 20 30\n1 2\n2 3\n3 1\n",
        "output": "13.333333\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 266,
        "task_id": 2082,
        "test_case_id": 3,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n",
        "output": "18.571429\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 267,
        "task_id": 2082,
        "test_case_id": 4,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "10 14\n594 965 90 327 549 206 514 993 803 635\n1 2\n1 3\n3 4\n2 5\n5 6\n5 7\n4 8\n4 9\n5 10\n10 4\n7 8\n2 6\n6 4\n5 4\n",
        "output": "326.088889\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 268,
        "task_id": 2082,
        "test_case_id": 5,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "10 19\n15704 19758 26631 25050 22778 15041 8487 26418 5136 4199\n1 2\n1 3\n1 4\n2 5\n1 6\n2 7\n2 8\n7 9\n6 10\n7 3\n4 7\n6 4\n6 8\n5 8\n6 9\n5 4\n1 8\n1 9\n5 3\n",
        "output": "11616.755556\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 269,
        "task_id": 2082,
        "test_case_id": 6,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "10 14\n296 371 507 807 102 558 199 500 553 150\n1 2\n2 3\n3 4\n1 5\n5 6\n3 7\n2 8\n5 9\n8 10\n7 2\n8 7\n4 6\n1 7\n5 4\n",
        "output": "213.933333\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 270,
        "task_id": 2082,
        "test_case_id": 7,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "10 19\n13637 26970 19043 3616 12880 19387 12539 25190 2452 1261\n1 2\n1 3\n1 4\n2 5\n3 6\n6 7\n3 8\n5 9\n3 10\n4 10\n9 3\n2 8\n4 3\n2 3\n7 10\n7 8\n5 10\n5 6\n7 4\n",
        "output": "8241.422222\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 271,
        "task_id": 2082,
        "test_case_id": 8,
        "question": "Of course our child likes walking in a zoo. The zoo has n areas, that are numbered from 1 to n. The i-th area contains a_{i} animals in it. Also there are m roads in the zoo, and each road connects two distinct areas. Naturally the zoo is connected, so you can reach any area of the zoo from any other area using the roads.\n\nOur child is very smart. Imagine the child want to go from area p to area q. Firstly he considers all the simple routes from p to q. For each route the child writes down the number, that is equal to the minimum number of animals among the route areas. Let's denote the largest of the written numbers as f(p, q). Finally, the child chooses one of the routes for which he writes down the value f(p, q).\n\nAfter the child has visited the zoo, he thinks about the question: what is the average value of f(p, q) for all pairs p, q (p ≠ q)? Can you answer his question?\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 10^5; 0 ≤ m ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each line contains two integers x_{i} and y_{i} (1 ≤ x_{i}, y_{i} ≤ n; x_{i} ≠ y_{i}), denoting the road between areas x_{i} and y_{i}.\n\nAll roads are bidirectional, each pair of areas is connected by at most one road.\n\n\n-----Output-----\n\nOutput a real number — the value of $\\frac{\\sum_{p, q, p \\neq q} f(p, q)}{n(n - 1)}$.\n\nThe answer will be considered correct if its relative or absolute error doesn't exceed 10^{ - 4}.\n\n\n-----Examples-----\nInput\n4 3\n10 20 30 40\n1 3\n2 3\n4 3\n\nOutput\n16.666667\n\nInput\n3 3\n10 20 30\n1 2\n2 3\n3 1\n\nOutput\n13.333333\n\nInput\n7 8\n40 20 10 30 20 50 40\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n1 4\n5 7\n\nOutput\n18.571429\n\n\n\n-----Note-----\n\nConsider the first sample. There are 12 possible situations:\n\n  p = 1, q = 3, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 4, q = 3, f(p, q) = 30.  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 4, f(p, q) = 20.  p = 4, q = 1, f(p, q) = 10. \n\nAnother 6 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 30 + 10 + 20 + 10) \\times 2}{12} \\approx 16.666667$.\n\nConsider the second sample. There are 6 possible situations:\n\n  p = 1, q = 2, f(p, q) = 10.  p = 2, q = 3, f(p, q) = 20.  p = 1, q = 3, f(p, q) = 10. \n\nAnother 3 cases are symmetrical to the above. The average is $\\frac{(10 + 20 + 10) \\times 2}{6} \\approx 13.333333$.",
        "solutions": "[\"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"n, m = map(int, input().split())\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = map(int, input().split())\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r:\\n        continue\\n    if len(q[l]) > len(q[r]):\\n        l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]:\\n        p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\", \"read = lambda: list(map(int, input().split()))\\nn, m = read()\\nv = list(read())\\n\\ne = []\\nfor i in range(m):\\n    x, y = [int(x) - 1 for x in input().split()]\\n    e.append((x, y, min(v[x], v[y])))\\ne.sort(key = lambda x: x[2], reverse = True)\\n\\nbelong = list(range(n))\\nunion = [[i] for i in belong]\\nsize = [1] * n\\n\\nans = 0\\n\\nfor i, j, k in e:\\n    fi, fj = belong[i], belong[j];\\n    if fi == fj: continue\\n    if not (len(union[fi]) > len(union[fj])):\\n        fi, fj = fj, fi\\n    ans += k * size[fi] * size[fj]\\n    size[fi] += size[fj]\\n    union[fi] += union[fj]\\n    for t in union[fj]: belong[t] = fi\\n    \\nprint(\\\"%.7lf\\\" % (ans * 2 / n / (n - 1)))\\n\", \"def main():\\n    n, m = list(map(int,input().split()))\\n    l = [int(i) for i in input().split()]\\n    rank, ans = [], 0\\n    for i in range(m):\\n        a,b = list(map(int, input().split()))\\n        a,b = a-1,b-1\\n        rank.append((min(l[a],l[b]),a,b))\\n\\n    rank = sorted(rank,key = lambda x: -x[0])\\n    par = list(range(n))\\n    siz = [1]*n\\n    ans = 0.0\\n\\n    def find(a):\\n        root = a\\n        while root != par[root]: root = par[root]\\n        while a != par[a]:\\n            newp = par[a]\\n            par[a] = root\\n            a = newp\\n        return root\\n\\n    def merge(a,b,i):\\n        a = find(a)\\n        b = find(b)\\n        if a == b: return 0\\n        add = siz[a]*siz[b]\\n        siz[a] += siz[b]\\n        par[b] = a\\n        return (add * i[0]) / (n * (n - 1))\\n\\n\\n\\n    for i in rank:\\n        ans += merge(i[1],i[2],i)\\n\\n    print(\\\"%.12f\\\"%(2*ans))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\", \"n, m = list(map(int, input().split()))\\np, c = list(range(n + 1)), [1] * (n + 1)\\nv = [0] + list(map(int, input().split()))\\ns, e = 0, [()] * m\\nfor i in range(m):\\n    x, y = list(map(int, input().split()))\\n    e[i] = (x, y, min(v[x], v[y]))\\ne.sort(key = lambda x: x[2], reverse = True)\\nq = [[i] for i in range(n + 1)]\\nfor l, r, v in e:\\n    l, r = p[l], p[r]\\n    if l == r: continue\\n    if len(q[l]) > len(q[r]): l, r = r, l\\n    q[r].extend(q[l])\\n    for t in q[l]: p[t] = r\\n    s += c[l] * c[r] * v\\n    c[r] += c[l]\\nprint(s * 2 / (n * (n - 1)))\\n\"]",
        "difficulty": "interview",
        "input": "2 1\n233 2333\n1 2\n",
        "output": "233.000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/437/D"
    },
    {
        "id": 272,
        "task_id": 2124,
        "test_case_id": 1,
        "question": "Recently Vladik discovered a new entertainment — coding bots for social networks. He would like to use machine learning in his bots so now he want to prepare some learning data for them.\n\nAt first, he need to download t chats. Vladik coded a script which should have downloaded the chats, however, something went wrong. In particular, some of the messages have no information of their sender. It is known that if a person sends several messages in a row, they all are merged into a single message. It means that there could not be two or more messages in a row with the same sender. Moreover, a sender never mention himself in his messages.\n\nVladik wants to recover senders of all the messages so that each two neighboring messages will have different senders and no sender will mention himself in his messages.\n\nHe has no idea of how to do this, and asks you for help. Help Vladik to recover senders in each of the chats!\n\n\n-----Input-----\n\nThe first line contains single integer t (1 ≤ t ≤ 10) — the number of chats. The t chats follow. Each chat is given in the following format.\n\nThe first line of each chat description contains single integer n (1 ≤ n ≤ 100) — the number of users in the chat.\n\nThe next line contains n space-separated distinct usernames. Each username consists of lowercase and uppercase English letters and digits. The usernames can't start with a digit. Two usernames are different even if they differ only with letters' case. The length of username is positive and doesn't exceed 10 characters.\n\nThe next line contains single integer m (1 ≤ m ≤ 100) — the number of messages in the chat. The next m line contain the messages in the following formats, one per line:   <username>:<text> — the format of a message with known sender. The username should appear in the list of usernames of the chat.  <?>:<text> — the format of a message with unknown sender. \n\nThe text of a message can consist of lowercase and uppercase English letter, digits, characters '.' (dot), ',' (comma), '!' (exclamation mark), '?' (question mark) and ' ' (space). The text doesn't contain trailing spaces. The length of the text is positive and doesn't exceed 100 characters.\n\nWe say that a text mention a user if his username appears in the text as a word. In other words, the username appears in a such a position that the two characters before and after its appearance either do not exist or are not English letters or digits. For example, the text \"Vasya, masha13 and Kate!\" can mention users \"Vasya\", \"masha13\", \"and\" and \"Kate\", but not \"masha\".\n\nIt is guaranteed that in each chat no known sender mention himself in his messages and there are no two neighboring messages with the same known sender.\n\n\n-----Output-----\n\nPrint the information about the t chats in the following format:\n\nIf it is not possible to recover senders, print single line \"Impossible\" for this chat. Otherwise print m messages in the following format:\n\n<username>:<text>\n\nIf there are multiple answers, print any of them.\n\n\n-----Examples-----\nInput\n1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n\nOutput\nnetman: Hello, Vladik!\nVladik: Hi\n\nInput\n1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n\nOutput\nImpossible\n\nInput\n2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n\nOutput\nImpossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.",
        "solutions": "[\"def pr( name , lvl , dp , u , tot ): \\n    if lvl == 0:\\n        print(name + ':' + tot[lvl])\\n        return\\n\\n    pr( u[lvl][name] , lvl - 1 , dp , u , tot )\\n    print(name + ':' + tot[lvl])\\n\\ndef solve(): \\n    n = int(input())\\n    users = input().split()\\n    m = int(input())\\n    dp = [] \\n    u = []\\n    tot = [] \\n    for i in range( m ) : \\n        dp.append( set() ) \\n        u.append( {} ) \\n        line = input().split(':')\\n        sender = line[0]\\n        tot.append( line[1] ) \\n        line[1] = line[1].replace( '?' , ' ' )\\n        line[1] = line[1].replace( '.' , ' ' )\\n        line[1] = line[1].replace( ',' , ' ' )\\n        line[1] = line[1].replace( '!' , ' ' )\\n        mess = line[1].split()\\n\\n        if sender == '?' : \\n            if i != 0:\\n                for name in users:\\n                    for x in dp[i-1]: \\n                        if x != name and name not in mess:\\n                            dp[i].add( name ) \\n                            u[i][name] = x\\n            else : \\n                for name in users: \\n                    if name not in mess:\\n                        dp[i].add( name ) \\n        else: \\n            if i != 0: \\n                for x in dp[i-1]: \\n                    if x != sender: \\n                        dp[i].add( sender ) \\n                        u[i][sender] = x\\n            else: \\n                dp[i].add( sender )\\n        \\n        \\n    if dp[m-1]: \\n        pr( list(dp[m-1])[0] , m-1 , dp , u , tot )\\n    else: \\n        print(\\\"Impossible\\\")\\n\\n\\nt = int(input())\\nfor i in range( t ) : \\n    solve()\\n\", \"#!/usr/bin/env python3\\n\\nimport re\\n\\ndef cont(line, user):\\n\\treturn re.search('[^a-zA-Z0-9]' + user + '[^a-zA-Z0-9]', '_' + line + '_')\\n\\ntests = int(input())\\nfor test in range(tests):\\n\\tinput()\\n\\tusers = input().rstrip('\\\\n').split(' ')\\n\\tlinecnt = int(input())\\n\\tlines = [input().rstrip('\\\\n') for _ in range(linecnt)]\\n\\tlines = [x.split(':', maxsplit=1) for x in lines]\\n\\tposs = [[]] * linecnt\\n\\tfor i, (user, line) in enumerate(lines):\\n\\t\\tif user != '?':\\n\\t\\t\\tposs[i] = [user]\\n\\t\\telse:\\n\\t\\t\\tposs[i] = [u for u in users if not cont(line, u)]\\n\\n\\tel = list(enumerate(lines))\\n\\trel = list(reversed(el))\\n\\n\\tchanged = False\\n\\twhile True:\\n\\t\\tfor i, (user, line) in el:\\n\\t\\t\\tif i > 0 and len(poss[i-1]) == 1:\\n\\t\\t\\t\\tif poss[i-1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i-1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tfor i, (user, line) in rel:\\n\\t\\t\\tif i < linecnt - 1 and len(poss[i+1]) == 1:\\n\\t\\t\\t\\tif poss[i+1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i+1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tif not changed:\\n\\t\\t\\tbreak\\n\\t\\tchanged = False\\n\\n\\tif all(len(p) > 0 for p in poss):\\n\\t\\tfor i, p, (user, line) in zip(list(range(linecnt)), poss, lines):\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\tif poss[i-1][0] in p:\\n\\t\\t\\t\\t\\tp.remove(poss[i-1][0])\\n\\t\\t\\tprint(p[0] + ':' + line)\\n\\telse:\\n\\t\\tprint('Impossible')\\n\", \"import re\\nimport sys\\n\\n\\ndef g(cts, names):\\n  la = dict([(p, None) for p in names.difference(cts[0])])\\n  if len(la) == 0:\\n    return None\\n  d = [la]\\n  for i in range(1, len(cts)):\\n    prev = d[-1]\\n    la = dict()\\n    for p in names.difference(cts[i]):\\n      for n in prev.keys():\\n        if n != p:\\n          la[p] = n\\n          break\\n    if len(la) == 0:\\n      return None\\n    d.append(la)\\n  cur = list(d[-1].keys())[0]\\n  result = []\\n  for i in range(len(cts) - 1, -1, -1):\\n    result.append(cur)\\n    cur = d[i][cur]\\n  result.reverse()\\n  return result\\n\\n\\ndef solve():\\n  n = int(input())\\n  names = input().split()\\n  set_names = set(names)\\n\\n  def get_names(text):\\n    result = []\\n    for p in re.split(\\\"\\\\W+\\\", text):\\n      if p in set_names:\\n        result.append(p)\\n    return result\\n\\n  m = int(input())\\n  messages = []\\n  for i in range(m):\\n    s = input()\\n    colon = s.find(\\\":\\\")\\n    name = s[:colon]\\n    text = s[colon + 1:]\\n    if name == \\\"?\\\":\\n      name = None\\n    messages.append([name, text, get_names(text)])\\n\\n  for i in range(m):\\n    if messages[i][0]:\\n      continue\\n    j = i\\n    cts = []\\n    while j < m and not messages[j][0]:\\n      cts.append(set(messages[j][2]))\\n      j += 1\\n    if i > 0:\\n      cts[0].add(messages[i - 1][0])\\n    if j < m:\\n      cts[-1].add(messages[j][0])\\n    sb = g(cts, set_names)\\n    if not sb:\\n      return None\\n    for k in range(i, j):\\n      messages[k][0] = sb[k - i]\\n\\n  for p in messages:\\n    print(\\\"%s:%s\\\" % (p[0], p[1]))\\n  return True;\\n\\n\\ndef main():\\n  tests = int(input())\\n  for i in range(tests):\\n    if not solve():\\n      print(\\\"Impossible\\\")\\n  return 0\\n\\ndef __starting_point():\\n  return(main())\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"def dfs(ind):\\n    if ind > 0:\\n        if possible[ind][0] in possible[ind - 1]:\\n            possible[ind - 1].remove(possible[ind][0])\\n            if len(possible[ind - 1]) == 1:\\n                dfs(ind - 1)\\n    if ind < m - 1:\\n        if possible[ind][0] in possible[ind + 1]:\\n            possible[ind + 1].remove(possible[ind][0])\\n            if len(possible[ind + 1]) == 1:\\n                dfs(ind + 1)\\ndef Check(st, ms):\\n    for i in range(0, len(ms) - len(st) + 1):\\n        if st == ms[i:i+len(st)]:\\n            t = True\\n            if i > 0:\\n                if ms[i - 1] == ' ' or ms[i - 1] == '.' or ms[i - 1] == ',' or ms[i - 1] == '?' or ms[i - 1] == '!':\\n                    e = 0\\n                else:\\n                    t = False\\n            if i < len(ms) - len(st):\\n                if (ms[i + len(st)] == ' ' or ms[i + len(st)] == '.' or ms[i + len(st)] == ',' or ms[i + len(st)] == '?' or ms[i + len(st)] == '!'):\\n                    e = 0\\n                else:\\n                    t = False\\n            if t:\\n                return True\\n    return False\\nR = lambda:list(map(int, input().split(' ')))\\n#r, w = open(\\\"input.txt\\\", \\\"r\\\"), open(\\\"output.txt\\\", \\\"w\\\")\\nT = int(input())\\nwhile T:\\n    n = int(input())\\n    users = input().split(' ')\\n    m = int(input())\\n    possible = [[] for i in range(m)]\\n    sender, message = [], []\\n    for i in range(m):\\n        s = input().split(':')\\n        sender.append(s[0])\\n        message.append(s[1])\\n    for i in range(m):\\n        if sender[i] == '?':\\n            unallow = \\\" \\\"\\n            if i > 0 and len(possible[i - 1]) == 1:\\n                unallow = possible[i - 1][0]\\n            for j in users:\\n                if j == unallow:\\n                    continue\\n                done = Check(j, message[i])\\n                if not done:\\n                    possible[i].append(j)\\n        else:\\n            possible[i].append(sender[i])\\n    used = [0 for i in range(m)]\\n    for i in range(m):\\n        if len(possible[i]) == 1 and used[i] == 0:\\n            dfs(i)\\n    for i in range(m):\\n        if len(possible[i]) > 1:\\n            possible[i] = [possible[i][0]]\\n            dfs(i)\\n    done = False\\n    for i in possible:\\n        if len(i) == 0:\\n            print(\\\"Impossible\\\")\\n            done = True\\n            break\\n    if not done:\\n        for i in range(m):\\n            print(possible[i][0]+':'+message[i])\\n    T -= 1\\n\\n\\n    \\n\\n\", \"# Bartek Kostka\\n#  You are not prepared!\\n\\nimport re\\n\\n\\ndef go(i):\\n    nonlocal counter\\n    counter += 1\\n    if counter > 2000:\\n        return False\\n    if i == len(E):\\n        return True\\n    if E[i][0] != \\\"?\\\":\\n        prop = E[i][0]\\n        if i == 0 or W[i-1] != prop:\\n            W[i] = prop\\n            return go(i+1)\\n        else:\\n            return False\\n    for pos in E[i][1]:\\n        if i == 0 or W[i-1] != pos:\\n            W[i] = pos\\n            if go(i+1):\\n                return True\\n    return False\\n\\n\\nt = int(input())\\nfor ttt in range(t):\\n    n = int(input())\\n    users = input().split(\\\" \\\")\\n    users_set = set(users)\\n    m = int(input())\\n    E = []\\n    S = []\\n    counter = 0\\n    W = [\\\"\\\" for x in range(m)]\\n    for mmm in range(m):\\n        line = input()\\n        (user, sentence) = line.split(\\\":\\\")\\n        S.append((user, sentence))\\n        words = [x.strip() for x in re.split('\\\\W+', sentence)]\\n        mentions = list([x for x in words if x in users_set])\\n        E.append([user, set(mentions)])\\n    for i in range(len(E)-1):\\n        if E[i][0] != \\\"?\\\":\\n            E[i+1][1].add(E[i][0])\\n    for i in range(1, len(E)):\\n        if E[i][0] != \\\"?\\\":\\n            E[i-1][1].add(E[i][0])\\n    for i in range(len(E)):\\n        E[i][1] = E[i][1].symmetric_difference(users_set)\\n    if go(0):\\n        for i in range(len(E)):\\n            print(str(W[i])+\\\":\\\"+str(S[i][1]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import re\\nimport copy\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\tn = int(input())\\n\\tsenderList = []\\n\\tfor x in input().split():\\n\\t\\tsenderList.append(x)\\n\\n\\tm = int(input())\\n\\tmsg = [None] * m\\n\\tresSenders = [None] * m\\n\\tpossibleSenders = []\\n\\tfor i in range(m):\\n\\t\\tpossibleSenders.append(copy.copy(senderList))\\n\\n\\tfor i in range(m):\\n\\t\\tline = input()\\n\\t\\tblocks = re.findall(r\\\"[\\\\w]+\\\", line)\\n\\n\\t\\tfor x in blocks:\\n\\t\\t\\ttry:\\n\\t\\t\\t\\tpossibleSenders[i].remove(x)\\n\\t\\t\\texcept:\\n\\t\\t\\t\\tpass\\n\\n\\t\\tif line[0] != '?':\\n\\t\\t\\tresSenders[i] = blocks[0]\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tmsg[i] = line[line.find(\\\":\\\")+1:]\\n\\n\\tresolved = False\\n\\twhile True:\\n\\t\\tdone = True\\n\\t\\tjustPick = True\\n\\t\\tfor i in range(m):\\n\\t\\t\\tif resSenders[i] != None:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tif len(possibleSenders[i]) == 0:\\n\\t\\t\\t\\tdone = True\\n\\t\\t\\t\\tresolved = True\\n\\t\\t\\t\\tprint(\\\"Impossible\\\")\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tif len(possibleSenders[i]) == 1:\\n\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\tif i > 0:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tjustPick = False\\n\\t\\t\\telse:\\n\\t\\t\\t\\tdone = False\\n\\t\\tif done:\\n\\t\\t\\tbreak\\n\\t\\tif justPick:\\n\\t\\t\\tfor i in range(m):\\n\\t\\t\\t\\tif resSenders[i] == None:\\n\\t\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\tbreak\\n\\n\\tif not resolved:\\n\\t\\tfor i in range(m):\\n\\t\\t\\tprint(resSenders[i] + \\\":\\\" + msg[i])\", \"import re\\n\\n\\ndef foo():\\n    n = int(input())\\n    who = input().split()\\n    m = int(input())\\n    msg = []\\n\\n    l = []\\n\\n    for num in range(m):\\n        a, t = input().split(':')\\n        msg.append(t)\\n\\n        st = set()\\n\\n        if a != '?':\\n            st.add(a)\\n        else:\\n\\n            names = re.split('\\\\W+', t)\\n\\n            for w in who:\\n                if not w in names:\\n                    st.add(w)\\n\\n        l.append(st)\\n\\n    d2 = []\\n\\n    for num in range(1, m):\\n        d = {}\\n        for w1 in l[num]:\\n            for w2 in l[num - 1]:\\n                if w1 != w2:\\n                    d[w1] = w2\\n                    break\\n\\n        l[num] = [x for x in d]\\n\\n        d2.append(d)\\n\\n    curr = None\\n\\n    for w in l[m - 1]:\\n        curr = w\\n\\n    res = [curr]\\n\\n    for num in list(reversed(list(range(1, m)))):\\n        curr = d2[num - 1].get(curr, None)  # d2[num - 1][curr]\\n        res.append(curr)\\n\\n    res = list(reversed(res))\\n\\n    if None in res:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for num in range(m):\\n            print(res[num] + ':' + msg[num])\\n\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    foo()\\n\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\", \"import re\\n\\nt = int(input())\\n\\ndelimiters = \\\"?\\\", \\\".\\\", \\\" \\\", \\\",\\\", \\\"!\\\", \\\":\\\"\\nregexPattern = '|'.join(map(re.escape, delimiters))\\n\\nfor i in range(t):\\n  n = int(input())\\n  usernames = {x for x in str.split(input(), ' ')}\\n  m = int(input())\\n  possibilities = []\\n  for j in range(m):\\n    possibilities.append({x for x in usernames})\\n  messages = []\\n  for j in range(m):\\n    messages.append(input())\\n\\n  for j in range(m):\\n    if (messages[j][0] != '?'):\\n      messageSplit = re.split(':', messages[j])\\n      possibilities[j] = {messageSplit[0]}\\n    else:\\n      messageSplit = re.split(regexPattern, messages[j])\\n      for token in messageSplit:\\n        if token in usernames:\\n          possibilities[j] = possibilities[j] - {token}\\n  changed = True\\n  while changed:\\n    changed = False\\n    for j in range(m):\\n      if len(possibilities[j]) == 1:\\n        (poss,) = possibilities[j]\\n        if j < m-1 and poss in possibilities[j+1]:\\n          changed = True\\n          possibilities[j+1] = possibilities[j+1] - possibilities[j]\\n        if j > 0 and poss in possibilities[j-1]:\\n          changed = True\\n          possibilities[j-1] = possibilities[j-1] - possibilities[j]\\n  worked = True\\n  for j in range(m):\\n    if len(possibilities[j]) == 0:\\n      worked = False\\n      \\n  if not worked:\\n    print(\\\"Impossible\\\")\\n  else :\\n    for j in range(m):\\n      poss = next(iter(possibilities[j]))\\n      if (messages[j][0] == '?'):\\n        print(poss + messages[j][1:])\\n      else:\\n        print(messages[j])\\n      if (j < m-1):\\n        possibilities[j+1] = possibilities[j+1] - {poss}\\n\\n\", \"import re\\n\\ndef add_used(id, s, used):\\n    used[id-1].add(s)\\n    used[id].add(s)\\n    used[id+1].add(s)\\n\\ntcase = int(input())\\nfor cas in range(tcase):\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    \\n    res = True\\n    ans = [''] * (m+1)\\n    msg = [''] * (m+1)\\n    used = [set() for i in range(m+1)]\\n    \\n    for i in range(m):\\n        ans[i], msg[i] = input().split(':')\\n        if ans[i] != '?':\\n            add_used(i, ans[i], used)\\n        \\n        mentioned = re.split('\\\\W+', msg[i])\\n        for s in mentioned:\\n            if s in names:\\n                used[i].add(s)\\n    \\n    for i in range(m):\\n        if ans[i] == '?' and len(used[i]) == n-1:\\n            ans[i] = list(set(names) - used[i])[0]\\n            add_used(i, ans[i], used)\\n    \\n    for i in range(m):\\n        if ans[i] == '?':\\n            if len(used[i]) == n:\\n                res = False\\n            else:\\n                ans[i] = list(set(names) - used[i])[0]\\n                add_used(i, ans[i], used)\\n    \\n    for i in range(m-1):\\n        if ans[i] == ans[i+1]:\\n            res = False\\n    \\n    if res:\\n        for i in range(m):\\n            print(ans[i] + ':' + msg[i])\\n    else:\\n        print('Impossible')\", \"from sys import stdin\\n\\nt = int(stdin.readline())\\n\\ndef c(l,r,msg):\\n    f = True\\n    if l > 0 and not msg[l-1].isdigit() and not msg[l-1].isalpha():\\n        pass\\n    elif l == 0:\\n        pass\\n    else:\\n        f = False\\n\\n    if r == len(msg) - 1:\\n        pass\\n    elif not msg[r+1].isdigit() and not msg[r+1].isalpha():\\n        pass\\n    else:\\n        f = False\\n\\n    return f\\n\\n\\ndef dd(i):\\n    if i - 1 in no_author_set:\\n        authors[i-1].discard(authors[i])\\n        if len(authors[i-1]) == 1:\\n            no_author_set.discard(i-1)\\n            for d in authors[i-1]:\\n                authors[i-1] = d\\n                break\\n            dd(i-1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    if i + 1 in no_author_set:\\n        authors[i+1].discard(authors[i])\\n        if len(authors[i+1]) == 1:\\n            no_author_set.discard(i+1)\\n            for d in authors[i+1]:\\n                authors[i+1] = d\\n                break\\n            dd(i+1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    return True\\n\\ntry:\\n    answers = []\\n    for i in range(t):\\n        ans = True\\n        n = int(stdin.readline())\\n        names = set(stdin.readline().strip().split())\\n        m = int(stdin.readline())\\n        authors = list()\\n        messages = list()\\n        no_author = list()\\n        no_author_set = set()\\n        for j in range(m):\\n            line = stdin.readline().strip()\\n            author, msg = line.split(':')\\n            messages.append(msg)\\n            if author == '?':\\n                no_author.append(j)\\n                no_author_set.add(j)\\n                a_set = set()\\n                for name in names:\\n                    l = msg.find(name)\\n                    while l != -1:\\n                        if c(l,l+len(name)-1, msg):\\n                            a_set.add(name)\\n                        l = msg.find(name,l+len(name))\\n                authors.append(names-a_set)\\n                if j-1 not in no_author_set:\\n                    authors[j].discard(authors[j-1])\\n            else:\\n                authors.append(author)\\n                if j - 1 in no_author_set:\\n                    authors[j-1].discard(author)\\n\\n        for j in no_author:\\n            if j in no_author_set:\\n                if len(authors[j]) == 1:\\n                    no_author_set.discard(j)\\n                    for d in authors[j]:\\n                        authors[j] = d\\n                        break\\n                    if not dd(j):\\n                        ans = False\\n                elif len(authors[j]) == 0:\\n                    ans = False\\n\\n        no_author = list()\\n        for j in no_author_set:\\n            no_author.append(j)\\n        no_author.sort()\\n        for i in no_author:\\n            for d in authors[i]:\\n                authors[i] = d\\n                break\\n            if i + 1 in no_author_set:\\n                authors[i+1].discard(authors[i])\\n        if not ans:\\n            answers.append('Impossible')\\n        else:\\n            for j in range(m):\\n                answers.append('%s:%s'%(authors[j],messages[j]))\\nexcept Exception as e:\\n    print(e)\\n\\nfor m in answers:\\n    print(m)\", \"import re\\nt = int(input())\\nfor _ in range(t):\\n    n = int(input())\\n    names = set(input().split())\\n    m = int(input())\\n    dp = []\\n    a = []\\n    for i in range(m): a.append(input())\\n    for i in range(m):\\n        sender, msg = a[i].split(':')\\n        ls = set([_f for _f in re.split('\\\\W+',msg) if _f])\\n        dp.append((names if sender == '?' else set([sender])) - ls)\\n        if i and len(dp[i-1]) == 1: dp[i] -= dp[i-1]\\n        # print(dp[i]);\\n    if any([len(p) == 0 for p in dp]): print(\\\"Impossible\\\")\\n    else:\\n        res = []\\n        for i in reversed(list(range(m))):\\n            res.append(dp[i].pop())\\n            if i > 0: dp[i-1].discard(res[-1])\\n        for i in range(m): \\n            sender, msg = a[i].split(':')\\n            sender = res[m-i-1]\\n            print(sender+':'+msg)\\n\", \"import re\\n\\nt = int( input() )\\nfor z in range(t):\\n\\tn = int( input() )\\n\\tusers = input().split();\\n\\t#print(users)\\n\\tm = int( input() )\\n\\tcan = [ [] for i in range(m) ]\\n\\tok = True\\n\\tL = []\\n\\tfor i in range(m):\\n\\t\\tl = input().split(':')\\n\\t\\t#print(l)\\n\\t\\tL += [l]\\n\\t\\tif l[0] != '?':\\n\\t\\t\\tcan[i] = [users.index(l[0])]\\n\\t\\telse:\\n\\t\\t\\tcan[i] = [x for x in range(n)]\\n\\t\\tll = re.sub(r'[^A-Za-z0-9]', ' ', l[1]).split()\\n\\t\\tfor j in ll:\\n\\t\\t\\t#print(j)\\n\\t\\t\\tif j in users:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tcan[i].remove( users.index(j) )\\n\\t\\t\\t\\texcept ValueError:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tif len(can[i]) == 0:\\n\\t\\t\\tok = False\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\twhile 1:\\n\\t\\tflag = True\\n\\t\\tfor i in range(m - 1):\\n\\t\\t\\tif (len(can[i]) == 0): ok = False\\n\\t\\t\\tif (len(can[i]) == 1 and can[i][0] in can[i+1]):\\n\\t\\t\\t\\tcan[i+1].remove( can[i][0] )\\n\\t\\t\\t\\tL[i][0] = users[ can[i][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\t\\tif (len(can[i + 1]) == 1 and can[i + 1][0] in can[i]):\\n\\t\\t\\t\\tcan[i].remove( can[i + 1][0] )\\n\\t\\t\\t\\tL[i + 1][0] = users[ can[i + 1][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\tif len(can[m - 1]) == 0: ok = False\\n\\t\\tif ok == False: break\\n\\t\\tif flag: break\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\tfor i in range(m):\\n\\t\\tif L[i][0] == '?':\\n\\t\\t\\tprint(users[ can[i][0] ], end='')\\n\\t\\t\\tif i < m - 1 and can[i][0] in can[i + 1]:\\n\\t\\t\\t\\tcan[i + 1].remove( can[i][0] )\\n\\t\\telse:\\n\\t\\t\\tprint(L[i][0], end='')\\n\\t\\tprint(':', L[i][1], sep = '');\\n\", \"3\\n\\nimport re\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    n = int(input())\\n    users = set(input().split(' '))\\n\\n    m = int(input())\\n    messages = []\\n\\n    guessed = [None for i in range(m)]\\n    denied = [set() for i in range(m)]\\n\\n    for i in range(m):\\n        mg = input()\\n        if not mg.startswith('?:'):\\n            guessed[i] = mg[:mg.find(':')]\\n\\n        mg = mg[mg.find(':'):]\\n        messages.append(mg)\\n\\n        m2 = mg + ' '\\n        for u in users:\\n            if re.search('[^a-zA-Z0-9]' + u + '[^a-zA-Z0-9]', m2):\\n                denied[i].add(u)\\n\\n    answer = True\\n\\n    for i in range(m):\\n        if guessed[i]:\\n            if i > 0: denied[i-1].add(guessed[i])\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n    for i in range(m):\\n        if guessed[i] in denied[i]:\\n            answer = False\\n\\n    #print(guessed)\\n    #print(denied)\\n\\n    changed = True\\n    while changed and answer:\\n        changed = False\\n        for i in range(m):\\n            if not guessed[i]:\\n                if len(users) - len(denied[i]) == 1:\\n                    changed = True\\n                    guessed[i] = (users - denied[i]).pop()\\n                    if i > 0: denied[i-1].add(guessed[i])\\n                    if i < m-1: denied[i+1].add(guessed[i])\\n                if len(users) == len(denied[i]):\\n                    answer = False\\n                    break\\n\\n    for i in range(m):\\n        if not guessed[i] and len(users) - len(denied[i]) >= 1:\\n            guessed[i] = (users - denied[i]).pop()\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n\\n    for i in guessed:\\n        if not i:\\n            answer = False\\n\\n    if not answer:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for i in range(m):\\n            print(guessed[i] + messages[i])\\n    \\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  t = int(input())\\n  for t in range(t):\\n    # input\\n    n = int(input())\\n    users = set(input().split())\\n    m = int(input())\\n    msg = []\\n    for i in range(m):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        alts = users - {x for x in re.split(r'[^A-Za-z0-9]+',text)}\\n      msg.append(dict(user=user, text=text, users=alts))\\n    # remove before and after\\n    for i in range(m-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i+1]['users'].difference_update(msg[i]['users'])\\n    for i in range(m-1,0,-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i-1]['users'].difference_update(msg[i]['users'])\\n    # compute answer\\n    last = ''\\n    impo = False\\n    for i in range(m):\\n      msg[i]['users'].discard(last)\\n      if len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n      last = next(iter(msg[i]['users']))\\n      msg[i]['user'] = last\\n    if impo:\\n      print('Impossible')\\n      continue\\n    for i in range(m):\\n      print(msg[i]['user']+':'+msg[i]['text'])\\n    '''\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(1,n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]'''\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    impo = False\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?': user = users.index(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split(r'[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n      if msg[i]['user'] == '?' and len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n    if impo:\\n      print('Impossible')\\n      continue\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      u = msg[i]['user']\\n      for j in range(n+1):\\n        if u != '?':\\n          if u != j and dp[i+1][u]: dp[i][j] = u\\n        else:\\n          for alt in msg[i]['users']:\\n            if alt != j and dp[i+1][alt]:\\n              dp[i][j] = alt\\n              break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        user = users.index(user)\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split('[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\"]",
        "difficulty": "interview",
        "input": "1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n",
        "output": "netman: Hello, Vladik!\nVladik: Hi\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/754/C"
    },
    {
        "id": 273,
        "task_id": 2124,
        "test_case_id": 2,
        "question": "Recently Vladik discovered a new entertainment — coding bots for social networks. He would like to use machine learning in his bots so now he want to prepare some learning data for them.\n\nAt first, he need to download t chats. Vladik coded a script which should have downloaded the chats, however, something went wrong. In particular, some of the messages have no information of their sender. It is known that if a person sends several messages in a row, they all are merged into a single message. It means that there could not be two or more messages in a row with the same sender. Moreover, a sender never mention himself in his messages.\n\nVladik wants to recover senders of all the messages so that each two neighboring messages will have different senders and no sender will mention himself in his messages.\n\nHe has no idea of how to do this, and asks you for help. Help Vladik to recover senders in each of the chats!\n\n\n-----Input-----\n\nThe first line contains single integer t (1 ≤ t ≤ 10) — the number of chats. The t chats follow. Each chat is given in the following format.\n\nThe first line of each chat description contains single integer n (1 ≤ n ≤ 100) — the number of users in the chat.\n\nThe next line contains n space-separated distinct usernames. Each username consists of lowercase and uppercase English letters and digits. The usernames can't start with a digit. Two usernames are different even if they differ only with letters' case. The length of username is positive and doesn't exceed 10 characters.\n\nThe next line contains single integer m (1 ≤ m ≤ 100) — the number of messages in the chat. The next m line contain the messages in the following formats, one per line:   <username>:<text> — the format of a message with known sender. The username should appear in the list of usernames of the chat.  <?>:<text> — the format of a message with unknown sender. \n\nThe text of a message can consist of lowercase and uppercase English letter, digits, characters '.' (dot), ',' (comma), '!' (exclamation mark), '?' (question mark) and ' ' (space). The text doesn't contain trailing spaces. The length of the text is positive and doesn't exceed 100 characters.\n\nWe say that a text mention a user if his username appears in the text as a word. In other words, the username appears in a such a position that the two characters before and after its appearance either do not exist or are not English letters or digits. For example, the text \"Vasya, masha13 and Kate!\" can mention users \"Vasya\", \"masha13\", \"and\" and \"Kate\", but not \"masha\".\n\nIt is guaranteed that in each chat no known sender mention himself in his messages and there are no two neighboring messages with the same known sender.\n\n\n-----Output-----\n\nPrint the information about the t chats in the following format:\n\nIf it is not possible to recover senders, print single line \"Impossible\" for this chat. Otherwise print m messages in the following format:\n\n<username>:<text>\n\nIf there are multiple answers, print any of them.\n\n\n-----Examples-----\nInput\n1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n\nOutput\nnetman: Hello, Vladik!\nVladik: Hi\n\nInput\n1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n\nOutput\nImpossible\n\nInput\n2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n\nOutput\nImpossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.",
        "solutions": "[\"def pr( name , lvl , dp , u , tot ): \\n    if lvl == 0:\\n        print(name + ':' + tot[lvl])\\n        return\\n\\n    pr( u[lvl][name] , lvl - 1 , dp , u , tot )\\n    print(name + ':' + tot[lvl])\\n\\ndef solve(): \\n    n = int(input())\\n    users = input().split()\\n    m = int(input())\\n    dp = [] \\n    u = []\\n    tot = [] \\n    for i in range( m ) : \\n        dp.append( set() ) \\n        u.append( {} ) \\n        line = input().split(':')\\n        sender = line[0]\\n        tot.append( line[1] ) \\n        line[1] = line[1].replace( '?' , ' ' )\\n        line[1] = line[1].replace( '.' , ' ' )\\n        line[1] = line[1].replace( ',' , ' ' )\\n        line[1] = line[1].replace( '!' , ' ' )\\n        mess = line[1].split()\\n\\n        if sender == '?' : \\n            if i != 0:\\n                for name in users:\\n                    for x in dp[i-1]: \\n                        if x != name and name not in mess:\\n                            dp[i].add( name ) \\n                            u[i][name] = x\\n            else : \\n                for name in users: \\n                    if name not in mess:\\n                        dp[i].add( name ) \\n        else: \\n            if i != 0: \\n                for x in dp[i-1]: \\n                    if x != sender: \\n                        dp[i].add( sender ) \\n                        u[i][sender] = x\\n            else: \\n                dp[i].add( sender )\\n        \\n        \\n    if dp[m-1]: \\n        pr( list(dp[m-1])[0] , m-1 , dp , u , tot )\\n    else: \\n        print(\\\"Impossible\\\")\\n\\n\\nt = int(input())\\nfor i in range( t ) : \\n    solve()\\n\", \"#!/usr/bin/env python3\\n\\nimport re\\n\\ndef cont(line, user):\\n\\treturn re.search('[^a-zA-Z0-9]' + user + '[^a-zA-Z0-9]', '_' + line + '_')\\n\\ntests = int(input())\\nfor test in range(tests):\\n\\tinput()\\n\\tusers = input().rstrip('\\\\n').split(' ')\\n\\tlinecnt = int(input())\\n\\tlines = [input().rstrip('\\\\n') for _ in range(linecnt)]\\n\\tlines = [x.split(':', maxsplit=1) for x in lines]\\n\\tposs = [[]] * linecnt\\n\\tfor i, (user, line) in enumerate(lines):\\n\\t\\tif user != '?':\\n\\t\\t\\tposs[i] = [user]\\n\\t\\telse:\\n\\t\\t\\tposs[i] = [u for u in users if not cont(line, u)]\\n\\n\\tel = list(enumerate(lines))\\n\\trel = list(reversed(el))\\n\\n\\tchanged = False\\n\\twhile True:\\n\\t\\tfor i, (user, line) in el:\\n\\t\\t\\tif i > 0 and len(poss[i-1]) == 1:\\n\\t\\t\\t\\tif poss[i-1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i-1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tfor i, (user, line) in rel:\\n\\t\\t\\tif i < linecnt - 1 and len(poss[i+1]) == 1:\\n\\t\\t\\t\\tif poss[i+1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i+1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tif not changed:\\n\\t\\t\\tbreak\\n\\t\\tchanged = False\\n\\n\\tif all(len(p) > 0 for p in poss):\\n\\t\\tfor i, p, (user, line) in zip(list(range(linecnt)), poss, lines):\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\tif poss[i-1][0] in p:\\n\\t\\t\\t\\t\\tp.remove(poss[i-1][0])\\n\\t\\t\\tprint(p[0] + ':' + line)\\n\\telse:\\n\\t\\tprint('Impossible')\\n\", \"import re\\nimport sys\\n\\n\\ndef g(cts, names):\\n  la = dict([(p, None) for p in names.difference(cts[0])])\\n  if len(la) == 0:\\n    return None\\n  d = [la]\\n  for i in range(1, len(cts)):\\n    prev = d[-1]\\n    la = dict()\\n    for p in names.difference(cts[i]):\\n      for n in prev.keys():\\n        if n != p:\\n          la[p] = n\\n          break\\n    if len(la) == 0:\\n      return None\\n    d.append(la)\\n  cur = list(d[-1].keys())[0]\\n  result = []\\n  for i in range(len(cts) - 1, -1, -1):\\n    result.append(cur)\\n    cur = d[i][cur]\\n  result.reverse()\\n  return result\\n\\n\\ndef solve():\\n  n = int(input())\\n  names = input().split()\\n  set_names = set(names)\\n\\n  def get_names(text):\\n    result = []\\n    for p in re.split(\\\"\\\\W+\\\", text):\\n      if p in set_names:\\n        result.append(p)\\n    return result\\n\\n  m = int(input())\\n  messages = []\\n  for i in range(m):\\n    s = input()\\n    colon = s.find(\\\":\\\")\\n    name = s[:colon]\\n    text = s[colon + 1:]\\n    if name == \\\"?\\\":\\n      name = None\\n    messages.append([name, text, get_names(text)])\\n\\n  for i in range(m):\\n    if messages[i][0]:\\n      continue\\n    j = i\\n    cts = []\\n    while j < m and not messages[j][0]:\\n      cts.append(set(messages[j][2]))\\n      j += 1\\n    if i > 0:\\n      cts[0].add(messages[i - 1][0])\\n    if j < m:\\n      cts[-1].add(messages[j][0])\\n    sb = g(cts, set_names)\\n    if not sb:\\n      return None\\n    for k in range(i, j):\\n      messages[k][0] = sb[k - i]\\n\\n  for p in messages:\\n    print(\\\"%s:%s\\\" % (p[0], p[1]))\\n  return True;\\n\\n\\ndef main():\\n  tests = int(input())\\n  for i in range(tests):\\n    if not solve():\\n      print(\\\"Impossible\\\")\\n  return 0\\n\\ndef __starting_point():\\n  return(main())\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"def dfs(ind):\\n    if ind > 0:\\n        if possible[ind][0] in possible[ind - 1]:\\n            possible[ind - 1].remove(possible[ind][0])\\n            if len(possible[ind - 1]) == 1:\\n                dfs(ind - 1)\\n    if ind < m - 1:\\n        if possible[ind][0] in possible[ind + 1]:\\n            possible[ind + 1].remove(possible[ind][0])\\n            if len(possible[ind + 1]) == 1:\\n                dfs(ind + 1)\\ndef Check(st, ms):\\n    for i in range(0, len(ms) - len(st) + 1):\\n        if st == ms[i:i+len(st)]:\\n            t = True\\n            if i > 0:\\n                if ms[i - 1] == ' ' or ms[i - 1] == '.' or ms[i - 1] == ',' or ms[i - 1] == '?' or ms[i - 1] == '!':\\n                    e = 0\\n                else:\\n                    t = False\\n            if i < len(ms) - len(st):\\n                if (ms[i + len(st)] == ' ' or ms[i + len(st)] == '.' or ms[i + len(st)] == ',' or ms[i + len(st)] == '?' or ms[i + len(st)] == '!'):\\n                    e = 0\\n                else:\\n                    t = False\\n            if t:\\n                return True\\n    return False\\nR = lambda:list(map(int, input().split(' ')))\\n#r, w = open(\\\"input.txt\\\", \\\"r\\\"), open(\\\"output.txt\\\", \\\"w\\\")\\nT = int(input())\\nwhile T:\\n    n = int(input())\\n    users = input().split(' ')\\n    m = int(input())\\n    possible = [[] for i in range(m)]\\n    sender, message = [], []\\n    for i in range(m):\\n        s = input().split(':')\\n        sender.append(s[0])\\n        message.append(s[1])\\n    for i in range(m):\\n        if sender[i] == '?':\\n            unallow = \\\" \\\"\\n            if i > 0 and len(possible[i - 1]) == 1:\\n                unallow = possible[i - 1][0]\\n            for j in users:\\n                if j == unallow:\\n                    continue\\n                done = Check(j, message[i])\\n                if not done:\\n                    possible[i].append(j)\\n        else:\\n            possible[i].append(sender[i])\\n    used = [0 for i in range(m)]\\n    for i in range(m):\\n        if len(possible[i]) == 1 and used[i] == 0:\\n            dfs(i)\\n    for i in range(m):\\n        if len(possible[i]) > 1:\\n            possible[i] = [possible[i][0]]\\n            dfs(i)\\n    done = False\\n    for i in possible:\\n        if len(i) == 0:\\n            print(\\\"Impossible\\\")\\n            done = True\\n            break\\n    if not done:\\n        for i in range(m):\\n            print(possible[i][0]+':'+message[i])\\n    T -= 1\\n\\n\\n    \\n\\n\", \"# Bartek Kostka\\n#  You are not prepared!\\n\\nimport re\\n\\n\\ndef go(i):\\n    nonlocal counter\\n    counter += 1\\n    if counter > 2000:\\n        return False\\n    if i == len(E):\\n        return True\\n    if E[i][0] != \\\"?\\\":\\n        prop = E[i][0]\\n        if i == 0 or W[i-1] != prop:\\n            W[i] = prop\\n            return go(i+1)\\n        else:\\n            return False\\n    for pos in E[i][1]:\\n        if i == 0 or W[i-1] != pos:\\n            W[i] = pos\\n            if go(i+1):\\n                return True\\n    return False\\n\\n\\nt = int(input())\\nfor ttt in range(t):\\n    n = int(input())\\n    users = input().split(\\\" \\\")\\n    users_set = set(users)\\n    m = int(input())\\n    E = []\\n    S = []\\n    counter = 0\\n    W = [\\\"\\\" for x in range(m)]\\n    for mmm in range(m):\\n        line = input()\\n        (user, sentence) = line.split(\\\":\\\")\\n        S.append((user, sentence))\\n        words = [x.strip() for x in re.split('\\\\W+', sentence)]\\n        mentions = list([x for x in words if x in users_set])\\n        E.append([user, set(mentions)])\\n    for i in range(len(E)-1):\\n        if E[i][0] != \\\"?\\\":\\n            E[i+1][1].add(E[i][0])\\n    for i in range(1, len(E)):\\n        if E[i][0] != \\\"?\\\":\\n            E[i-1][1].add(E[i][0])\\n    for i in range(len(E)):\\n        E[i][1] = E[i][1].symmetric_difference(users_set)\\n    if go(0):\\n        for i in range(len(E)):\\n            print(str(W[i])+\\\":\\\"+str(S[i][1]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import re\\nimport copy\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\tn = int(input())\\n\\tsenderList = []\\n\\tfor x in input().split():\\n\\t\\tsenderList.append(x)\\n\\n\\tm = int(input())\\n\\tmsg = [None] * m\\n\\tresSenders = [None] * m\\n\\tpossibleSenders = []\\n\\tfor i in range(m):\\n\\t\\tpossibleSenders.append(copy.copy(senderList))\\n\\n\\tfor i in range(m):\\n\\t\\tline = input()\\n\\t\\tblocks = re.findall(r\\\"[\\\\w]+\\\", line)\\n\\n\\t\\tfor x in blocks:\\n\\t\\t\\ttry:\\n\\t\\t\\t\\tpossibleSenders[i].remove(x)\\n\\t\\t\\texcept:\\n\\t\\t\\t\\tpass\\n\\n\\t\\tif line[0] != '?':\\n\\t\\t\\tresSenders[i] = blocks[0]\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tmsg[i] = line[line.find(\\\":\\\")+1:]\\n\\n\\tresolved = False\\n\\twhile True:\\n\\t\\tdone = True\\n\\t\\tjustPick = True\\n\\t\\tfor i in range(m):\\n\\t\\t\\tif resSenders[i] != None:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tif len(possibleSenders[i]) == 0:\\n\\t\\t\\t\\tdone = True\\n\\t\\t\\t\\tresolved = True\\n\\t\\t\\t\\tprint(\\\"Impossible\\\")\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tif len(possibleSenders[i]) == 1:\\n\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\tif i > 0:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tjustPick = False\\n\\t\\t\\telse:\\n\\t\\t\\t\\tdone = False\\n\\t\\tif done:\\n\\t\\t\\tbreak\\n\\t\\tif justPick:\\n\\t\\t\\tfor i in range(m):\\n\\t\\t\\t\\tif resSenders[i] == None:\\n\\t\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\tbreak\\n\\n\\tif not resolved:\\n\\t\\tfor i in range(m):\\n\\t\\t\\tprint(resSenders[i] + \\\":\\\" + msg[i])\", \"import re\\n\\n\\ndef foo():\\n    n = int(input())\\n    who = input().split()\\n    m = int(input())\\n    msg = []\\n\\n    l = []\\n\\n    for num in range(m):\\n        a, t = input().split(':')\\n        msg.append(t)\\n\\n        st = set()\\n\\n        if a != '?':\\n            st.add(a)\\n        else:\\n\\n            names = re.split('\\\\W+', t)\\n\\n            for w in who:\\n                if not w in names:\\n                    st.add(w)\\n\\n        l.append(st)\\n\\n    d2 = []\\n\\n    for num in range(1, m):\\n        d = {}\\n        for w1 in l[num]:\\n            for w2 in l[num - 1]:\\n                if w1 != w2:\\n                    d[w1] = w2\\n                    break\\n\\n        l[num] = [x for x in d]\\n\\n        d2.append(d)\\n\\n    curr = None\\n\\n    for w in l[m - 1]:\\n        curr = w\\n\\n    res = [curr]\\n\\n    for num in list(reversed(list(range(1, m)))):\\n        curr = d2[num - 1].get(curr, None)  # d2[num - 1][curr]\\n        res.append(curr)\\n\\n    res = list(reversed(res))\\n\\n    if None in res:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for num in range(m):\\n            print(res[num] + ':' + msg[num])\\n\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    foo()\\n\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\", \"import re\\n\\nt = int(input())\\n\\ndelimiters = \\\"?\\\", \\\".\\\", \\\" \\\", \\\",\\\", \\\"!\\\", \\\":\\\"\\nregexPattern = '|'.join(map(re.escape, delimiters))\\n\\nfor i in range(t):\\n  n = int(input())\\n  usernames = {x for x in str.split(input(), ' ')}\\n  m = int(input())\\n  possibilities = []\\n  for j in range(m):\\n    possibilities.append({x for x in usernames})\\n  messages = []\\n  for j in range(m):\\n    messages.append(input())\\n\\n  for j in range(m):\\n    if (messages[j][0] != '?'):\\n      messageSplit = re.split(':', messages[j])\\n      possibilities[j] = {messageSplit[0]}\\n    else:\\n      messageSplit = re.split(regexPattern, messages[j])\\n      for token in messageSplit:\\n        if token in usernames:\\n          possibilities[j] = possibilities[j] - {token}\\n  changed = True\\n  while changed:\\n    changed = False\\n    for j in range(m):\\n      if len(possibilities[j]) == 1:\\n        (poss,) = possibilities[j]\\n        if j < m-1 and poss in possibilities[j+1]:\\n          changed = True\\n          possibilities[j+1] = possibilities[j+1] - possibilities[j]\\n        if j > 0 and poss in possibilities[j-1]:\\n          changed = True\\n          possibilities[j-1] = possibilities[j-1] - possibilities[j]\\n  worked = True\\n  for j in range(m):\\n    if len(possibilities[j]) == 0:\\n      worked = False\\n      \\n  if not worked:\\n    print(\\\"Impossible\\\")\\n  else :\\n    for j in range(m):\\n      poss = next(iter(possibilities[j]))\\n      if (messages[j][0] == '?'):\\n        print(poss + messages[j][1:])\\n      else:\\n        print(messages[j])\\n      if (j < m-1):\\n        possibilities[j+1] = possibilities[j+1] - {poss}\\n\\n\", \"import re\\n\\ndef add_used(id, s, used):\\n    used[id-1].add(s)\\n    used[id].add(s)\\n    used[id+1].add(s)\\n\\ntcase = int(input())\\nfor cas in range(tcase):\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    \\n    res = True\\n    ans = [''] * (m+1)\\n    msg = [''] * (m+1)\\n    used = [set() for i in range(m+1)]\\n    \\n    for i in range(m):\\n        ans[i], msg[i] = input().split(':')\\n        if ans[i] != '?':\\n            add_used(i, ans[i], used)\\n        \\n        mentioned = re.split('\\\\W+', msg[i])\\n        for s in mentioned:\\n            if s in names:\\n                used[i].add(s)\\n    \\n    for i in range(m):\\n        if ans[i] == '?' and len(used[i]) == n-1:\\n            ans[i] = list(set(names) - used[i])[0]\\n            add_used(i, ans[i], used)\\n    \\n    for i in range(m):\\n        if ans[i] == '?':\\n            if len(used[i]) == n:\\n                res = False\\n            else:\\n                ans[i] = list(set(names) - used[i])[0]\\n                add_used(i, ans[i], used)\\n    \\n    for i in range(m-1):\\n        if ans[i] == ans[i+1]:\\n            res = False\\n    \\n    if res:\\n        for i in range(m):\\n            print(ans[i] + ':' + msg[i])\\n    else:\\n        print('Impossible')\", \"from sys import stdin\\n\\nt = int(stdin.readline())\\n\\ndef c(l,r,msg):\\n    f = True\\n    if l > 0 and not msg[l-1].isdigit() and not msg[l-1].isalpha():\\n        pass\\n    elif l == 0:\\n        pass\\n    else:\\n        f = False\\n\\n    if r == len(msg) - 1:\\n        pass\\n    elif not msg[r+1].isdigit() and not msg[r+1].isalpha():\\n        pass\\n    else:\\n        f = False\\n\\n    return f\\n\\n\\ndef dd(i):\\n    if i - 1 in no_author_set:\\n        authors[i-1].discard(authors[i])\\n        if len(authors[i-1]) == 1:\\n            no_author_set.discard(i-1)\\n            for d in authors[i-1]:\\n                authors[i-1] = d\\n                break\\n            dd(i-1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    if i + 1 in no_author_set:\\n        authors[i+1].discard(authors[i])\\n        if len(authors[i+1]) == 1:\\n            no_author_set.discard(i+1)\\n            for d in authors[i+1]:\\n                authors[i+1] = d\\n                break\\n            dd(i+1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    return True\\n\\ntry:\\n    answers = []\\n    for i in range(t):\\n        ans = True\\n        n = int(stdin.readline())\\n        names = set(stdin.readline().strip().split())\\n        m = int(stdin.readline())\\n        authors = list()\\n        messages = list()\\n        no_author = list()\\n        no_author_set = set()\\n        for j in range(m):\\n            line = stdin.readline().strip()\\n            author, msg = line.split(':')\\n            messages.append(msg)\\n            if author == '?':\\n                no_author.append(j)\\n                no_author_set.add(j)\\n                a_set = set()\\n                for name in names:\\n                    l = msg.find(name)\\n                    while l != -1:\\n                        if c(l,l+len(name)-1, msg):\\n                            a_set.add(name)\\n                        l = msg.find(name,l+len(name))\\n                authors.append(names-a_set)\\n                if j-1 not in no_author_set:\\n                    authors[j].discard(authors[j-1])\\n            else:\\n                authors.append(author)\\n                if j - 1 in no_author_set:\\n                    authors[j-1].discard(author)\\n\\n        for j in no_author:\\n            if j in no_author_set:\\n                if len(authors[j]) == 1:\\n                    no_author_set.discard(j)\\n                    for d in authors[j]:\\n                        authors[j] = d\\n                        break\\n                    if not dd(j):\\n                        ans = False\\n                elif len(authors[j]) == 0:\\n                    ans = False\\n\\n        no_author = list()\\n        for j in no_author_set:\\n            no_author.append(j)\\n        no_author.sort()\\n        for i in no_author:\\n            for d in authors[i]:\\n                authors[i] = d\\n                break\\n            if i + 1 in no_author_set:\\n                authors[i+1].discard(authors[i])\\n        if not ans:\\n            answers.append('Impossible')\\n        else:\\n            for j in range(m):\\n                answers.append('%s:%s'%(authors[j],messages[j]))\\nexcept Exception as e:\\n    print(e)\\n\\nfor m in answers:\\n    print(m)\", \"import re\\nt = int(input())\\nfor _ in range(t):\\n    n = int(input())\\n    names = set(input().split())\\n    m = int(input())\\n    dp = []\\n    a = []\\n    for i in range(m): a.append(input())\\n    for i in range(m):\\n        sender, msg = a[i].split(':')\\n        ls = set([_f for _f in re.split('\\\\W+',msg) if _f])\\n        dp.append((names if sender == '?' else set([sender])) - ls)\\n        if i and len(dp[i-1]) == 1: dp[i] -= dp[i-1]\\n        # print(dp[i]);\\n    if any([len(p) == 0 for p in dp]): print(\\\"Impossible\\\")\\n    else:\\n        res = []\\n        for i in reversed(list(range(m))):\\n            res.append(dp[i].pop())\\n            if i > 0: dp[i-1].discard(res[-1])\\n        for i in range(m): \\n            sender, msg = a[i].split(':')\\n            sender = res[m-i-1]\\n            print(sender+':'+msg)\\n\", \"import re\\n\\nt = int( input() )\\nfor z in range(t):\\n\\tn = int( input() )\\n\\tusers = input().split();\\n\\t#print(users)\\n\\tm = int( input() )\\n\\tcan = [ [] for i in range(m) ]\\n\\tok = True\\n\\tL = []\\n\\tfor i in range(m):\\n\\t\\tl = input().split(':')\\n\\t\\t#print(l)\\n\\t\\tL += [l]\\n\\t\\tif l[0] != '?':\\n\\t\\t\\tcan[i] = [users.index(l[0])]\\n\\t\\telse:\\n\\t\\t\\tcan[i] = [x for x in range(n)]\\n\\t\\tll = re.sub(r'[^A-Za-z0-9]', ' ', l[1]).split()\\n\\t\\tfor j in ll:\\n\\t\\t\\t#print(j)\\n\\t\\t\\tif j in users:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tcan[i].remove( users.index(j) )\\n\\t\\t\\t\\texcept ValueError:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tif len(can[i]) == 0:\\n\\t\\t\\tok = False\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\twhile 1:\\n\\t\\tflag = True\\n\\t\\tfor i in range(m - 1):\\n\\t\\t\\tif (len(can[i]) == 0): ok = False\\n\\t\\t\\tif (len(can[i]) == 1 and can[i][0] in can[i+1]):\\n\\t\\t\\t\\tcan[i+1].remove( can[i][0] )\\n\\t\\t\\t\\tL[i][0] = users[ can[i][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\t\\tif (len(can[i + 1]) == 1 and can[i + 1][0] in can[i]):\\n\\t\\t\\t\\tcan[i].remove( can[i + 1][0] )\\n\\t\\t\\t\\tL[i + 1][0] = users[ can[i + 1][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\tif len(can[m - 1]) == 0: ok = False\\n\\t\\tif ok == False: break\\n\\t\\tif flag: break\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\tfor i in range(m):\\n\\t\\tif L[i][0] == '?':\\n\\t\\t\\tprint(users[ can[i][0] ], end='')\\n\\t\\t\\tif i < m - 1 and can[i][0] in can[i + 1]:\\n\\t\\t\\t\\tcan[i + 1].remove( can[i][0] )\\n\\t\\telse:\\n\\t\\t\\tprint(L[i][0], end='')\\n\\t\\tprint(':', L[i][1], sep = '');\\n\", \"3\\n\\nimport re\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    n = int(input())\\n    users = set(input().split(' '))\\n\\n    m = int(input())\\n    messages = []\\n\\n    guessed = [None for i in range(m)]\\n    denied = [set() for i in range(m)]\\n\\n    for i in range(m):\\n        mg = input()\\n        if not mg.startswith('?:'):\\n            guessed[i] = mg[:mg.find(':')]\\n\\n        mg = mg[mg.find(':'):]\\n        messages.append(mg)\\n\\n        m2 = mg + ' '\\n        for u in users:\\n            if re.search('[^a-zA-Z0-9]' + u + '[^a-zA-Z0-9]', m2):\\n                denied[i].add(u)\\n\\n    answer = True\\n\\n    for i in range(m):\\n        if guessed[i]:\\n            if i > 0: denied[i-1].add(guessed[i])\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n    for i in range(m):\\n        if guessed[i] in denied[i]:\\n            answer = False\\n\\n    #print(guessed)\\n    #print(denied)\\n\\n    changed = True\\n    while changed and answer:\\n        changed = False\\n        for i in range(m):\\n            if not guessed[i]:\\n                if len(users) - len(denied[i]) == 1:\\n                    changed = True\\n                    guessed[i] = (users - denied[i]).pop()\\n                    if i > 0: denied[i-1].add(guessed[i])\\n                    if i < m-1: denied[i+1].add(guessed[i])\\n                if len(users) == len(denied[i]):\\n                    answer = False\\n                    break\\n\\n    for i in range(m):\\n        if not guessed[i] and len(users) - len(denied[i]) >= 1:\\n            guessed[i] = (users - denied[i]).pop()\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n\\n    for i in guessed:\\n        if not i:\\n            answer = False\\n\\n    if not answer:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for i in range(m):\\n            print(guessed[i] + messages[i])\\n    \\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  t = int(input())\\n  for t in range(t):\\n    # input\\n    n = int(input())\\n    users = set(input().split())\\n    m = int(input())\\n    msg = []\\n    for i in range(m):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        alts = users - {x for x in re.split(r'[^A-Za-z0-9]+',text)}\\n      msg.append(dict(user=user, text=text, users=alts))\\n    # remove before and after\\n    for i in range(m-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i+1]['users'].difference_update(msg[i]['users'])\\n    for i in range(m-1,0,-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i-1]['users'].difference_update(msg[i]['users'])\\n    # compute answer\\n    last = ''\\n    impo = False\\n    for i in range(m):\\n      msg[i]['users'].discard(last)\\n      if len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n      last = next(iter(msg[i]['users']))\\n      msg[i]['user'] = last\\n    if impo:\\n      print('Impossible')\\n      continue\\n    for i in range(m):\\n      print(msg[i]['user']+':'+msg[i]['text'])\\n    '''\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(1,n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]'''\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    impo = False\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?': user = users.index(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split(r'[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n      if msg[i]['user'] == '?' and len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n    if impo:\\n      print('Impossible')\\n      continue\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      u = msg[i]['user']\\n      for j in range(n+1):\\n        if u != '?':\\n          if u != j and dp[i+1][u]: dp[i][j] = u\\n        else:\\n          for alt in msg[i]['users']:\\n            if alt != j and dp[i+1][alt]:\\n              dp[i][j] = alt\\n              break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        user = users.index(user)\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split('[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\"]",
        "difficulty": "interview",
        "input": "1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n",
        "output": "Impossible\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/754/C"
    },
    {
        "id": 274,
        "task_id": 2124,
        "test_case_id": 3,
        "question": "Recently Vladik discovered a new entertainment — coding bots for social networks. He would like to use machine learning in his bots so now he want to prepare some learning data for them.\n\nAt first, he need to download t chats. Vladik coded a script which should have downloaded the chats, however, something went wrong. In particular, some of the messages have no information of their sender. It is known that if a person sends several messages in a row, they all are merged into a single message. It means that there could not be two or more messages in a row with the same sender. Moreover, a sender never mention himself in his messages.\n\nVladik wants to recover senders of all the messages so that each two neighboring messages will have different senders and no sender will mention himself in his messages.\n\nHe has no idea of how to do this, and asks you for help. Help Vladik to recover senders in each of the chats!\n\n\n-----Input-----\n\nThe first line contains single integer t (1 ≤ t ≤ 10) — the number of chats. The t chats follow. Each chat is given in the following format.\n\nThe first line of each chat description contains single integer n (1 ≤ n ≤ 100) — the number of users in the chat.\n\nThe next line contains n space-separated distinct usernames. Each username consists of lowercase and uppercase English letters and digits. The usernames can't start with a digit. Two usernames are different even if they differ only with letters' case. The length of username is positive and doesn't exceed 10 characters.\n\nThe next line contains single integer m (1 ≤ m ≤ 100) — the number of messages in the chat. The next m line contain the messages in the following formats, one per line:   <username>:<text> — the format of a message with known sender. The username should appear in the list of usernames of the chat.  <?>:<text> — the format of a message with unknown sender. \n\nThe text of a message can consist of lowercase and uppercase English letter, digits, characters '.' (dot), ',' (comma), '!' (exclamation mark), '?' (question mark) and ' ' (space). The text doesn't contain trailing spaces. The length of the text is positive and doesn't exceed 100 characters.\n\nWe say that a text mention a user if his username appears in the text as a word. In other words, the username appears in a such a position that the two characters before and after its appearance either do not exist or are not English letters or digits. For example, the text \"Vasya, masha13 and Kate!\" can mention users \"Vasya\", \"masha13\", \"and\" and \"Kate\", but not \"masha\".\n\nIt is guaranteed that in each chat no known sender mention himself in his messages and there are no two neighboring messages with the same known sender.\n\n\n-----Output-----\n\nPrint the information about the t chats in the following format:\n\nIf it is not possible to recover senders, print single line \"Impossible\" for this chat. Otherwise print m messages in the following format:\n\n<username>:<text>\n\nIf there are multiple answers, print any of them.\n\n\n-----Examples-----\nInput\n1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n\nOutput\nnetman: Hello, Vladik!\nVladik: Hi\n\nInput\n1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n\nOutput\nImpossible\n\nInput\n2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n\nOutput\nImpossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.",
        "solutions": "[\"def pr( name , lvl , dp , u , tot ): \\n    if lvl == 0:\\n        print(name + ':' + tot[lvl])\\n        return\\n\\n    pr( u[lvl][name] , lvl - 1 , dp , u , tot )\\n    print(name + ':' + tot[lvl])\\n\\ndef solve(): \\n    n = int(input())\\n    users = input().split()\\n    m = int(input())\\n    dp = [] \\n    u = []\\n    tot = [] \\n    for i in range( m ) : \\n        dp.append( set() ) \\n        u.append( {} ) \\n        line = input().split(':')\\n        sender = line[0]\\n        tot.append( line[1] ) \\n        line[1] = line[1].replace( '?' , ' ' )\\n        line[1] = line[1].replace( '.' , ' ' )\\n        line[1] = line[1].replace( ',' , ' ' )\\n        line[1] = line[1].replace( '!' , ' ' )\\n        mess = line[1].split()\\n\\n        if sender == '?' : \\n            if i != 0:\\n                for name in users:\\n                    for x in dp[i-1]: \\n                        if x != name and name not in mess:\\n                            dp[i].add( name ) \\n                            u[i][name] = x\\n            else : \\n                for name in users: \\n                    if name not in mess:\\n                        dp[i].add( name ) \\n        else: \\n            if i != 0: \\n                for x in dp[i-1]: \\n                    if x != sender: \\n                        dp[i].add( sender ) \\n                        u[i][sender] = x\\n            else: \\n                dp[i].add( sender )\\n        \\n        \\n    if dp[m-1]: \\n        pr( list(dp[m-1])[0] , m-1 , dp , u , tot )\\n    else: \\n        print(\\\"Impossible\\\")\\n\\n\\nt = int(input())\\nfor i in range( t ) : \\n    solve()\\n\", \"#!/usr/bin/env python3\\n\\nimport re\\n\\ndef cont(line, user):\\n\\treturn re.search('[^a-zA-Z0-9]' + user + '[^a-zA-Z0-9]', '_' + line + '_')\\n\\ntests = int(input())\\nfor test in range(tests):\\n\\tinput()\\n\\tusers = input().rstrip('\\\\n').split(' ')\\n\\tlinecnt = int(input())\\n\\tlines = [input().rstrip('\\\\n') for _ in range(linecnt)]\\n\\tlines = [x.split(':', maxsplit=1) for x in lines]\\n\\tposs = [[]] * linecnt\\n\\tfor i, (user, line) in enumerate(lines):\\n\\t\\tif user != '?':\\n\\t\\t\\tposs[i] = [user]\\n\\t\\telse:\\n\\t\\t\\tposs[i] = [u for u in users if not cont(line, u)]\\n\\n\\tel = list(enumerate(lines))\\n\\trel = list(reversed(el))\\n\\n\\tchanged = False\\n\\twhile True:\\n\\t\\tfor i, (user, line) in el:\\n\\t\\t\\tif i > 0 and len(poss[i-1]) == 1:\\n\\t\\t\\t\\tif poss[i-1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i-1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tfor i, (user, line) in rel:\\n\\t\\t\\tif i < linecnt - 1 and len(poss[i+1]) == 1:\\n\\t\\t\\t\\tif poss[i+1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i+1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tif not changed:\\n\\t\\t\\tbreak\\n\\t\\tchanged = False\\n\\n\\tif all(len(p) > 0 for p in poss):\\n\\t\\tfor i, p, (user, line) in zip(list(range(linecnt)), poss, lines):\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\tif poss[i-1][0] in p:\\n\\t\\t\\t\\t\\tp.remove(poss[i-1][0])\\n\\t\\t\\tprint(p[0] + ':' + line)\\n\\telse:\\n\\t\\tprint('Impossible')\\n\", \"import re\\nimport sys\\n\\n\\ndef g(cts, names):\\n  la = dict([(p, None) for p in names.difference(cts[0])])\\n  if len(la) == 0:\\n    return None\\n  d = [la]\\n  for i in range(1, len(cts)):\\n    prev = d[-1]\\n    la = dict()\\n    for p in names.difference(cts[i]):\\n      for n in prev.keys():\\n        if n != p:\\n          la[p] = n\\n          break\\n    if len(la) == 0:\\n      return None\\n    d.append(la)\\n  cur = list(d[-1].keys())[0]\\n  result = []\\n  for i in range(len(cts) - 1, -1, -1):\\n    result.append(cur)\\n    cur = d[i][cur]\\n  result.reverse()\\n  return result\\n\\n\\ndef solve():\\n  n = int(input())\\n  names = input().split()\\n  set_names = set(names)\\n\\n  def get_names(text):\\n    result = []\\n    for p in re.split(\\\"\\\\W+\\\", text):\\n      if p in set_names:\\n        result.append(p)\\n    return result\\n\\n  m = int(input())\\n  messages = []\\n  for i in range(m):\\n    s = input()\\n    colon = s.find(\\\":\\\")\\n    name = s[:colon]\\n    text = s[colon + 1:]\\n    if name == \\\"?\\\":\\n      name = None\\n    messages.append([name, text, get_names(text)])\\n\\n  for i in range(m):\\n    if messages[i][0]:\\n      continue\\n    j = i\\n    cts = []\\n    while j < m and not messages[j][0]:\\n      cts.append(set(messages[j][2]))\\n      j += 1\\n    if i > 0:\\n      cts[0].add(messages[i - 1][0])\\n    if j < m:\\n      cts[-1].add(messages[j][0])\\n    sb = g(cts, set_names)\\n    if not sb:\\n      return None\\n    for k in range(i, j):\\n      messages[k][0] = sb[k - i]\\n\\n  for p in messages:\\n    print(\\\"%s:%s\\\" % (p[0], p[1]))\\n  return True;\\n\\n\\ndef main():\\n  tests = int(input())\\n  for i in range(tests):\\n    if not solve():\\n      print(\\\"Impossible\\\")\\n  return 0\\n\\ndef __starting_point():\\n  return(main())\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"def dfs(ind):\\n    if ind > 0:\\n        if possible[ind][0] in possible[ind - 1]:\\n            possible[ind - 1].remove(possible[ind][0])\\n            if len(possible[ind - 1]) == 1:\\n                dfs(ind - 1)\\n    if ind < m - 1:\\n        if possible[ind][0] in possible[ind + 1]:\\n            possible[ind + 1].remove(possible[ind][0])\\n            if len(possible[ind + 1]) == 1:\\n                dfs(ind + 1)\\ndef Check(st, ms):\\n    for i in range(0, len(ms) - len(st) + 1):\\n        if st == ms[i:i+len(st)]:\\n            t = True\\n            if i > 0:\\n                if ms[i - 1] == ' ' or ms[i - 1] == '.' or ms[i - 1] == ',' or ms[i - 1] == '?' or ms[i - 1] == '!':\\n                    e = 0\\n                else:\\n                    t = False\\n            if i < len(ms) - len(st):\\n                if (ms[i + len(st)] == ' ' or ms[i + len(st)] == '.' or ms[i + len(st)] == ',' or ms[i + len(st)] == '?' or ms[i + len(st)] == '!'):\\n                    e = 0\\n                else:\\n                    t = False\\n            if t:\\n                return True\\n    return False\\nR = lambda:list(map(int, input().split(' ')))\\n#r, w = open(\\\"input.txt\\\", \\\"r\\\"), open(\\\"output.txt\\\", \\\"w\\\")\\nT = int(input())\\nwhile T:\\n    n = int(input())\\n    users = input().split(' ')\\n    m = int(input())\\n    possible = [[] for i in range(m)]\\n    sender, message = [], []\\n    for i in range(m):\\n        s = input().split(':')\\n        sender.append(s[0])\\n        message.append(s[1])\\n    for i in range(m):\\n        if sender[i] == '?':\\n            unallow = \\\" \\\"\\n            if i > 0 and len(possible[i - 1]) == 1:\\n                unallow = possible[i - 1][0]\\n            for j in users:\\n                if j == unallow:\\n                    continue\\n                done = Check(j, message[i])\\n                if not done:\\n                    possible[i].append(j)\\n        else:\\n            possible[i].append(sender[i])\\n    used = [0 for i in range(m)]\\n    for i in range(m):\\n        if len(possible[i]) == 1 and used[i] == 0:\\n            dfs(i)\\n    for i in range(m):\\n        if len(possible[i]) > 1:\\n            possible[i] = [possible[i][0]]\\n            dfs(i)\\n    done = False\\n    for i in possible:\\n        if len(i) == 0:\\n            print(\\\"Impossible\\\")\\n            done = True\\n            break\\n    if not done:\\n        for i in range(m):\\n            print(possible[i][0]+':'+message[i])\\n    T -= 1\\n\\n\\n    \\n\\n\", \"# Bartek Kostka\\n#  You are not prepared!\\n\\nimport re\\n\\n\\ndef go(i):\\n    nonlocal counter\\n    counter += 1\\n    if counter > 2000:\\n        return False\\n    if i == len(E):\\n        return True\\n    if E[i][0] != \\\"?\\\":\\n        prop = E[i][0]\\n        if i == 0 or W[i-1] != prop:\\n            W[i] = prop\\n            return go(i+1)\\n        else:\\n            return False\\n    for pos in E[i][1]:\\n        if i == 0 or W[i-1] != pos:\\n            W[i] = pos\\n            if go(i+1):\\n                return True\\n    return False\\n\\n\\nt = int(input())\\nfor ttt in range(t):\\n    n = int(input())\\n    users = input().split(\\\" \\\")\\n    users_set = set(users)\\n    m = int(input())\\n    E = []\\n    S = []\\n    counter = 0\\n    W = [\\\"\\\" for x in range(m)]\\n    for mmm in range(m):\\n        line = input()\\n        (user, sentence) = line.split(\\\":\\\")\\n        S.append((user, sentence))\\n        words = [x.strip() for x in re.split('\\\\W+', sentence)]\\n        mentions = list([x for x in words if x in users_set])\\n        E.append([user, set(mentions)])\\n    for i in range(len(E)-1):\\n        if E[i][0] != \\\"?\\\":\\n            E[i+1][1].add(E[i][0])\\n    for i in range(1, len(E)):\\n        if E[i][0] != \\\"?\\\":\\n            E[i-1][1].add(E[i][0])\\n    for i in range(len(E)):\\n        E[i][1] = E[i][1].symmetric_difference(users_set)\\n    if go(0):\\n        for i in range(len(E)):\\n            print(str(W[i])+\\\":\\\"+str(S[i][1]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import re\\nimport copy\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\tn = int(input())\\n\\tsenderList = []\\n\\tfor x in input().split():\\n\\t\\tsenderList.append(x)\\n\\n\\tm = int(input())\\n\\tmsg = [None] * m\\n\\tresSenders = [None] * m\\n\\tpossibleSenders = []\\n\\tfor i in range(m):\\n\\t\\tpossibleSenders.append(copy.copy(senderList))\\n\\n\\tfor i in range(m):\\n\\t\\tline = input()\\n\\t\\tblocks = re.findall(r\\\"[\\\\w]+\\\", line)\\n\\n\\t\\tfor x in blocks:\\n\\t\\t\\ttry:\\n\\t\\t\\t\\tpossibleSenders[i].remove(x)\\n\\t\\t\\texcept:\\n\\t\\t\\t\\tpass\\n\\n\\t\\tif line[0] != '?':\\n\\t\\t\\tresSenders[i] = blocks[0]\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tmsg[i] = line[line.find(\\\":\\\")+1:]\\n\\n\\tresolved = False\\n\\twhile True:\\n\\t\\tdone = True\\n\\t\\tjustPick = True\\n\\t\\tfor i in range(m):\\n\\t\\t\\tif resSenders[i] != None:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tif len(possibleSenders[i]) == 0:\\n\\t\\t\\t\\tdone = True\\n\\t\\t\\t\\tresolved = True\\n\\t\\t\\t\\tprint(\\\"Impossible\\\")\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tif len(possibleSenders[i]) == 1:\\n\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\tif i > 0:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tjustPick = False\\n\\t\\t\\telse:\\n\\t\\t\\t\\tdone = False\\n\\t\\tif done:\\n\\t\\t\\tbreak\\n\\t\\tif justPick:\\n\\t\\t\\tfor i in range(m):\\n\\t\\t\\t\\tif resSenders[i] == None:\\n\\t\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\tbreak\\n\\n\\tif not resolved:\\n\\t\\tfor i in range(m):\\n\\t\\t\\tprint(resSenders[i] + \\\":\\\" + msg[i])\", \"import re\\n\\n\\ndef foo():\\n    n = int(input())\\n    who = input().split()\\n    m = int(input())\\n    msg = []\\n\\n    l = []\\n\\n    for num in range(m):\\n        a, t = input().split(':')\\n        msg.append(t)\\n\\n        st = set()\\n\\n        if a != '?':\\n            st.add(a)\\n        else:\\n\\n            names = re.split('\\\\W+', t)\\n\\n            for w in who:\\n                if not w in names:\\n                    st.add(w)\\n\\n        l.append(st)\\n\\n    d2 = []\\n\\n    for num in range(1, m):\\n        d = {}\\n        for w1 in l[num]:\\n            for w2 in l[num - 1]:\\n                if w1 != w2:\\n                    d[w1] = w2\\n                    break\\n\\n        l[num] = [x for x in d]\\n\\n        d2.append(d)\\n\\n    curr = None\\n\\n    for w in l[m - 1]:\\n        curr = w\\n\\n    res = [curr]\\n\\n    for num in list(reversed(list(range(1, m)))):\\n        curr = d2[num - 1].get(curr, None)  # d2[num - 1][curr]\\n        res.append(curr)\\n\\n    res = list(reversed(res))\\n\\n    if None in res:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for num in range(m):\\n            print(res[num] + ':' + msg[num])\\n\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    foo()\\n\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\", \"import re\\n\\nt = int(input())\\n\\ndelimiters = \\\"?\\\", \\\".\\\", \\\" \\\", \\\",\\\", \\\"!\\\", \\\":\\\"\\nregexPattern = '|'.join(map(re.escape, delimiters))\\n\\nfor i in range(t):\\n  n = int(input())\\n  usernames = {x for x in str.split(input(), ' ')}\\n  m = int(input())\\n  possibilities = []\\n  for j in range(m):\\n    possibilities.append({x for x in usernames})\\n  messages = []\\n  for j in range(m):\\n    messages.append(input())\\n\\n  for j in range(m):\\n    if (messages[j][0] != '?'):\\n      messageSplit = re.split(':', messages[j])\\n      possibilities[j] = {messageSplit[0]}\\n    else:\\n      messageSplit = re.split(regexPattern, messages[j])\\n      for token in messageSplit:\\n        if token in usernames:\\n          possibilities[j] = possibilities[j] - {token}\\n  changed = True\\n  while changed:\\n    changed = False\\n    for j in range(m):\\n      if len(possibilities[j]) == 1:\\n        (poss,) = possibilities[j]\\n        if j < m-1 and poss in possibilities[j+1]:\\n          changed = True\\n          possibilities[j+1] = possibilities[j+1] - possibilities[j]\\n        if j > 0 and poss in possibilities[j-1]:\\n          changed = True\\n          possibilities[j-1] = possibilities[j-1] - possibilities[j]\\n  worked = True\\n  for j in range(m):\\n    if len(possibilities[j]) == 0:\\n      worked = False\\n      \\n  if not worked:\\n    print(\\\"Impossible\\\")\\n  else :\\n    for j in range(m):\\n      poss = next(iter(possibilities[j]))\\n      if (messages[j][0] == '?'):\\n        print(poss + messages[j][1:])\\n      else:\\n        print(messages[j])\\n      if (j < m-1):\\n        possibilities[j+1] = possibilities[j+1] - {poss}\\n\\n\", \"import re\\n\\ndef add_used(id, s, used):\\n    used[id-1].add(s)\\n    used[id].add(s)\\n    used[id+1].add(s)\\n\\ntcase = int(input())\\nfor cas in range(tcase):\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    \\n    res = True\\n    ans = [''] * (m+1)\\n    msg = [''] * (m+1)\\n    used = [set() for i in range(m+1)]\\n    \\n    for i in range(m):\\n        ans[i], msg[i] = input().split(':')\\n        if ans[i] != '?':\\n            add_used(i, ans[i], used)\\n        \\n        mentioned = re.split('\\\\W+', msg[i])\\n        for s in mentioned:\\n            if s in names:\\n                used[i].add(s)\\n    \\n    for i in range(m):\\n        if ans[i] == '?' and len(used[i]) == n-1:\\n            ans[i] = list(set(names) - used[i])[0]\\n            add_used(i, ans[i], used)\\n    \\n    for i in range(m):\\n        if ans[i] == '?':\\n            if len(used[i]) == n:\\n                res = False\\n            else:\\n                ans[i] = list(set(names) - used[i])[0]\\n                add_used(i, ans[i], used)\\n    \\n    for i in range(m-1):\\n        if ans[i] == ans[i+1]:\\n            res = False\\n    \\n    if res:\\n        for i in range(m):\\n            print(ans[i] + ':' + msg[i])\\n    else:\\n        print('Impossible')\", \"from sys import stdin\\n\\nt = int(stdin.readline())\\n\\ndef c(l,r,msg):\\n    f = True\\n    if l > 0 and not msg[l-1].isdigit() and not msg[l-1].isalpha():\\n        pass\\n    elif l == 0:\\n        pass\\n    else:\\n        f = False\\n\\n    if r == len(msg) - 1:\\n        pass\\n    elif not msg[r+1].isdigit() and not msg[r+1].isalpha():\\n        pass\\n    else:\\n        f = False\\n\\n    return f\\n\\n\\ndef dd(i):\\n    if i - 1 in no_author_set:\\n        authors[i-1].discard(authors[i])\\n        if len(authors[i-1]) == 1:\\n            no_author_set.discard(i-1)\\n            for d in authors[i-1]:\\n                authors[i-1] = d\\n                break\\n            dd(i-1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    if i + 1 in no_author_set:\\n        authors[i+1].discard(authors[i])\\n        if len(authors[i+1]) == 1:\\n            no_author_set.discard(i+1)\\n            for d in authors[i+1]:\\n                authors[i+1] = d\\n                break\\n            dd(i+1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    return True\\n\\ntry:\\n    answers = []\\n    for i in range(t):\\n        ans = True\\n        n = int(stdin.readline())\\n        names = set(stdin.readline().strip().split())\\n        m = int(stdin.readline())\\n        authors = list()\\n        messages = list()\\n        no_author = list()\\n        no_author_set = set()\\n        for j in range(m):\\n            line = stdin.readline().strip()\\n            author, msg = line.split(':')\\n            messages.append(msg)\\n            if author == '?':\\n                no_author.append(j)\\n                no_author_set.add(j)\\n                a_set = set()\\n                for name in names:\\n                    l = msg.find(name)\\n                    while l != -1:\\n                        if c(l,l+len(name)-1, msg):\\n                            a_set.add(name)\\n                        l = msg.find(name,l+len(name))\\n                authors.append(names-a_set)\\n                if j-1 not in no_author_set:\\n                    authors[j].discard(authors[j-1])\\n            else:\\n                authors.append(author)\\n                if j - 1 in no_author_set:\\n                    authors[j-1].discard(author)\\n\\n        for j in no_author:\\n            if j in no_author_set:\\n                if len(authors[j]) == 1:\\n                    no_author_set.discard(j)\\n                    for d in authors[j]:\\n                        authors[j] = d\\n                        break\\n                    if not dd(j):\\n                        ans = False\\n                elif len(authors[j]) == 0:\\n                    ans = False\\n\\n        no_author = list()\\n        for j in no_author_set:\\n            no_author.append(j)\\n        no_author.sort()\\n        for i in no_author:\\n            for d in authors[i]:\\n                authors[i] = d\\n                break\\n            if i + 1 in no_author_set:\\n                authors[i+1].discard(authors[i])\\n        if not ans:\\n            answers.append('Impossible')\\n        else:\\n            for j in range(m):\\n                answers.append('%s:%s'%(authors[j],messages[j]))\\nexcept Exception as e:\\n    print(e)\\n\\nfor m in answers:\\n    print(m)\", \"import re\\nt = int(input())\\nfor _ in range(t):\\n    n = int(input())\\n    names = set(input().split())\\n    m = int(input())\\n    dp = []\\n    a = []\\n    for i in range(m): a.append(input())\\n    for i in range(m):\\n        sender, msg = a[i].split(':')\\n        ls = set([_f for _f in re.split('\\\\W+',msg) if _f])\\n        dp.append((names if sender == '?' else set([sender])) - ls)\\n        if i and len(dp[i-1]) == 1: dp[i] -= dp[i-1]\\n        # print(dp[i]);\\n    if any([len(p) == 0 for p in dp]): print(\\\"Impossible\\\")\\n    else:\\n        res = []\\n        for i in reversed(list(range(m))):\\n            res.append(dp[i].pop())\\n            if i > 0: dp[i-1].discard(res[-1])\\n        for i in range(m): \\n            sender, msg = a[i].split(':')\\n            sender = res[m-i-1]\\n            print(sender+':'+msg)\\n\", \"import re\\n\\nt = int( input() )\\nfor z in range(t):\\n\\tn = int( input() )\\n\\tusers = input().split();\\n\\t#print(users)\\n\\tm = int( input() )\\n\\tcan = [ [] for i in range(m) ]\\n\\tok = True\\n\\tL = []\\n\\tfor i in range(m):\\n\\t\\tl = input().split(':')\\n\\t\\t#print(l)\\n\\t\\tL += [l]\\n\\t\\tif l[0] != '?':\\n\\t\\t\\tcan[i] = [users.index(l[0])]\\n\\t\\telse:\\n\\t\\t\\tcan[i] = [x for x in range(n)]\\n\\t\\tll = re.sub(r'[^A-Za-z0-9]', ' ', l[1]).split()\\n\\t\\tfor j in ll:\\n\\t\\t\\t#print(j)\\n\\t\\t\\tif j in users:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tcan[i].remove( users.index(j) )\\n\\t\\t\\t\\texcept ValueError:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tif len(can[i]) == 0:\\n\\t\\t\\tok = False\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\twhile 1:\\n\\t\\tflag = True\\n\\t\\tfor i in range(m - 1):\\n\\t\\t\\tif (len(can[i]) == 0): ok = False\\n\\t\\t\\tif (len(can[i]) == 1 and can[i][0] in can[i+1]):\\n\\t\\t\\t\\tcan[i+1].remove( can[i][0] )\\n\\t\\t\\t\\tL[i][0] = users[ can[i][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\t\\tif (len(can[i + 1]) == 1 and can[i + 1][0] in can[i]):\\n\\t\\t\\t\\tcan[i].remove( can[i + 1][0] )\\n\\t\\t\\t\\tL[i + 1][0] = users[ can[i + 1][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\tif len(can[m - 1]) == 0: ok = False\\n\\t\\tif ok == False: break\\n\\t\\tif flag: break\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\tfor i in range(m):\\n\\t\\tif L[i][0] == '?':\\n\\t\\t\\tprint(users[ can[i][0] ], end='')\\n\\t\\t\\tif i < m - 1 and can[i][0] in can[i + 1]:\\n\\t\\t\\t\\tcan[i + 1].remove( can[i][0] )\\n\\t\\telse:\\n\\t\\t\\tprint(L[i][0], end='')\\n\\t\\tprint(':', L[i][1], sep = '');\\n\", \"3\\n\\nimport re\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    n = int(input())\\n    users = set(input().split(' '))\\n\\n    m = int(input())\\n    messages = []\\n\\n    guessed = [None for i in range(m)]\\n    denied = [set() for i in range(m)]\\n\\n    for i in range(m):\\n        mg = input()\\n        if not mg.startswith('?:'):\\n            guessed[i] = mg[:mg.find(':')]\\n\\n        mg = mg[mg.find(':'):]\\n        messages.append(mg)\\n\\n        m2 = mg + ' '\\n        for u in users:\\n            if re.search('[^a-zA-Z0-9]' + u + '[^a-zA-Z0-9]', m2):\\n                denied[i].add(u)\\n\\n    answer = True\\n\\n    for i in range(m):\\n        if guessed[i]:\\n            if i > 0: denied[i-1].add(guessed[i])\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n    for i in range(m):\\n        if guessed[i] in denied[i]:\\n            answer = False\\n\\n    #print(guessed)\\n    #print(denied)\\n\\n    changed = True\\n    while changed and answer:\\n        changed = False\\n        for i in range(m):\\n            if not guessed[i]:\\n                if len(users) - len(denied[i]) == 1:\\n                    changed = True\\n                    guessed[i] = (users - denied[i]).pop()\\n                    if i > 0: denied[i-1].add(guessed[i])\\n                    if i < m-1: denied[i+1].add(guessed[i])\\n                if len(users) == len(denied[i]):\\n                    answer = False\\n                    break\\n\\n    for i in range(m):\\n        if not guessed[i] and len(users) - len(denied[i]) >= 1:\\n            guessed[i] = (users - denied[i]).pop()\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n\\n    for i in guessed:\\n        if not i:\\n            answer = False\\n\\n    if not answer:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for i in range(m):\\n            print(guessed[i] + messages[i])\\n    \\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  t = int(input())\\n  for t in range(t):\\n    # input\\n    n = int(input())\\n    users = set(input().split())\\n    m = int(input())\\n    msg = []\\n    for i in range(m):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        alts = users - {x for x in re.split(r'[^A-Za-z0-9]+',text)}\\n      msg.append(dict(user=user, text=text, users=alts))\\n    # remove before and after\\n    for i in range(m-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i+1]['users'].difference_update(msg[i]['users'])\\n    for i in range(m-1,0,-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i-1]['users'].difference_update(msg[i]['users'])\\n    # compute answer\\n    last = ''\\n    impo = False\\n    for i in range(m):\\n      msg[i]['users'].discard(last)\\n      if len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n      last = next(iter(msg[i]['users']))\\n      msg[i]['user'] = last\\n    if impo:\\n      print('Impossible')\\n      continue\\n    for i in range(m):\\n      print(msg[i]['user']+':'+msg[i]['text'])\\n    '''\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(1,n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]'''\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    impo = False\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?': user = users.index(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split(r'[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n      if msg[i]['user'] == '?' and len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n    if impo:\\n      print('Impossible')\\n      continue\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      u = msg[i]['user']\\n      for j in range(n+1):\\n        if u != '?':\\n          if u != j and dp[i+1][u]: dp[i][j] = u\\n        else:\\n          for alt in msg[i]['users']:\\n            if alt != j and dp[i+1][alt]:\\n              dp[i][j] = alt\\n              break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        user = users.index(user)\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split('[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\"]",
        "difficulty": "interview",
        "input": "2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n",
        "output": "Impossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/754/C"
    },
    {
        "id": 275,
        "task_id": 2124,
        "test_case_id": 4,
        "question": "Recently Vladik discovered a new entertainment — coding bots for social networks. He would like to use machine learning in his bots so now he want to prepare some learning data for them.\n\nAt first, he need to download t chats. Vladik coded a script which should have downloaded the chats, however, something went wrong. In particular, some of the messages have no information of their sender. It is known that if a person sends several messages in a row, they all are merged into a single message. It means that there could not be two or more messages in a row with the same sender. Moreover, a sender never mention himself in his messages.\n\nVladik wants to recover senders of all the messages so that each two neighboring messages will have different senders and no sender will mention himself in his messages.\n\nHe has no idea of how to do this, and asks you for help. Help Vladik to recover senders in each of the chats!\n\n\n-----Input-----\n\nThe first line contains single integer t (1 ≤ t ≤ 10) — the number of chats. The t chats follow. Each chat is given in the following format.\n\nThe first line of each chat description contains single integer n (1 ≤ n ≤ 100) — the number of users in the chat.\n\nThe next line contains n space-separated distinct usernames. Each username consists of lowercase and uppercase English letters and digits. The usernames can't start with a digit. Two usernames are different even if they differ only with letters' case. The length of username is positive and doesn't exceed 10 characters.\n\nThe next line contains single integer m (1 ≤ m ≤ 100) — the number of messages in the chat. The next m line contain the messages in the following formats, one per line:   <username>:<text> — the format of a message with known sender. The username should appear in the list of usernames of the chat.  <?>:<text> — the format of a message with unknown sender. \n\nThe text of a message can consist of lowercase and uppercase English letter, digits, characters '.' (dot), ',' (comma), '!' (exclamation mark), '?' (question mark) and ' ' (space). The text doesn't contain trailing spaces. The length of the text is positive and doesn't exceed 100 characters.\n\nWe say that a text mention a user if his username appears in the text as a word. In other words, the username appears in a such a position that the two characters before and after its appearance either do not exist or are not English letters or digits. For example, the text \"Vasya, masha13 and Kate!\" can mention users \"Vasya\", \"masha13\", \"and\" and \"Kate\", but not \"masha\".\n\nIt is guaranteed that in each chat no known sender mention himself in his messages and there are no two neighboring messages with the same known sender.\n\n\n-----Output-----\n\nPrint the information about the t chats in the following format:\n\nIf it is not possible to recover senders, print single line \"Impossible\" for this chat. Otherwise print m messages in the following format:\n\n<username>:<text>\n\nIf there are multiple answers, print any of them.\n\n\n-----Examples-----\nInput\n1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n\nOutput\nnetman: Hello, Vladik!\nVladik: Hi\n\nInput\n1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n\nOutput\nImpossible\n\nInput\n2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n\nOutput\nImpossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.",
        "solutions": "[\"def pr( name , lvl , dp , u , tot ): \\n    if lvl == 0:\\n        print(name + ':' + tot[lvl])\\n        return\\n\\n    pr( u[lvl][name] , lvl - 1 , dp , u , tot )\\n    print(name + ':' + tot[lvl])\\n\\ndef solve(): \\n    n = int(input())\\n    users = input().split()\\n    m = int(input())\\n    dp = [] \\n    u = []\\n    tot = [] \\n    for i in range( m ) : \\n        dp.append( set() ) \\n        u.append( {} ) \\n        line = input().split(':')\\n        sender = line[0]\\n        tot.append( line[1] ) \\n        line[1] = line[1].replace( '?' , ' ' )\\n        line[1] = line[1].replace( '.' , ' ' )\\n        line[1] = line[1].replace( ',' , ' ' )\\n        line[1] = line[1].replace( '!' , ' ' )\\n        mess = line[1].split()\\n\\n        if sender == '?' : \\n            if i != 0:\\n                for name in users:\\n                    for x in dp[i-1]: \\n                        if x != name and name not in mess:\\n                            dp[i].add( name ) \\n                            u[i][name] = x\\n            else : \\n                for name in users: \\n                    if name not in mess:\\n                        dp[i].add( name ) \\n        else: \\n            if i != 0: \\n                for x in dp[i-1]: \\n                    if x != sender: \\n                        dp[i].add( sender ) \\n                        u[i][sender] = x\\n            else: \\n                dp[i].add( sender )\\n        \\n        \\n    if dp[m-1]: \\n        pr( list(dp[m-1])[0] , m-1 , dp , u , tot )\\n    else: \\n        print(\\\"Impossible\\\")\\n\\n\\nt = int(input())\\nfor i in range( t ) : \\n    solve()\\n\", \"#!/usr/bin/env python3\\n\\nimport re\\n\\ndef cont(line, user):\\n\\treturn re.search('[^a-zA-Z0-9]' + user + '[^a-zA-Z0-9]', '_' + line + '_')\\n\\ntests = int(input())\\nfor test in range(tests):\\n\\tinput()\\n\\tusers = input().rstrip('\\\\n').split(' ')\\n\\tlinecnt = int(input())\\n\\tlines = [input().rstrip('\\\\n') for _ in range(linecnt)]\\n\\tlines = [x.split(':', maxsplit=1) for x in lines]\\n\\tposs = [[]] * linecnt\\n\\tfor i, (user, line) in enumerate(lines):\\n\\t\\tif user != '?':\\n\\t\\t\\tposs[i] = [user]\\n\\t\\telse:\\n\\t\\t\\tposs[i] = [u for u in users if not cont(line, u)]\\n\\n\\tel = list(enumerate(lines))\\n\\trel = list(reversed(el))\\n\\n\\tchanged = False\\n\\twhile True:\\n\\t\\tfor i, (user, line) in el:\\n\\t\\t\\tif i > 0 and len(poss[i-1]) == 1:\\n\\t\\t\\t\\tif poss[i-1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i-1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tfor i, (user, line) in rel:\\n\\t\\t\\tif i < linecnt - 1 and len(poss[i+1]) == 1:\\n\\t\\t\\t\\tif poss[i+1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i+1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tif not changed:\\n\\t\\t\\tbreak\\n\\t\\tchanged = False\\n\\n\\tif all(len(p) > 0 for p in poss):\\n\\t\\tfor i, p, (user, line) in zip(list(range(linecnt)), poss, lines):\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\tif poss[i-1][0] in p:\\n\\t\\t\\t\\t\\tp.remove(poss[i-1][0])\\n\\t\\t\\tprint(p[0] + ':' + line)\\n\\telse:\\n\\t\\tprint('Impossible')\\n\", \"import re\\nimport sys\\n\\n\\ndef g(cts, names):\\n  la = dict([(p, None) for p in names.difference(cts[0])])\\n  if len(la) == 0:\\n    return None\\n  d = [la]\\n  for i in range(1, len(cts)):\\n    prev = d[-1]\\n    la = dict()\\n    for p in names.difference(cts[i]):\\n      for n in prev.keys():\\n        if n != p:\\n          la[p] = n\\n          break\\n    if len(la) == 0:\\n      return None\\n    d.append(la)\\n  cur = list(d[-1].keys())[0]\\n  result = []\\n  for i in range(len(cts) - 1, -1, -1):\\n    result.append(cur)\\n    cur = d[i][cur]\\n  result.reverse()\\n  return result\\n\\n\\ndef solve():\\n  n = int(input())\\n  names = input().split()\\n  set_names = set(names)\\n\\n  def get_names(text):\\n    result = []\\n    for p in re.split(\\\"\\\\W+\\\", text):\\n      if p in set_names:\\n        result.append(p)\\n    return result\\n\\n  m = int(input())\\n  messages = []\\n  for i in range(m):\\n    s = input()\\n    colon = s.find(\\\":\\\")\\n    name = s[:colon]\\n    text = s[colon + 1:]\\n    if name == \\\"?\\\":\\n      name = None\\n    messages.append([name, text, get_names(text)])\\n\\n  for i in range(m):\\n    if messages[i][0]:\\n      continue\\n    j = i\\n    cts = []\\n    while j < m and not messages[j][0]:\\n      cts.append(set(messages[j][2]))\\n      j += 1\\n    if i > 0:\\n      cts[0].add(messages[i - 1][0])\\n    if j < m:\\n      cts[-1].add(messages[j][0])\\n    sb = g(cts, set_names)\\n    if not sb:\\n      return None\\n    for k in range(i, j):\\n      messages[k][0] = sb[k - i]\\n\\n  for p in messages:\\n    print(\\\"%s:%s\\\" % (p[0], p[1]))\\n  return True;\\n\\n\\ndef main():\\n  tests = int(input())\\n  for i in range(tests):\\n    if not solve():\\n      print(\\\"Impossible\\\")\\n  return 0\\n\\ndef __starting_point():\\n  return(main())\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"def dfs(ind):\\n    if ind > 0:\\n        if possible[ind][0] in possible[ind - 1]:\\n            possible[ind - 1].remove(possible[ind][0])\\n            if len(possible[ind - 1]) == 1:\\n                dfs(ind - 1)\\n    if ind < m - 1:\\n        if possible[ind][0] in possible[ind + 1]:\\n            possible[ind + 1].remove(possible[ind][0])\\n            if len(possible[ind + 1]) == 1:\\n                dfs(ind + 1)\\ndef Check(st, ms):\\n    for i in range(0, len(ms) - len(st) + 1):\\n        if st == ms[i:i+len(st)]:\\n            t = True\\n            if i > 0:\\n                if ms[i - 1] == ' ' or ms[i - 1] == '.' or ms[i - 1] == ',' or ms[i - 1] == '?' or ms[i - 1] == '!':\\n                    e = 0\\n                else:\\n                    t = False\\n            if i < len(ms) - len(st):\\n                if (ms[i + len(st)] == ' ' or ms[i + len(st)] == '.' or ms[i + len(st)] == ',' or ms[i + len(st)] == '?' or ms[i + len(st)] == '!'):\\n                    e = 0\\n                else:\\n                    t = False\\n            if t:\\n                return True\\n    return False\\nR = lambda:list(map(int, input().split(' ')))\\n#r, w = open(\\\"input.txt\\\", \\\"r\\\"), open(\\\"output.txt\\\", \\\"w\\\")\\nT = int(input())\\nwhile T:\\n    n = int(input())\\n    users = input().split(' ')\\n    m = int(input())\\n    possible = [[] for i in range(m)]\\n    sender, message = [], []\\n    for i in range(m):\\n        s = input().split(':')\\n        sender.append(s[0])\\n        message.append(s[1])\\n    for i in range(m):\\n        if sender[i] == '?':\\n            unallow = \\\" \\\"\\n            if i > 0 and len(possible[i - 1]) == 1:\\n                unallow = possible[i - 1][0]\\n            for j in users:\\n                if j == unallow:\\n                    continue\\n                done = Check(j, message[i])\\n                if not done:\\n                    possible[i].append(j)\\n        else:\\n            possible[i].append(sender[i])\\n    used = [0 for i in range(m)]\\n    for i in range(m):\\n        if len(possible[i]) == 1 and used[i] == 0:\\n            dfs(i)\\n    for i in range(m):\\n        if len(possible[i]) > 1:\\n            possible[i] = [possible[i][0]]\\n            dfs(i)\\n    done = False\\n    for i in possible:\\n        if len(i) == 0:\\n            print(\\\"Impossible\\\")\\n            done = True\\n            break\\n    if not done:\\n        for i in range(m):\\n            print(possible[i][0]+':'+message[i])\\n    T -= 1\\n\\n\\n    \\n\\n\", \"# Bartek Kostka\\n#  You are not prepared!\\n\\nimport re\\n\\n\\ndef go(i):\\n    nonlocal counter\\n    counter += 1\\n    if counter > 2000:\\n        return False\\n    if i == len(E):\\n        return True\\n    if E[i][0] != \\\"?\\\":\\n        prop = E[i][0]\\n        if i == 0 or W[i-1] != prop:\\n            W[i] = prop\\n            return go(i+1)\\n        else:\\n            return False\\n    for pos in E[i][1]:\\n        if i == 0 or W[i-1] != pos:\\n            W[i] = pos\\n            if go(i+1):\\n                return True\\n    return False\\n\\n\\nt = int(input())\\nfor ttt in range(t):\\n    n = int(input())\\n    users = input().split(\\\" \\\")\\n    users_set = set(users)\\n    m = int(input())\\n    E = []\\n    S = []\\n    counter = 0\\n    W = [\\\"\\\" for x in range(m)]\\n    for mmm in range(m):\\n        line = input()\\n        (user, sentence) = line.split(\\\":\\\")\\n        S.append((user, sentence))\\n        words = [x.strip() for x in re.split('\\\\W+', sentence)]\\n        mentions = list([x for x in words if x in users_set])\\n        E.append([user, set(mentions)])\\n    for i in range(len(E)-1):\\n        if E[i][0] != \\\"?\\\":\\n            E[i+1][1].add(E[i][0])\\n    for i in range(1, len(E)):\\n        if E[i][0] != \\\"?\\\":\\n            E[i-1][1].add(E[i][0])\\n    for i in range(len(E)):\\n        E[i][1] = E[i][1].symmetric_difference(users_set)\\n    if go(0):\\n        for i in range(len(E)):\\n            print(str(W[i])+\\\":\\\"+str(S[i][1]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import re\\nimport copy\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\tn = int(input())\\n\\tsenderList = []\\n\\tfor x in input().split():\\n\\t\\tsenderList.append(x)\\n\\n\\tm = int(input())\\n\\tmsg = [None] * m\\n\\tresSenders = [None] * m\\n\\tpossibleSenders = []\\n\\tfor i in range(m):\\n\\t\\tpossibleSenders.append(copy.copy(senderList))\\n\\n\\tfor i in range(m):\\n\\t\\tline = input()\\n\\t\\tblocks = re.findall(r\\\"[\\\\w]+\\\", line)\\n\\n\\t\\tfor x in blocks:\\n\\t\\t\\ttry:\\n\\t\\t\\t\\tpossibleSenders[i].remove(x)\\n\\t\\t\\texcept:\\n\\t\\t\\t\\tpass\\n\\n\\t\\tif line[0] != '?':\\n\\t\\t\\tresSenders[i] = blocks[0]\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tmsg[i] = line[line.find(\\\":\\\")+1:]\\n\\n\\tresolved = False\\n\\twhile True:\\n\\t\\tdone = True\\n\\t\\tjustPick = True\\n\\t\\tfor i in range(m):\\n\\t\\t\\tif resSenders[i] != None:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tif len(possibleSenders[i]) == 0:\\n\\t\\t\\t\\tdone = True\\n\\t\\t\\t\\tresolved = True\\n\\t\\t\\t\\tprint(\\\"Impossible\\\")\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tif len(possibleSenders[i]) == 1:\\n\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\tif i > 0:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tjustPick = False\\n\\t\\t\\telse:\\n\\t\\t\\t\\tdone = False\\n\\t\\tif done:\\n\\t\\t\\tbreak\\n\\t\\tif justPick:\\n\\t\\t\\tfor i in range(m):\\n\\t\\t\\t\\tif resSenders[i] == None:\\n\\t\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\tbreak\\n\\n\\tif not resolved:\\n\\t\\tfor i in range(m):\\n\\t\\t\\tprint(resSenders[i] + \\\":\\\" + msg[i])\", \"import re\\n\\n\\ndef foo():\\n    n = int(input())\\n    who = input().split()\\n    m = int(input())\\n    msg = []\\n\\n    l = []\\n\\n    for num in range(m):\\n        a, t = input().split(':')\\n        msg.append(t)\\n\\n        st = set()\\n\\n        if a != '?':\\n            st.add(a)\\n        else:\\n\\n            names = re.split('\\\\W+', t)\\n\\n            for w in who:\\n                if not w in names:\\n                    st.add(w)\\n\\n        l.append(st)\\n\\n    d2 = []\\n\\n    for num in range(1, m):\\n        d = {}\\n        for w1 in l[num]:\\n            for w2 in l[num - 1]:\\n                if w1 != w2:\\n                    d[w1] = w2\\n                    break\\n\\n        l[num] = [x for x in d]\\n\\n        d2.append(d)\\n\\n    curr = None\\n\\n    for w in l[m - 1]:\\n        curr = w\\n\\n    res = [curr]\\n\\n    for num in list(reversed(list(range(1, m)))):\\n        curr = d2[num - 1].get(curr, None)  # d2[num - 1][curr]\\n        res.append(curr)\\n\\n    res = list(reversed(res))\\n\\n    if None in res:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for num in range(m):\\n            print(res[num] + ':' + msg[num])\\n\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    foo()\\n\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\", \"import re\\n\\nt = int(input())\\n\\ndelimiters = \\\"?\\\", \\\".\\\", \\\" \\\", \\\",\\\", \\\"!\\\", \\\":\\\"\\nregexPattern = '|'.join(map(re.escape, delimiters))\\n\\nfor i in range(t):\\n  n = int(input())\\n  usernames = {x for x in str.split(input(), ' ')}\\n  m = int(input())\\n  possibilities = []\\n  for j in range(m):\\n    possibilities.append({x for x in usernames})\\n  messages = []\\n  for j in range(m):\\n    messages.append(input())\\n\\n  for j in range(m):\\n    if (messages[j][0] != '?'):\\n      messageSplit = re.split(':', messages[j])\\n      possibilities[j] = {messageSplit[0]}\\n    else:\\n      messageSplit = re.split(regexPattern, messages[j])\\n      for token in messageSplit:\\n        if token in usernames:\\n          possibilities[j] = possibilities[j] - {token}\\n  changed = True\\n  while changed:\\n    changed = False\\n    for j in range(m):\\n      if len(possibilities[j]) == 1:\\n        (poss,) = possibilities[j]\\n        if j < m-1 and poss in possibilities[j+1]:\\n          changed = True\\n          possibilities[j+1] = possibilities[j+1] - possibilities[j]\\n        if j > 0 and poss in possibilities[j-1]:\\n          changed = True\\n          possibilities[j-1] = possibilities[j-1] - possibilities[j]\\n  worked = True\\n  for j in range(m):\\n    if len(possibilities[j]) == 0:\\n      worked = False\\n      \\n  if not worked:\\n    print(\\\"Impossible\\\")\\n  else :\\n    for j in range(m):\\n      poss = next(iter(possibilities[j]))\\n      if (messages[j][0] == '?'):\\n        print(poss + messages[j][1:])\\n      else:\\n        print(messages[j])\\n      if (j < m-1):\\n        possibilities[j+1] = possibilities[j+1] - {poss}\\n\\n\", \"import re\\n\\ndef add_used(id, s, used):\\n    used[id-1].add(s)\\n    used[id].add(s)\\n    used[id+1].add(s)\\n\\ntcase = int(input())\\nfor cas in range(tcase):\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    \\n    res = True\\n    ans = [''] * (m+1)\\n    msg = [''] * (m+1)\\n    used = [set() for i in range(m+1)]\\n    \\n    for i in range(m):\\n        ans[i], msg[i] = input().split(':')\\n        if ans[i] != '?':\\n            add_used(i, ans[i], used)\\n        \\n        mentioned = re.split('\\\\W+', msg[i])\\n        for s in mentioned:\\n            if s in names:\\n                used[i].add(s)\\n    \\n    for i in range(m):\\n        if ans[i] == '?' and len(used[i]) == n-1:\\n            ans[i] = list(set(names) - used[i])[0]\\n            add_used(i, ans[i], used)\\n    \\n    for i in range(m):\\n        if ans[i] == '?':\\n            if len(used[i]) == n:\\n                res = False\\n            else:\\n                ans[i] = list(set(names) - used[i])[0]\\n                add_used(i, ans[i], used)\\n    \\n    for i in range(m-1):\\n        if ans[i] == ans[i+1]:\\n            res = False\\n    \\n    if res:\\n        for i in range(m):\\n            print(ans[i] + ':' + msg[i])\\n    else:\\n        print('Impossible')\", \"from sys import stdin\\n\\nt = int(stdin.readline())\\n\\ndef c(l,r,msg):\\n    f = True\\n    if l > 0 and not msg[l-1].isdigit() and not msg[l-1].isalpha():\\n        pass\\n    elif l == 0:\\n        pass\\n    else:\\n        f = False\\n\\n    if r == len(msg) - 1:\\n        pass\\n    elif not msg[r+1].isdigit() and not msg[r+1].isalpha():\\n        pass\\n    else:\\n        f = False\\n\\n    return f\\n\\n\\ndef dd(i):\\n    if i - 1 in no_author_set:\\n        authors[i-1].discard(authors[i])\\n        if len(authors[i-1]) == 1:\\n            no_author_set.discard(i-1)\\n            for d in authors[i-1]:\\n                authors[i-1] = d\\n                break\\n            dd(i-1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    if i + 1 in no_author_set:\\n        authors[i+1].discard(authors[i])\\n        if len(authors[i+1]) == 1:\\n            no_author_set.discard(i+1)\\n            for d in authors[i+1]:\\n                authors[i+1] = d\\n                break\\n            dd(i+1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    return True\\n\\ntry:\\n    answers = []\\n    for i in range(t):\\n        ans = True\\n        n = int(stdin.readline())\\n        names = set(stdin.readline().strip().split())\\n        m = int(stdin.readline())\\n        authors = list()\\n        messages = list()\\n        no_author = list()\\n        no_author_set = set()\\n        for j in range(m):\\n            line = stdin.readline().strip()\\n            author, msg = line.split(':')\\n            messages.append(msg)\\n            if author == '?':\\n                no_author.append(j)\\n                no_author_set.add(j)\\n                a_set = set()\\n                for name in names:\\n                    l = msg.find(name)\\n                    while l != -1:\\n                        if c(l,l+len(name)-1, msg):\\n                            a_set.add(name)\\n                        l = msg.find(name,l+len(name))\\n                authors.append(names-a_set)\\n                if j-1 not in no_author_set:\\n                    authors[j].discard(authors[j-1])\\n            else:\\n                authors.append(author)\\n                if j - 1 in no_author_set:\\n                    authors[j-1].discard(author)\\n\\n        for j in no_author:\\n            if j in no_author_set:\\n                if len(authors[j]) == 1:\\n                    no_author_set.discard(j)\\n                    for d in authors[j]:\\n                        authors[j] = d\\n                        break\\n                    if not dd(j):\\n                        ans = False\\n                elif len(authors[j]) == 0:\\n                    ans = False\\n\\n        no_author = list()\\n        for j in no_author_set:\\n            no_author.append(j)\\n        no_author.sort()\\n        for i in no_author:\\n            for d in authors[i]:\\n                authors[i] = d\\n                break\\n            if i + 1 in no_author_set:\\n                authors[i+1].discard(authors[i])\\n        if not ans:\\n            answers.append('Impossible')\\n        else:\\n            for j in range(m):\\n                answers.append('%s:%s'%(authors[j],messages[j]))\\nexcept Exception as e:\\n    print(e)\\n\\nfor m in answers:\\n    print(m)\", \"import re\\nt = int(input())\\nfor _ in range(t):\\n    n = int(input())\\n    names = set(input().split())\\n    m = int(input())\\n    dp = []\\n    a = []\\n    for i in range(m): a.append(input())\\n    for i in range(m):\\n        sender, msg = a[i].split(':')\\n        ls = set([_f for _f in re.split('\\\\W+',msg) if _f])\\n        dp.append((names if sender == '?' else set([sender])) - ls)\\n        if i and len(dp[i-1]) == 1: dp[i] -= dp[i-1]\\n        # print(dp[i]);\\n    if any([len(p) == 0 for p in dp]): print(\\\"Impossible\\\")\\n    else:\\n        res = []\\n        for i in reversed(list(range(m))):\\n            res.append(dp[i].pop())\\n            if i > 0: dp[i-1].discard(res[-1])\\n        for i in range(m): \\n            sender, msg = a[i].split(':')\\n            sender = res[m-i-1]\\n            print(sender+':'+msg)\\n\", \"import re\\n\\nt = int( input() )\\nfor z in range(t):\\n\\tn = int( input() )\\n\\tusers = input().split();\\n\\t#print(users)\\n\\tm = int( input() )\\n\\tcan = [ [] for i in range(m) ]\\n\\tok = True\\n\\tL = []\\n\\tfor i in range(m):\\n\\t\\tl = input().split(':')\\n\\t\\t#print(l)\\n\\t\\tL += [l]\\n\\t\\tif l[0] != '?':\\n\\t\\t\\tcan[i] = [users.index(l[0])]\\n\\t\\telse:\\n\\t\\t\\tcan[i] = [x for x in range(n)]\\n\\t\\tll = re.sub(r'[^A-Za-z0-9]', ' ', l[1]).split()\\n\\t\\tfor j in ll:\\n\\t\\t\\t#print(j)\\n\\t\\t\\tif j in users:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tcan[i].remove( users.index(j) )\\n\\t\\t\\t\\texcept ValueError:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tif len(can[i]) == 0:\\n\\t\\t\\tok = False\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\twhile 1:\\n\\t\\tflag = True\\n\\t\\tfor i in range(m - 1):\\n\\t\\t\\tif (len(can[i]) == 0): ok = False\\n\\t\\t\\tif (len(can[i]) == 1 and can[i][0] in can[i+1]):\\n\\t\\t\\t\\tcan[i+1].remove( can[i][0] )\\n\\t\\t\\t\\tL[i][0] = users[ can[i][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\t\\tif (len(can[i + 1]) == 1 and can[i + 1][0] in can[i]):\\n\\t\\t\\t\\tcan[i].remove( can[i + 1][0] )\\n\\t\\t\\t\\tL[i + 1][0] = users[ can[i + 1][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\tif len(can[m - 1]) == 0: ok = False\\n\\t\\tif ok == False: break\\n\\t\\tif flag: break\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\tfor i in range(m):\\n\\t\\tif L[i][0] == '?':\\n\\t\\t\\tprint(users[ can[i][0] ], end='')\\n\\t\\t\\tif i < m - 1 and can[i][0] in can[i + 1]:\\n\\t\\t\\t\\tcan[i + 1].remove( can[i][0] )\\n\\t\\telse:\\n\\t\\t\\tprint(L[i][0], end='')\\n\\t\\tprint(':', L[i][1], sep = '');\\n\", \"3\\n\\nimport re\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    n = int(input())\\n    users = set(input().split(' '))\\n\\n    m = int(input())\\n    messages = []\\n\\n    guessed = [None for i in range(m)]\\n    denied = [set() for i in range(m)]\\n\\n    for i in range(m):\\n        mg = input()\\n        if not mg.startswith('?:'):\\n            guessed[i] = mg[:mg.find(':')]\\n\\n        mg = mg[mg.find(':'):]\\n        messages.append(mg)\\n\\n        m2 = mg + ' '\\n        for u in users:\\n            if re.search('[^a-zA-Z0-9]' + u + '[^a-zA-Z0-9]', m2):\\n                denied[i].add(u)\\n\\n    answer = True\\n\\n    for i in range(m):\\n        if guessed[i]:\\n            if i > 0: denied[i-1].add(guessed[i])\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n    for i in range(m):\\n        if guessed[i] in denied[i]:\\n            answer = False\\n\\n    #print(guessed)\\n    #print(denied)\\n\\n    changed = True\\n    while changed and answer:\\n        changed = False\\n        for i in range(m):\\n            if not guessed[i]:\\n                if len(users) - len(denied[i]) == 1:\\n                    changed = True\\n                    guessed[i] = (users - denied[i]).pop()\\n                    if i > 0: denied[i-1].add(guessed[i])\\n                    if i < m-1: denied[i+1].add(guessed[i])\\n                if len(users) == len(denied[i]):\\n                    answer = False\\n                    break\\n\\n    for i in range(m):\\n        if not guessed[i] and len(users) - len(denied[i]) >= 1:\\n            guessed[i] = (users - denied[i]).pop()\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n\\n    for i in guessed:\\n        if not i:\\n            answer = False\\n\\n    if not answer:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for i in range(m):\\n            print(guessed[i] + messages[i])\\n    \\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  t = int(input())\\n  for t in range(t):\\n    # input\\n    n = int(input())\\n    users = set(input().split())\\n    m = int(input())\\n    msg = []\\n    for i in range(m):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        alts = users - {x for x in re.split(r'[^A-Za-z0-9]+',text)}\\n      msg.append(dict(user=user, text=text, users=alts))\\n    # remove before and after\\n    for i in range(m-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i+1]['users'].difference_update(msg[i]['users'])\\n    for i in range(m-1,0,-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i-1]['users'].difference_update(msg[i]['users'])\\n    # compute answer\\n    last = ''\\n    impo = False\\n    for i in range(m):\\n      msg[i]['users'].discard(last)\\n      if len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n      last = next(iter(msg[i]['users']))\\n      msg[i]['user'] = last\\n    if impo:\\n      print('Impossible')\\n      continue\\n    for i in range(m):\\n      print(msg[i]['user']+':'+msg[i]['text'])\\n    '''\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(1,n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]'''\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    impo = False\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?': user = users.index(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split(r'[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n      if msg[i]['user'] == '?' and len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n    if impo:\\n      print('Impossible')\\n      continue\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      u = msg[i]['user']\\n      for j in range(n+1):\\n        if u != '?':\\n          if u != j and dp[i+1][u]: dp[i][j] = u\\n        else:\\n          for alt in msg[i]['users']:\\n            if alt != j and dp[i+1][alt]:\\n              dp[i][j] = alt\\n              break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        user = users.index(user)\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split('[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\"]",
        "difficulty": "interview",
        "input": "1\n1\nb\n1\nb:lala!\n",
        "output": "b:lala!\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/754/C"
    },
    {
        "id": 276,
        "task_id": 2124,
        "test_case_id": 5,
        "question": "Recently Vladik discovered a new entertainment — coding bots for social networks. He would like to use machine learning in his bots so now he want to prepare some learning data for them.\n\nAt first, he need to download t chats. Vladik coded a script which should have downloaded the chats, however, something went wrong. In particular, some of the messages have no information of their sender. It is known that if a person sends several messages in a row, they all are merged into a single message. It means that there could not be two or more messages in a row with the same sender. Moreover, a sender never mention himself in his messages.\n\nVladik wants to recover senders of all the messages so that each two neighboring messages will have different senders and no sender will mention himself in his messages.\n\nHe has no idea of how to do this, and asks you for help. Help Vladik to recover senders in each of the chats!\n\n\n-----Input-----\n\nThe first line contains single integer t (1 ≤ t ≤ 10) — the number of chats. The t chats follow. Each chat is given in the following format.\n\nThe first line of each chat description contains single integer n (1 ≤ n ≤ 100) — the number of users in the chat.\n\nThe next line contains n space-separated distinct usernames. Each username consists of lowercase and uppercase English letters and digits. The usernames can't start with a digit. Two usernames are different even if they differ only with letters' case. The length of username is positive and doesn't exceed 10 characters.\n\nThe next line contains single integer m (1 ≤ m ≤ 100) — the number of messages in the chat. The next m line contain the messages in the following formats, one per line:   <username>:<text> — the format of a message with known sender. The username should appear in the list of usernames of the chat.  <?>:<text> — the format of a message with unknown sender. \n\nThe text of a message can consist of lowercase and uppercase English letter, digits, characters '.' (dot), ',' (comma), '!' (exclamation mark), '?' (question mark) and ' ' (space). The text doesn't contain trailing spaces. The length of the text is positive and doesn't exceed 100 characters.\n\nWe say that a text mention a user if his username appears in the text as a word. In other words, the username appears in a such a position that the two characters before and after its appearance either do not exist or are not English letters or digits. For example, the text \"Vasya, masha13 and Kate!\" can mention users \"Vasya\", \"masha13\", \"and\" and \"Kate\", but not \"masha\".\n\nIt is guaranteed that in each chat no known sender mention himself in his messages and there are no two neighboring messages with the same known sender.\n\n\n-----Output-----\n\nPrint the information about the t chats in the following format:\n\nIf it is not possible to recover senders, print single line \"Impossible\" for this chat. Otherwise print m messages in the following format:\n\n<username>:<text>\n\nIf there are multiple answers, print any of them.\n\n\n-----Examples-----\nInput\n1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n\nOutput\nnetman: Hello, Vladik!\nVladik: Hi\n\nInput\n1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n\nOutput\nImpossible\n\nInput\n2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n\nOutput\nImpossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.",
        "solutions": "[\"def pr( name , lvl , dp , u , tot ): \\n    if lvl == 0:\\n        print(name + ':' + tot[lvl])\\n        return\\n\\n    pr( u[lvl][name] , lvl - 1 , dp , u , tot )\\n    print(name + ':' + tot[lvl])\\n\\ndef solve(): \\n    n = int(input())\\n    users = input().split()\\n    m = int(input())\\n    dp = [] \\n    u = []\\n    tot = [] \\n    for i in range( m ) : \\n        dp.append( set() ) \\n        u.append( {} ) \\n        line = input().split(':')\\n        sender = line[0]\\n        tot.append( line[1] ) \\n        line[1] = line[1].replace( '?' , ' ' )\\n        line[1] = line[1].replace( '.' , ' ' )\\n        line[1] = line[1].replace( ',' , ' ' )\\n        line[1] = line[1].replace( '!' , ' ' )\\n        mess = line[1].split()\\n\\n        if sender == '?' : \\n            if i != 0:\\n                for name in users:\\n                    for x in dp[i-1]: \\n                        if x != name and name not in mess:\\n                            dp[i].add( name ) \\n                            u[i][name] = x\\n            else : \\n                for name in users: \\n                    if name not in mess:\\n                        dp[i].add( name ) \\n        else: \\n            if i != 0: \\n                for x in dp[i-1]: \\n                    if x != sender: \\n                        dp[i].add( sender ) \\n                        u[i][sender] = x\\n            else: \\n                dp[i].add( sender )\\n        \\n        \\n    if dp[m-1]: \\n        pr( list(dp[m-1])[0] , m-1 , dp , u , tot )\\n    else: \\n        print(\\\"Impossible\\\")\\n\\n\\nt = int(input())\\nfor i in range( t ) : \\n    solve()\\n\", \"#!/usr/bin/env python3\\n\\nimport re\\n\\ndef cont(line, user):\\n\\treturn re.search('[^a-zA-Z0-9]' + user + '[^a-zA-Z0-9]', '_' + line + '_')\\n\\ntests = int(input())\\nfor test in range(tests):\\n\\tinput()\\n\\tusers = input().rstrip('\\\\n').split(' ')\\n\\tlinecnt = int(input())\\n\\tlines = [input().rstrip('\\\\n') for _ in range(linecnt)]\\n\\tlines = [x.split(':', maxsplit=1) for x in lines]\\n\\tposs = [[]] * linecnt\\n\\tfor i, (user, line) in enumerate(lines):\\n\\t\\tif user != '?':\\n\\t\\t\\tposs[i] = [user]\\n\\t\\telse:\\n\\t\\t\\tposs[i] = [u for u in users if not cont(line, u)]\\n\\n\\tel = list(enumerate(lines))\\n\\trel = list(reversed(el))\\n\\n\\tchanged = False\\n\\twhile True:\\n\\t\\tfor i, (user, line) in el:\\n\\t\\t\\tif i > 0 and len(poss[i-1]) == 1:\\n\\t\\t\\t\\tif poss[i-1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i-1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tfor i, (user, line) in rel:\\n\\t\\t\\tif i < linecnt - 1 and len(poss[i+1]) == 1:\\n\\t\\t\\t\\tif poss[i+1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i+1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tif not changed:\\n\\t\\t\\tbreak\\n\\t\\tchanged = False\\n\\n\\tif all(len(p) > 0 for p in poss):\\n\\t\\tfor i, p, (user, line) in zip(list(range(linecnt)), poss, lines):\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\tif poss[i-1][0] in p:\\n\\t\\t\\t\\t\\tp.remove(poss[i-1][0])\\n\\t\\t\\tprint(p[0] + ':' + line)\\n\\telse:\\n\\t\\tprint('Impossible')\\n\", \"import re\\nimport sys\\n\\n\\ndef g(cts, names):\\n  la = dict([(p, None) for p in names.difference(cts[0])])\\n  if len(la) == 0:\\n    return None\\n  d = [la]\\n  for i in range(1, len(cts)):\\n    prev = d[-1]\\n    la = dict()\\n    for p in names.difference(cts[i]):\\n      for n in prev.keys():\\n        if n != p:\\n          la[p] = n\\n          break\\n    if len(la) == 0:\\n      return None\\n    d.append(la)\\n  cur = list(d[-1].keys())[0]\\n  result = []\\n  for i in range(len(cts) - 1, -1, -1):\\n    result.append(cur)\\n    cur = d[i][cur]\\n  result.reverse()\\n  return result\\n\\n\\ndef solve():\\n  n = int(input())\\n  names = input().split()\\n  set_names = set(names)\\n\\n  def get_names(text):\\n    result = []\\n    for p in re.split(\\\"\\\\W+\\\", text):\\n      if p in set_names:\\n        result.append(p)\\n    return result\\n\\n  m = int(input())\\n  messages = []\\n  for i in range(m):\\n    s = input()\\n    colon = s.find(\\\":\\\")\\n    name = s[:colon]\\n    text = s[colon + 1:]\\n    if name == \\\"?\\\":\\n      name = None\\n    messages.append([name, text, get_names(text)])\\n\\n  for i in range(m):\\n    if messages[i][0]:\\n      continue\\n    j = i\\n    cts = []\\n    while j < m and not messages[j][0]:\\n      cts.append(set(messages[j][2]))\\n      j += 1\\n    if i > 0:\\n      cts[0].add(messages[i - 1][0])\\n    if j < m:\\n      cts[-1].add(messages[j][0])\\n    sb = g(cts, set_names)\\n    if not sb:\\n      return None\\n    for k in range(i, j):\\n      messages[k][0] = sb[k - i]\\n\\n  for p in messages:\\n    print(\\\"%s:%s\\\" % (p[0], p[1]))\\n  return True;\\n\\n\\ndef main():\\n  tests = int(input())\\n  for i in range(tests):\\n    if not solve():\\n      print(\\\"Impossible\\\")\\n  return 0\\n\\ndef __starting_point():\\n  return(main())\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"def dfs(ind):\\n    if ind > 0:\\n        if possible[ind][0] in possible[ind - 1]:\\n            possible[ind - 1].remove(possible[ind][0])\\n            if len(possible[ind - 1]) == 1:\\n                dfs(ind - 1)\\n    if ind < m - 1:\\n        if possible[ind][0] in possible[ind + 1]:\\n            possible[ind + 1].remove(possible[ind][0])\\n            if len(possible[ind + 1]) == 1:\\n                dfs(ind + 1)\\ndef Check(st, ms):\\n    for i in range(0, len(ms) - len(st) + 1):\\n        if st == ms[i:i+len(st)]:\\n            t = True\\n            if i > 0:\\n                if ms[i - 1] == ' ' or ms[i - 1] == '.' or ms[i - 1] == ',' or ms[i - 1] == '?' or ms[i - 1] == '!':\\n                    e = 0\\n                else:\\n                    t = False\\n            if i < len(ms) - len(st):\\n                if (ms[i + len(st)] == ' ' or ms[i + len(st)] == '.' or ms[i + len(st)] == ',' or ms[i + len(st)] == '?' or ms[i + len(st)] == '!'):\\n                    e = 0\\n                else:\\n                    t = False\\n            if t:\\n                return True\\n    return False\\nR = lambda:list(map(int, input().split(' ')))\\n#r, w = open(\\\"input.txt\\\", \\\"r\\\"), open(\\\"output.txt\\\", \\\"w\\\")\\nT = int(input())\\nwhile T:\\n    n = int(input())\\n    users = input().split(' ')\\n    m = int(input())\\n    possible = [[] for i in range(m)]\\n    sender, message = [], []\\n    for i in range(m):\\n        s = input().split(':')\\n        sender.append(s[0])\\n        message.append(s[1])\\n    for i in range(m):\\n        if sender[i] == '?':\\n            unallow = \\\" \\\"\\n            if i > 0 and len(possible[i - 1]) == 1:\\n                unallow = possible[i - 1][0]\\n            for j in users:\\n                if j == unallow:\\n                    continue\\n                done = Check(j, message[i])\\n                if not done:\\n                    possible[i].append(j)\\n        else:\\n            possible[i].append(sender[i])\\n    used = [0 for i in range(m)]\\n    for i in range(m):\\n        if len(possible[i]) == 1 and used[i] == 0:\\n            dfs(i)\\n    for i in range(m):\\n        if len(possible[i]) > 1:\\n            possible[i] = [possible[i][0]]\\n            dfs(i)\\n    done = False\\n    for i in possible:\\n        if len(i) == 0:\\n            print(\\\"Impossible\\\")\\n            done = True\\n            break\\n    if not done:\\n        for i in range(m):\\n            print(possible[i][0]+':'+message[i])\\n    T -= 1\\n\\n\\n    \\n\\n\", \"# Bartek Kostka\\n#  You are not prepared!\\n\\nimport re\\n\\n\\ndef go(i):\\n    nonlocal counter\\n    counter += 1\\n    if counter > 2000:\\n        return False\\n    if i == len(E):\\n        return True\\n    if E[i][0] != \\\"?\\\":\\n        prop = E[i][0]\\n        if i == 0 or W[i-1] != prop:\\n            W[i] = prop\\n            return go(i+1)\\n        else:\\n            return False\\n    for pos in E[i][1]:\\n        if i == 0 or W[i-1] != pos:\\n            W[i] = pos\\n            if go(i+1):\\n                return True\\n    return False\\n\\n\\nt = int(input())\\nfor ttt in range(t):\\n    n = int(input())\\n    users = input().split(\\\" \\\")\\n    users_set = set(users)\\n    m = int(input())\\n    E = []\\n    S = []\\n    counter = 0\\n    W = [\\\"\\\" for x in range(m)]\\n    for mmm in range(m):\\n        line = input()\\n        (user, sentence) = line.split(\\\":\\\")\\n        S.append((user, sentence))\\n        words = [x.strip() for x in re.split('\\\\W+', sentence)]\\n        mentions = list([x for x in words if x in users_set])\\n        E.append([user, set(mentions)])\\n    for i in range(len(E)-1):\\n        if E[i][0] != \\\"?\\\":\\n            E[i+1][1].add(E[i][0])\\n    for i in range(1, len(E)):\\n        if E[i][0] != \\\"?\\\":\\n            E[i-1][1].add(E[i][0])\\n    for i in range(len(E)):\\n        E[i][1] = E[i][1].symmetric_difference(users_set)\\n    if go(0):\\n        for i in range(len(E)):\\n            print(str(W[i])+\\\":\\\"+str(S[i][1]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import re\\nimport copy\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\tn = int(input())\\n\\tsenderList = []\\n\\tfor x in input().split():\\n\\t\\tsenderList.append(x)\\n\\n\\tm = int(input())\\n\\tmsg = [None] * m\\n\\tresSenders = [None] * m\\n\\tpossibleSenders = []\\n\\tfor i in range(m):\\n\\t\\tpossibleSenders.append(copy.copy(senderList))\\n\\n\\tfor i in range(m):\\n\\t\\tline = input()\\n\\t\\tblocks = re.findall(r\\\"[\\\\w]+\\\", line)\\n\\n\\t\\tfor x in blocks:\\n\\t\\t\\ttry:\\n\\t\\t\\t\\tpossibleSenders[i].remove(x)\\n\\t\\t\\texcept:\\n\\t\\t\\t\\tpass\\n\\n\\t\\tif line[0] != '?':\\n\\t\\t\\tresSenders[i] = blocks[0]\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tmsg[i] = line[line.find(\\\":\\\")+1:]\\n\\n\\tresolved = False\\n\\twhile True:\\n\\t\\tdone = True\\n\\t\\tjustPick = True\\n\\t\\tfor i in range(m):\\n\\t\\t\\tif resSenders[i] != None:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tif len(possibleSenders[i]) == 0:\\n\\t\\t\\t\\tdone = True\\n\\t\\t\\t\\tresolved = True\\n\\t\\t\\t\\tprint(\\\"Impossible\\\")\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tif len(possibleSenders[i]) == 1:\\n\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\tif i > 0:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tjustPick = False\\n\\t\\t\\telse:\\n\\t\\t\\t\\tdone = False\\n\\t\\tif done:\\n\\t\\t\\tbreak\\n\\t\\tif justPick:\\n\\t\\t\\tfor i in range(m):\\n\\t\\t\\t\\tif resSenders[i] == None:\\n\\t\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\tbreak\\n\\n\\tif not resolved:\\n\\t\\tfor i in range(m):\\n\\t\\t\\tprint(resSenders[i] + \\\":\\\" + msg[i])\", \"import re\\n\\n\\ndef foo():\\n    n = int(input())\\n    who = input().split()\\n    m = int(input())\\n    msg = []\\n\\n    l = []\\n\\n    for num in range(m):\\n        a, t = input().split(':')\\n        msg.append(t)\\n\\n        st = set()\\n\\n        if a != '?':\\n            st.add(a)\\n        else:\\n\\n            names = re.split('\\\\W+', t)\\n\\n            for w in who:\\n                if not w in names:\\n                    st.add(w)\\n\\n        l.append(st)\\n\\n    d2 = []\\n\\n    for num in range(1, m):\\n        d = {}\\n        for w1 in l[num]:\\n            for w2 in l[num - 1]:\\n                if w1 != w2:\\n                    d[w1] = w2\\n                    break\\n\\n        l[num] = [x for x in d]\\n\\n        d2.append(d)\\n\\n    curr = None\\n\\n    for w in l[m - 1]:\\n        curr = w\\n\\n    res = [curr]\\n\\n    for num in list(reversed(list(range(1, m)))):\\n        curr = d2[num - 1].get(curr, None)  # d2[num - 1][curr]\\n        res.append(curr)\\n\\n    res = list(reversed(res))\\n\\n    if None in res:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for num in range(m):\\n            print(res[num] + ':' + msg[num])\\n\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    foo()\\n\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\", \"import re\\n\\nt = int(input())\\n\\ndelimiters = \\\"?\\\", \\\".\\\", \\\" \\\", \\\",\\\", \\\"!\\\", \\\":\\\"\\nregexPattern = '|'.join(map(re.escape, delimiters))\\n\\nfor i in range(t):\\n  n = int(input())\\n  usernames = {x for x in str.split(input(), ' ')}\\n  m = int(input())\\n  possibilities = []\\n  for j in range(m):\\n    possibilities.append({x for x in usernames})\\n  messages = []\\n  for j in range(m):\\n    messages.append(input())\\n\\n  for j in range(m):\\n    if (messages[j][0] != '?'):\\n      messageSplit = re.split(':', messages[j])\\n      possibilities[j] = {messageSplit[0]}\\n    else:\\n      messageSplit = re.split(regexPattern, messages[j])\\n      for token in messageSplit:\\n        if token in usernames:\\n          possibilities[j] = possibilities[j] - {token}\\n  changed = True\\n  while changed:\\n    changed = False\\n    for j in range(m):\\n      if len(possibilities[j]) == 1:\\n        (poss,) = possibilities[j]\\n        if j < m-1 and poss in possibilities[j+1]:\\n          changed = True\\n          possibilities[j+1] = possibilities[j+1] - possibilities[j]\\n        if j > 0 and poss in possibilities[j-1]:\\n          changed = True\\n          possibilities[j-1] = possibilities[j-1] - possibilities[j]\\n  worked = True\\n  for j in range(m):\\n    if len(possibilities[j]) == 0:\\n      worked = False\\n      \\n  if not worked:\\n    print(\\\"Impossible\\\")\\n  else :\\n    for j in range(m):\\n      poss = next(iter(possibilities[j]))\\n      if (messages[j][0] == '?'):\\n        print(poss + messages[j][1:])\\n      else:\\n        print(messages[j])\\n      if (j < m-1):\\n        possibilities[j+1] = possibilities[j+1] - {poss}\\n\\n\", \"import re\\n\\ndef add_used(id, s, used):\\n    used[id-1].add(s)\\n    used[id].add(s)\\n    used[id+1].add(s)\\n\\ntcase = int(input())\\nfor cas in range(tcase):\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    \\n    res = True\\n    ans = [''] * (m+1)\\n    msg = [''] * (m+1)\\n    used = [set() for i in range(m+1)]\\n    \\n    for i in range(m):\\n        ans[i], msg[i] = input().split(':')\\n        if ans[i] != '?':\\n            add_used(i, ans[i], used)\\n        \\n        mentioned = re.split('\\\\W+', msg[i])\\n        for s in mentioned:\\n            if s in names:\\n                used[i].add(s)\\n    \\n    for i in range(m):\\n        if ans[i] == '?' and len(used[i]) == n-1:\\n            ans[i] = list(set(names) - used[i])[0]\\n            add_used(i, ans[i], used)\\n    \\n    for i in range(m):\\n        if ans[i] == '?':\\n            if len(used[i]) == n:\\n                res = False\\n            else:\\n                ans[i] = list(set(names) - used[i])[0]\\n                add_used(i, ans[i], used)\\n    \\n    for i in range(m-1):\\n        if ans[i] == ans[i+1]:\\n            res = False\\n    \\n    if res:\\n        for i in range(m):\\n            print(ans[i] + ':' + msg[i])\\n    else:\\n        print('Impossible')\", \"from sys import stdin\\n\\nt = int(stdin.readline())\\n\\ndef c(l,r,msg):\\n    f = True\\n    if l > 0 and not msg[l-1].isdigit() and not msg[l-1].isalpha():\\n        pass\\n    elif l == 0:\\n        pass\\n    else:\\n        f = False\\n\\n    if r == len(msg) - 1:\\n        pass\\n    elif not msg[r+1].isdigit() and not msg[r+1].isalpha():\\n        pass\\n    else:\\n        f = False\\n\\n    return f\\n\\n\\ndef dd(i):\\n    if i - 1 in no_author_set:\\n        authors[i-1].discard(authors[i])\\n        if len(authors[i-1]) == 1:\\n            no_author_set.discard(i-1)\\n            for d in authors[i-1]:\\n                authors[i-1] = d\\n                break\\n            dd(i-1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    if i + 1 in no_author_set:\\n        authors[i+1].discard(authors[i])\\n        if len(authors[i+1]) == 1:\\n            no_author_set.discard(i+1)\\n            for d in authors[i+1]:\\n                authors[i+1] = d\\n                break\\n            dd(i+1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    return True\\n\\ntry:\\n    answers = []\\n    for i in range(t):\\n        ans = True\\n        n = int(stdin.readline())\\n        names = set(stdin.readline().strip().split())\\n        m = int(stdin.readline())\\n        authors = list()\\n        messages = list()\\n        no_author = list()\\n        no_author_set = set()\\n        for j in range(m):\\n            line = stdin.readline().strip()\\n            author, msg = line.split(':')\\n            messages.append(msg)\\n            if author == '?':\\n                no_author.append(j)\\n                no_author_set.add(j)\\n                a_set = set()\\n                for name in names:\\n                    l = msg.find(name)\\n                    while l != -1:\\n                        if c(l,l+len(name)-1, msg):\\n                            a_set.add(name)\\n                        l = msg.find(name,l+len(name))\\n                authors.append(names-a_set)\\n                if j-1 not in no_author_set:\\n                    authors[j].discard(authors[j-1])\\n            else:\\n                authors.append(author)\\n                if j - 1 in no_author_set:\\n                    authors[j-1].discard(author)\\n\\n        for j in no_author:\\n            if j in no_author_set:\\n                if len(authors[j]) == 1:\\n                    no_author_set.discard(j)\\n                    for d in authors[j]:\\n                        authors[j] = d\\n                        break\\n                    if not dd(j):\\n                        ans = False\\n                elif len(authors[j]) == 0:\\n                    ans = False\\n\\n        no_author = list()\\n        for j in no_author_set:\\n            no_author.append(j)\\n        no_author.sort()\\n        for i in no_author:\\n            for d in authors[i]:\\n                authors[i] = d\\n                break\\n            if i + 1 in no_author_set:\\n                authors[i+1].discard(authors[i])\\n        if not ans:\\n            answers.append('Impossible')\\n        else:\\n            for j in range(m):\\n                answers.append('%s:%s'%(authors[j],messages[j]))\\nexcept Exception as e:\\n    print(e)\\n\\nfor m in answers:\\n    print(m)\", \"import re\\nt = int(input())\\nfor _ in range(t):\\n    n = int(input())\\n    names = set(input().split())\\n    m = int(input())\\n    dp = []\\n    a = []\\n    for i in range(m): a.append(input())\\n    for i in range(m):\\n        sender, msg = a[i].split(':')\\n        ls = set([_f for _f in re.split('\\\\W+',msg) if _f])\\n        dp.append((names if sender == '?' else set([sender])) - ls)\\n        if i and len(dp[i-1]) == 1: dp[i] -= dp[i-1]\\n        # print(dp[i]);\\n    if any([len(p) == 0 for p in dp]): print(\\\"Impossible\\\")\\n    else:\\n        res = []\\n        for i in reversed(list(range(m))):\\n            res.append(dp[i].pop())\\n            if i > 0: dp[i-1].discard(res[-1])\\n        for i in range(m): \\n            sender, msg = a[i].split(':')\\n            sender = res[m-i-1]\\n            print(sender+':'+msg)\\n\", \"import re\\n\\nt = int( input() )\\nfor z in range(t):\\n\\tn = int( input() )\\n\\tusers = input().split();\\n\\t#print(users)\\n\\tm = int( input() )\\n\\tcan = [ [] for i in range(m) ]\\n\\tok = True\\n\\tL = []\\n\\tfor i in range(m):\\n\\t\\tl = input().split(':')\\n\\t\\t#print(l)\\n\\t\\tL += [l]\\n\\t\\tif l[0] != '?':\\n\\t\\t\\tcan[i] = [users.index(l[0])]\\n\\t\\telse:\\n\\t\\t\\tcan[i] = [x for x in range(n)]\\n\\t\\tll = re.sub(r'[^A-Za-z0-9]', ' ', l[1]).split()\\n\\t\\tfor j in ll:\\n\\t\\t\\t#print(j)\\n\\t\\t\\tif j in users:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tcan[i].remove( users.index(j) )\\n\\t\\t\\t\\texcept ValueError:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tif len(can[i]) == 0:\\n\\t\\t\\tok = False\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\twhile 1:\\n\\t\\tflag = True\\n\\t\\tfor i in range(m - 1):\\n\\t\\t\\tif (len(can[i]) == 0): ok = False\\n\\t\\t\\tif (len(can[i]) == 1 and can[i][0] in can[i+1]):\\n\\t\\t\\t\\tcan[i+1].remove( can[i][0] )\\n\\t\\t\\t\\tL[i][0] = users[ can[i][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\t\\tif (len(can[i + 1]) == 1 and can[i + 1][0] in can[i]):\\n\\t\\t\\t\\tcan[i].remove( can[i + 1][0] )\\n\\t\\t\\t\\tL[i + 1][0] = users[ can[i + 1][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\tif len(can[m - 1]) == 0: ok = False\\n\\t\\tif ok == False: break\\n\\t\\tif flag: break\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\tfor i in range(m):\\n\\t\\tif L[i][0] == '?':\\n\\t\\t\\tprint(users[ can[i][0] ], end='')\\n\\t\\t\\tif i < m - 1 and can[i][0] in can[i + 1]:\\n\\t\\t\\t\\tcan[i + 1].remove( can[i][0] )\\n\\t\\telse:\\n\\t\\t\\tprint(L[i][0], end='')\\n\\t\\tprint(':', L[i][1], sep = '');\\n\", \"3\\n\\nimport re\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    n = int(input())\\n    users = set(input().split(' '))\\n\\n    m = int(input())\\n    messages = []\\n\\n    guessed = [None for i in range(m)]\\n    denied = [set() for i in range(m)]\\n\\n    for i in range(m):\\n        mg = input()\\n        if not mg.startswith('?:'):\\n            guessed[i] = mg[:mg.find(':')]\\n\\n        mg = mg[mg.find(':'):]\\n        messages.append(mg)\\n\\n        m2 = mg + ' '\\n        for u in users:\\n            if re.search('[^a-zA-Z0-9]' + u + '[^a-zA-Z0-9]', m2):\\n                denied[i].add(u)\\n\\n    answer = True\\n\\n    for i in range(m):\\n        if guessed[i]:\\n            if i > 0: denied[i-1].add(guessed[i])\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n    for i in range(m):\\n        if guessed[i] in denied[i]:\\n            answer = False\\n\\n    #print(guessed)\\n    #print(denied)\\n\\n    changed = True\\n    while changed and answer:\\n        changed = False\\n        for i in range(m):\\n            if not guessed[i]:\\n                if len(users) - len(denied[i]) == 1:\\n                    changed = True\\n                    guessed[i] = (users - denied[i]).pop()\\n                    if i > 0: denied[i-1].add(guessed[i])\\n                    if i < m-1: denied[i+1].add(guessed[i])\\n                if len(users) == len(denied[i]):\\n                    answer = False\\n                    break\\n\\n    for i in range(m):\\n        if not guessed[i] and len(users) - len(denied[i]) >= 1:\\n            guessed[i] = (users - denied[i]).pop()\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n\\n    for i in guessed:\\n        if not i:\\n            answer = False\\n\\n    if not answer:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for i in range(m):\\n            print(guessed[i] + messages[i])\\n    \\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  t = int(input())\\n  for t in range(t):\\n    # input\\n    n = int(input())\\n    users = set(input().split())\\n    m = int(input())\\n    msg = []\\n    for i in range(m):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        alts = users - {x for x in re.split(r'[^A-Za-z0-9]+',text)}\\n      msg.append(dict(user=user, text=text, users=alts))\\n    # remove before and after\\n    for i in range(m-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i+1]['users'].difference_update(msg[i]['users'])\\n    for i in range(m-1,0,-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i-1]['users'].difference_update(msg[i]['users'])\\n    # compute answer\\n    last = ''\\n    impo = False\\n    for i in range(m):\\n      msg[i]['users'].discard(last)\\n      if len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n      last = next(iter(msg[i]['users']))\\n      msg[i]['user'] = last\\n    if impo:\\n      print('Impossible')\\n      continue\\n    for i in range(m):\\n      print(msg[i]['user']+':'+msg[i]['text'])\\n    '''\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(1,n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]'''\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    impo = False\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?': user = users.index(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split(r'[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n      if msg[i]['user'] == '?' and len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n    if impo:\\n      print('Impossible')\\n      continue\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      u = msg[i]['user']\\n      for j in range(n+1):\\n        if u != '?':\\n          if u != j and dp[i+1][u]: dp[i][j] = u\\n        else:\\n          for alt in msg[i]['users']:\\n            if alt != j and dp[i+1][alt]:\\n              dp[i][j] = alt\\n              break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        user = users.index(user)\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split('[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\"]",
        "difficulty": "interview",
        "input": "1\n1\nb\n1\n?:lala b!\n",
        "output": "Impossible\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/754/C"
    },
    {
        "id": 277,
        "task_id": 2124,
        "test_case_id": 6,
        "question": "Recently Vladik discovered a new entertainment — coding bots for social networks. He would like to use machine learning in his bots so now he want to prepare some learning data for them.\n\nAt first, he need to download t chats. Vladik coded a script which should have downloaded the chats, however, something went wrong. In particular, some of the messages have no information of their sender. It is known that if a person sends several messages in a row, they all are merged into a single message. It means that there could not be two or more messages in a row with the same sender. Moreover, a sender never mention himself in his messages.\n\nVladik wants to recover senders of all the messages so that each two neighboring messages will have different senders and no sender will mention himself in his messages.\n\nHe has no idea of how to do this, and asks you for help. Help Vladik to recover senders in each of the chats!\n\n\n-----Input-----\n\nThe first line contains single integer t (1 ≤ t ≤ 10) — the number of chats. The t chats follow. Each chat is given in the following format.\n\nThe first line of each chat description contains single integer n (1 ≤ n ≤ 100) — the number of users in the chat.\n\nThe next line contains n space-separated distinct usernames. Each username consists of lowercase and uppercase English letters and digits. The usernames can't start with a digit. Two usernames are different even if they differ only with letters' case. The length of username is positive and doesn't exceed 10 characters.\n\nThe next line contains single integer m (1 ≤ m ≤ 100) — the number of messages in the chat. The next m line contain the messages in the following formats, one per line:   <username>:<text> — the format of a message with known sender. The username should appear in the list of usernames of the chat.  <?>:<text> — the format of a message with unknown sender. \n\nThe text of a message can consist of lowercase and uppercase English letter, digits, characters '.' (dot), ',' (comma), '!' (exclamation mark), '?' (question mark) and ' ' (space). The text doesn't contain trailing spaces. The length of the text is positive and doesn't exceed 100 characters.\n\nWe say that a text mention a user if his username appears in the text as a word. In other words, the username appears in a such a position that the two characters before and after its appearance either do not exist or are not English letters or digits. For example, the text \"Vasya, masha13 and Kate!\" can mention users \"Vasya\", \"masha13\", \"and\" and \"Kate\", but not \"masha\".\n\nIt is guaranteed that in each chat no known sender mention himself in his messages and there are no two neighboring messages with the same known sender.\n\n\n-----Output-----\n\nPrint the information about the t chats in the following format:\n\nIf it is not possible to recover senders, print single line \"Impossible\" for this chat. Otherwise print m messages in the following format:\n\n<username>:<text>\n\nIf there are multiple answers, print any of them.\n\n\n-----Examples-----\nInput\n1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n\nOutput\nnetman: Hello, Vladik!\nVladik: Hi\n\nInput\n1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n\nOutput\nImpossible\n\nInput\n2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n\nOutput\nImpossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.",
        "solutions": "[\"def pr( name , lvl , dp , u , tot ): \\n    if lvl == 0:\\n        print(name + ':' + tot[lvl])\\n        return\\n\\n    pr( u[lvl][name] , lvl - 1 , dp , u , tot )\\n    print(name + ':' + tot[lvl])\\n\\ndef solve(): \\n    n = int(input())\\n    users = input().split()\\n    m = int(input())\\n    dp = [] \\n    u = []\\n    tot = [] \\n    for i in range( m ) : \\n        dp.append( set() ) \\n        u.append( {} ) \\n        line = input().split(':')\\n        sender = line[0]\\n        tot.append( line[1] ) \\n        line[1] = line[1].replace( '?' , ' ' )\\n        line[1] = line[1].replace( '.' , ' ' )\\n        line[1] = line[1].replace( ',' , ' ' )\\n        line[1] = line[1].replace( '!' , ' ' )\\n        mess = line[1].split()\\n\\n        if sender == '?' : \\n            if i != 0:\\n                for name in users:\\n                    for x in dp[i-1]: \\n                        if x != name and name not in mess:\\n                            dp[i].add( name ) \\n                            u[i][name] = x\\n            else : \\n                for name in users: \\n                    if name not in mess:\\n                        dp[i].add( name ) \\n        else: \\n            if i != 0: \\n                for x in dp[i-1]: \\n                    if x != sender: \\n                        dp[i].add( sender ) \\n                        u[i][sender] = x\\n            else: \\n                dp[i].add( sender )\\n        \\n        \\n    if dp[m-1]: \\n        pr( list(dp[m-1])[0] , m-1 , dp , u , tot )\\n    else: \\n        print(\\\"Impossible\\\")\\n\\n\\nt = int(input())\\nfor i in range( t ) : \\n    solve()\\n\", \"#!/usr/bin/env python3\\n\\nimport re\\n\\ndef cont(line, user):\\n\\treturn re.search('[^a-zA-Z0-9]' + user + '[^a-zA-Z0-9]', '_' + line + '_')\\n\\ntests = int(input())\\nfor test in range(tests):\\n\\tinput()\\n\\tusers = input().rstrip('\\\\n').split(' ')\\n\\tlinecnt = int(input())\\n\\tlines = [input().rstrip('\\\\n') for _ in range(linecnt)]\\n\\tlines = [x.split(':', maxsplit=1) for x in lines]\\n\\tposs = [[]] * linecnt\\n\\tfor i, (user, line) in enumerate(lines):\\n\\t\\tif user != '?':\\n\\t\\t\\tposs[i] = [user]\\n\\t\\telse:\\n\\t\\t\\tposs[i] = [u for u in users if not cont(line, u)]\\n\\n\\tel = list(enumerate(lines))\\n\\trel = list(reversed(el))\\n\\n\\tchanged = False\\n\\twhile True:\\n\\t\\tfor i, (user, line) in el:\\n\\t\\t\\tif i > 0 and len(poss[i-1]) == 1:\\n\\t\\t\\t\\tif poss[i-1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i-1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tfor i, (user, line) in rel:\\n\\t\\t\\tif i < linecnt - 1 and len(poss[i+1]) == 1:\\n\\t\\t\\t\\tif poss[i+1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i+1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tif not changed:\\n\\t\\t\\tbreak\\n\\t\\tchanged = False\\n\\n\\tif all(len(p) > 0 for p in poss):\\n\\t\\tfor i, p, (user, line) in zip(list(range(linecnt)), poss, lines):\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\tif poss[i-1][0] in p:\\n\\t\\t\\t\\t\\tp.remove(poss[i-1][0])\\n\\t\\t\\tprint(p[0] + ':' + line)\\n\\telse:\\n\\t\\tprint('Impossible')\\n\", \"import re\\nimport sys\\n\\n\\ndef g(cts, names):\\n  la = dict([(p, None) for p in names.difference(cts[0])])\\n  if len(la) == 0:\\n    return None\\n  d = [la]\\n  for i in range(1, len(cts)):\\n    prev = d[-1]\\n    la = dict()\\n    for p in names.difference(cts[i]):\\n      for n in prev.keys():\\n        if n != p:\\n          la[p] = n\\n          break\\n    if len(la) == 0:\\n      return None\\n    d.append(la)\\n  cur = list(d[-1].keys())[0]\\n  result = []\\n  for i in range(len(cts) - 1, -1, -1):\\n    result.append(cur)\\n    cur = d[i][cur]\\n  result.reverse()\\n  return result\\n\\n\\ndef solve():\\n  n = int(input())\\n  names = input().split()\\n  set_names = set(names)\\n\\n  def get_names(text):\\n    result = []\\n    for p in re.split(\\\"\\\\W+\\\", text):\\n      if p in set_names:\\n        result.append(p)\\n    return result\\n\\n  m = int(input())\\n  messages = []\\n  for i in range(m):\\n    s = input()\\n    colon = s.find(\\\":\\\")\\n    name = s[:colon]\\n    text = s[colon + 1:]\\n    if name == \\\"?\\\":\\n      name = None\\n    messages.append([name, text, get_names(text)])\\n\\n  for i in range(m):\\n    if messages[i][0]:\\n      continue\\n    j = i\\n    cts = []\\n    while j < m and not messages[j][0]:\\n      cts.append(set(messages[j][2]))\\n      j += 1\\n    if i > 0:\\n      cts[0].add(messages[i - 1][0])\\n    if j < m:\\n      cts[-1].add(messages[j][0])\\n    sb = g(cts, set_names)\\n    if not sb:\\n      return None\\n    for k in range(i, j):\\n      messages[k][0] = sb[k - i]\\n\\n  for p in messages:\\n    print(\\\"%s:%s\\\" % (p[0], p[1]))\\n  return True;\\n\\n\\ndef main():\\n  tests = int(input())\\n  for i in range(tests):\\n    if not solve():\\n      print(\\\"Impossible\\\")\\n  return 0\\n\\ndef __starting_point():\\n  return(main())\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"def dfs(ind):\\n    if ind > 0:\\n        if possible[ind][0] in possible[ind - 1]:\\n            possible[ind - 1].remove(possible[ind][0])\\n            if len(possible[ind - 1]) == 1:\\n                dfs(ind - 1)\\n    if ind < m - 1:\\n        if possible[ind][0] in possible[ind + 1]:\\n            possible[ind + 1].remove(possible[ind][0])\\n            if len(possible[ind + 1]) == 1:\\n                dfs(ind + 1)\\ndef Check(st, ms):\\n    for i in range(0, len(ms) - len(st) + 1):\\n        if st == ms[i:i+len(st)]:\\n            t = True\\n            if i > 0:\\n                if ms[i - 1] == ' ' or ms[i - 1] == '.' or ms[i - 1] == ',' or ms[i - 1] == '?' or ms[i - 1] == '!':\\n                    e = 0\\n                else:\\n                    t = False\\n            if i < len(ms) - len(st):\\n                if (ms[i + len(st)] == ' ' or ms[i + len(st)] == '.' or ms[i + len(st)] == ',' or ms[i + len(st)] == '?' or ms[i + len(st)] == '!'):\\n                    e = 0\\n                else:\\n                    t = False\\n            if t:\\n                return True\\n    return False\\nR = lambda:list(map(int, input().split(' ')))\\n#r, w = open(\\\"input.txt\\\", \\\"r\\\"), open(\\\"output.txt\\\", \\\"w\\\")\\nT = int(input())\\nwhile T:\\n    n = int(input())\\n    users = input().split(' ')\\n    m = int(input())\\n    possible = [[] for i in range(m)]\\n    sender, message = [], []\\n    for i in range(m):\\n        s = input().split(':')\\n        sender.append(s[0])\\n        message.append(s[1])\\n    for i in range(m):\\n        if sender[i] == '?':\\n            unallow = \\\" \\\"\\n            if i > 0 and len(possible[i - 1]) == 1:\\n                unallow = possible[i - 1][0]\\n            for j in users:\\n                if j == unallow:\\n                    continue\\n                done = Check(j, message[i])\\n                if not done:\\n                    possible[i].append(j)\\n        else:\\n            possible[i].append(sender[i])\\n    used = [0 for i in range(m)]\\n    for i in range(m):\\n        if len(possible[i]) == 1 and used[i] == 0:\\n            dfs(i)\\n    for i in range(m):\\n        if len(possible[i]) > 1:\\n            possible[i] = [possible[i][0]]\\n            dfs(i)\\n    done = False\\n    for i in possible:\\n        if len(i) == 0:\\n            print(\\\"Impossible\\\")\\n            done = True\\n            break\\n    if not done:\\n        for i in range(m):\\n            print(possible[i][0]+':'+message[i])\\n    T -= 1\\n\\n\\n    \\n\\n\", \"# Bartek Kostka\\n#  You are not prepared!\\n\\nimport re\\n\\n\\ndef go(i):\\n    nonlocal counter\\n    counter += 1\\n    if counter > 2000:\\n        return False\\n    if i == len(E):\\n        return True\\n    if E[i][0] != \\\"?\\\":\\n        prop = E[i][0]\\n        if i == 0 or W[i-1] != prop:\\n            W[i] = prop\\n            return go(i+1)\\n        else:\\n            return False\\n    for pos in E[i][1]:\\n        if i == 0 or W[i-1] != pos:\\n            W[i] = pos\\n            if go(i+1):\\n                return True\\n    return False\\n\\n\\nt = int(input())\\nfor ttt in range(t):\\n    n = int(input())\\n    users = input().split(\\\" \\\")\\n    users_set = set(users)\\n    m = int(input())\\n    E = []\\n    S = []\\n    counter = 0\\n    W = [\\\"\\\" for x in range(m)]\\n    for mmm in range(m):\\n        line = input()\\n        (user, sentence) = line.split(\\\":\\\")\\n        S.append((user, sentence))\\n        words = [x.strip() for x in re.split('\\\\W+', sentence)]\\n        mentions = list([x for x in words if x in users_set])\\n        E.append([user, set(mentions)])\\n    for i in range(len(E)-1):\\n        if E[i][0] != \\\"?\\\":\\n            E[i+1][1].add(E[i][0])\\n    for i in range(1, len(E)):\\n        if E[i][0] != \\\"?\\\":\\n            E[i-1][1].add(E[i][0])\\n    for i in range(len(E)):\\n        E[i][1] = E[i][1].symmetric_difference(users_set)\\n    if go(0):\\n        for i in range(len(E)):\\n            print(str(W[i])+\\\":\\\"+str(S[i][1]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import re\\nimport copy\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\tn = int(input())\\n\\tsenderList = []\\n\\tfor x in input().split():\\n\\t\\tsenderList.append(x)\\n\\n\\tm = int(input())\\n\\tmsg = [None] * m\\n\\tresSenders = [None] * m\\n\\tpossibleSenders = []\\n\\tfor i in range(m):\\n\\t\\tpossibleSenders.append(copy.copy(senderList))\\n\\n\\tfor i in range(m):\\n\\t\\tline = input()\\n\\t\\tblocks = re.findall(r\\\"[\\\\w]+\\\", line)\\n\\n\\t\\tfor x in blocks:\\n\\t\\t\\ttry:\\n\\t\\t\\t\\tpossibleSenders[i].remove(x)\\n\\t\\t\\texcept:\\n\\t\\t\\t\\tpass\\n\\n\\t\\tif line[0] != '?':\\n\\t\\t\\tresSenders[i] = blocks[0]\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tmsg[i] = line[line.find(\\\":\\\")+1:]\\n\\n\\tresolved = False\\n\\twhile True:\\n\\t\\tdone = True\\n\\t\\tjustPick = True\\n\\t\\tfor i in range(m):\\n\\t\\t\\tif resSenders[i] != None:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tif len(possibleSenders[i]) == 0:\\n\\t\\t\\t\\tdone = True\\n\\t\\t\\t\\tresolved = True\\n\\t\\t\\t\\tprint(\\\"Impossible\\\")\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tif len(possibleSenders[i]) == 1:\\n\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\tif i > 0:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tjustPick = False\\n\\t\\t\\telse:\\n\\t\\t\\t\\tdone = False\\n\\t\\tif done:\\n\\t\\t\\tbreak\\n\\t\\tif justPick:\\n\\t\\t\\tfor i in range(m):\\n\\t\\t\\t\\tif resSenders[i] == None:\\n\\t\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\tbreak\\n\\n\\tif not resolved:\\n\\t\\tfor i in range(m):\\n\\t\\t\\tprint(resSenders[i] + \\\":\\\" + msg[i])\", \"import re\\n\\n\\ndef foo():\\n    n = int(input())\\n    who = input().split()\\n    m = int(input())\\n    msg = []\\n\\n    l = []\\n\\n    for num in range(m):\\n        a, t = input().split(':')\\n        msg.append(t)\\n\\n        st = set()\\n\\n        if a != '?':\\n            st.add(a)\\n        else:\\n\\n            names = re.split('\\\\W+', t)\\n\\n            for w in who:\\n                if not w in names:\\n                    st.add(w)\\n\\n        l.append(st)\\n\\n    d2 = []\\n\\n    for num in range(1, m):\\n        d = {}\\n        for w1 in l[num]:\\n            for w2 in l[num - 1]:\\n                if w1 != w2:\\n                    d[w1] = w2\\n                    break\\n\\n        l[num] = [x for x in d]\\n\\n        d2.append(d)\\n\\n    curr = None\\n\\n    for w in l[m - 1]:\\n        curr = w\\n\\n    res = [curr]\\n\\n    for num in list(reversed(list(range(1, m)))):\\n        curr = d2[num - 1].get(curr, None)  # d2[num - 1][curr]\\n        res.append(curr)\\n\\n    res = list(reversed(res))\\n\\n    if None in res:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for num in range(m):\\n            print(res[num] + ':' + msg[num])\\n\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    foo()\\n\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\", \"import re\\n\\nt = int(input())\\n\\ndelimiters = \\\"?\\\", \\\".\\\", \\\" \\\", \\\",\\\", \\\"!\\\", \\\":\\\"\\nregexPattern = '|'.join(map(re.escape, delimiters))\\n\\nfor i in range(t):\\n  n = int(input())\\n  usernames = {x for x in str.split(input(), ' ')}\\n  m = int(input())\\n  possibilities = []\\n  for j in range(m):\\n    possibilities.append({x for x in usernames})\\n  messages = []\\n  for j in range(m):\\n    messages.append(input())\\n\\n  for j in range(m):\\n    if (messages[j][0] != '?'):\\n      messageSplit = re.split(':', messages[j])\\n      possibilities[j] = {messageSplit[0]}\\n    else:\\n      messageSplit = re.split(regexPattern, messages[j])\\n      for token in messageSplit:\\n        if token in usernames:\\n          possibilities[j] = possibilities[j] - {token}\\n  changed = True\\n  while changed:\\n    changed = False\\n    for j in range(m):\\n      if len(possibilities[j]) == 1:\\n        (poss,) = possibilities[j]\\n        if j < m-1 and poss in possibilities[j+1]:\\n          changed = True\\n          possibilities[j+1] = possibilities[j+1] - possibilities[j]\\n        if j > 0 and poss in possibilities[j-1]:\\n          changed = True\\n          possibilities[j-1] = possibilities[j-1] - possibilities[j]\\n  worked = True\\n  for j in range(m):\\n    if len(possibilities[j]) == 0:\\n      worked = False\\n      \\n  if not worked:\\n    print(\\\"Impossible\\\")\\n  else :\\n    for j in range(m):\\n      poss = next(iter(possibilities[j]))\\n      if (messages[j][0] == '?'):\\n        print(poss + messages[j][1:])\\n      else:\\n        print(messages[j])\\n      if (j < m-1):\\n        possibilities[j+1] = possibilities[j+1] - {poss}\\n\\n\", \"import re\\n\\ndef add_used(id, s, used):\\n    used[id-1].add(s)\\n    used[id].add(s)\\n    used[id+1].add(s)\\n\\ntcase = int(input())\\nfor cas in range(tcase):\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    \\n    res = True\\n    ans = [''] * (m+1)\\n    msg = [''] * (m+1)\\n    used = [set() for i in range(m+1)]\\n    \\n    for i in range(m):\\n        ans[i], msg[i] = input().split(':')\\n        if ans[i] != '?':\\n            add_used(i, ans[i], used)\\n        \\n        mentioned = re.split('\\\\W+', msg[i])\\n        for s in mentioned:\\n            if s in names:\\n                used[i].add(s)\\n    \\n    for i in range(m):\\n        if ans[i] == '?' and len(used[i]) == n-1:\\n            ans[i] = list(set(names) - used[i])[0]\\n            add_used(i, ans[i], used)\\n    \\n    for i in range(m):\\n        if ans[i] == '?':\\n            if len(used[i]) == n:\\n                res = False\\n            else:\\n                ans[i] = list(set(names) - used[i])[0]\\n                add_used(i, ans[i], used)\\n    \\n    for i in range(m-1):\\n        if ans[i] == ans[i+1]:\\n            res = False\\n    \\n    if res:\\n        for i in range(m):\\n            print(ans[i] + ':' + msg[i])\\n    else:\\n        print('Impossible')\", \"from sys import stdin\\n\\nt = int(stdin.readline())\\n\\ndef c(l,r,msg):\\n    f = True\\n    if l > 0 and not msg[l-1].isdigit() and not msg[l-1].isalpha():\\n        pass\\n    elif l == 0:\\n        pass\\n    else:\\n        f = False\\n\\n    if r == len(msg) - 1:\\n        pass\\n    elif not msg[r+1].isdigit() and not msg[r+1].isalpha():\\n        pass\\n    else:\\n        f = False\\n\\n    return f\\n\\n\\ndef dd(i):\\n    if i - 1 in no_author_set:\\n        authors[i-1].discard(authors[i])\\n        if len(authors[i-1]) == 1:\\n            no_author_set.discard(i-1)\\n            for d in authors[i-1]:\\n                authors[i-1] = d\\n                break\\n            dd(i-1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    if i + 1 in no_author_set:\\n        authors[i+1].discard(authors[i])\\n        if len(authors[i+1]) == 1:\\n            no_author_set.discard(i+1)\\n            for d in authors[i+1]:\\n                authors[i+1] = d\\n                break\\n            dd(i+1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    return True\\n\\ntry:\\n    answers = []\\n    for i in range(t):\\n        ans = True\\n        n = int(stdin.readline())\\n        names = set(stdin.readline().strip().split())\\n        m = int(stdin.readline())\\n        authors = list()\\n        messages = list()\\n        no_author = list()\\n        no_author_set = set()\\n        for j in range(m):\\n            line = stdin.readline().strip()\\n            author, msg = line.split(':')\\n            messages.append(msg)\\n            if author == '?':\\n                no_author.append(j)\\n                no_author_set.add(j)\\n                a_set = set()\\n                for name in names:\\n                    l = msg.find(name)\\n                    while l != -1:\\n                        if c(l,l+len(name)-1, msg):\\n                            a_set.add(name)\\n                        l = msg.find(name,l+len(name))\\n                authors.append(names-a_set)\\n                if j-1 not in no_author_set:\\n                    authors[j].discard(authors[j-1])\\n            else:\\n                authors.append(author)\\n                if j - 1 in no_author_set:\\n                    authors[j-1].discard(author)\\n\\n        for j in no_author:\\n            if j in no_author_set:\\n                if len(authors[j]) == 1:\\n                    no_author_set.discard(j)\\n                    for d in authors[j]:\\n                        authors[j] = d\\n                        break\\n                    if not dd(j):\\n                        ans = False\\n                elif len(authors[j]) == 0:\\n                    ans = False\\n\\n        no_author = list()\\n        for j in no_author_set:\\n            no_author.append(j)\\n        no_author.sort()\\n        for i in no_author:\\n            for d in authors[i]:\\n                authors[i] = d\\n                break\\n            if i + 1 in no_author_set:\\n                authors[i+1].discard(authors[i])\\n        if not ans:\\n            answers.append('Impossible')\\n        else:\\n            for j in range(m):\\n                answers.append('%s:%s'%(authors[j],messages[j]))\\nexcept Exception as e:\\n    print(e)\\n\\nfor m in answers:\\n    print(m)\", \"import re\\nt = int(input())\\nfor _ in range(t):\\n    n = int(input())\\n    names = set(input().split())\\n    m = int(input())\\n    dp = []\\n    a = []\\n    for i in range(m): a.append(input())\\n    for i in range(m):\\n        sender, msg = a[i].split(':')\\n        ls = set([_f for _f in re.split('\\\\W+',msg) if _f])\\n        dp.append((names if sender == '?' else set([sender])) - ls)\\n        if i and len(dp[i-1]) == 1: dp[i] -= dp[i-1]\\n        # print(dp[i]);\\n    if any([len(p) == 0 for p in dp]): print(\\\"Impossible\\\")\\n    else:\\n        res = []\\n        for i in reversed(list(range(m))):\\n            res.append(dp[i].pop())\\n            if i > 0: dp[i-1].discard(res[-1])\\n        for i in range(m): \\n            sender, msg = a[i].split(':')\\n            sender = res[m-i-1]\\n            print(sender+':'+msg)\\n\", \"import re\\n\\nt = int( input() )\\nfor z in range(t):\\n\\tn = int( input() )\\n\\tusers = input().split();\\n\\t#print(users)\\n\\tm = int( input() )\\n\\tcan = [ [] for i in range(m) ]\\n\\tok = True\\n\\tL = []\\n\\tfor i in range(m):\\n\\t\\tl = input().split(':')\\n\\t\\t#print(l)\\n\\t\\tL += [l]\\n\\t\\tif l[0] != '?':\\n\\t\\t\\tcan[i] = [users.index(l[0])]\\n\\t\\telse:\\n\\t\\t\\tcan[i] = [x for x in range(n)]\\n\\t\\tll = re.sub(r'[^A-Za-z0-9]', ' ', l[1]).split()\\n\\t\\tfor j in ll:\\n\\t\\t\\t#print(j)\\n\\t\\t\\tif j in users:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tcan[i].remove( users.index(j) )\\n\\t\\t\\t\\texcept ValueError:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tif len(can[i]) == 0:\\n\\t\\t\\tok = False\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\twhile 1:\\n\\t\\tflag = True\\n\\t\\tfor i in range(m - 1):\\n\\t\\t\\tif (len(can[i]) == 0): ok = False\\n\\t\\t\\tif (len(can[i]) == 1 and can[i][0] in can[i+1]):\\n\\t\\t\\t\\tcan[i+1].remove( can[i][0] )\\n\\t\\t\\t\\tL[i][0] = users[ can[i][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\t\\tif (len(can[i + 1]) == 1 and can[i + 1][0] in can[i]):\\n\\t\\t\\t\\tcan[i].remove( can[i + 1][0] )\\n\\t\\t\\t\\tL[i + 1][0] = users[ can[i + 1][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\tif len(can[m - 1]) == 0: ok = False\\n\\t\\tif ok == False: break\\n\\t\\tif flag: break\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\tfor i in range(m):\\n\\t\\tif L[i][0] == '?':\\n\\t\\t\\tprint(users[ can[i][0] ], end='')\\n\\t\\t\\tif i < m - 1 and can[i][0] in can[i + 1]:\\n\\t\\t\\t\\tcan[i + 1].remove( can[i][0] )\\n\\t\\telse:\\n\\t\\t\\tprint(L[i][0], end='')\\n\\t\\tprint(':', L[i][1], sep = '');\\n\", \"3\\n\\nimport re\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    n = int(input())\\n    users = set(input().split(' '))\\n\\n    m = int(input())\\n    messages = []\\n\\n    guessed = [None for i in range(m)]\\n    denied = [set() for i in range(m)]\\n\\n    for i in range(m):\\n        mg = input()\\n        if not mg.startswith('?:'):\\n            guessed[i] = mg[:mg.find(':')]\\n\\n        mg = mg[mg.find(':'):]\\n        messages.append(mg)\\n\\n        m2 = mg + ' '\\n        for u in users:\\n            if re.search('[^a-zA-Z0-9]' + u + '[^a-zA-Z0-9]', m2):\\n                denied[i].add(u)\\n\\n    answer = True\\n\\n    for i in range(m):\\n        if guessed[i]:\\n            if i > 0: denied[i-1].add(guessed[i])\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n    for i in range(m):\\n        if guessed[i] in denied[i]:\\n            answer = False\\n\\n    #print(guessed)\\n    #print(denied)\\n\\n    changed = True\\n    while changed and answer:\\n        changed = False\\n        for i in range(m):\\n            if not guessed[i]:\\n                if len(users) - len(denied[i]) == 1:\\n                    changed = True\\n                    guessed[i] = (users - denied[i]).pop()\\n                    if i > 0: denied[i-1].add(guessed[i])\\n                    if i < m-1: denied[i+1].add(guessed[i])\\n                if len(users) == len(denied[i]):\\n                    answer = False\\n                    break\\n\\n    for i in range(m):\\n        if not guessed[i] and len(users) - len(denied[i]) >= 1:\\n            guessed[i] = (users - denied[i]).pop()\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n\\n    for i in guessed:\\n        if not i:\\n            answer = False\\n\\n    if not answer:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for i in range(m):\\n            print(guessed[i] + messages[i])\\n    \\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  t = int(input())\\n  for t in range(t):\\n    # input\\n    n = int(input())\\n    users = set(input().split())\\n    m = int(input())\\n    msg = []\\n    for i in range(m):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        alts = users - {x for x in re.split(r'[^A-Za-z0-9]+',text)}\\n      msg.append(dict(user=user, text=text, users=alts))\\n    # remove before and after\\n    for i in range(m-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i+1]['users'].difference_update(msg[i]['users'])\\n    for i in range(m-1,0,-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i-1]['users'].difference_update(msg[i]['users'])\\n    # compute answer\\n    last = ''\\n    impo = False\\n    for i in range(m):\\n      msg[i]['users'].discard(last)\\n      if len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n      last = next(iter(msg[i]['users']))\\n      msg[i]['user'] = last\\n    if impo:\\n      print('Impossible')\\n      continue\\n    for i in range(m):\\n      print(msg[i]['user']+':'+msg[i]['text'])\\n    '''\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(1,n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]'''\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    impo = False\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?': user = users.index(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split(r'[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n      if msg[i]['user'] == '?' and len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n    if impo:\\n      print('Impossible')\\n      continue\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      u = msg[i]['user']\\n      for j in range(n+1):\\n        if u != '?':\\n          if u != j and dp[i+1][u]: dp[i][j] = u\\n        else:\\n          for alt in msg[i]['users']:\\n            if alt != j and dp[i+1][alt]:\\n              dp[i][j] = alt\\n              break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        user = users.index(user)\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split('[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\"]",
        "difficulty": "interview",
        "input": "1\n1\nb\n2\n?:lala hhe!\nb:wat?\n",
        "output": "Impossible\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/754/C"
    },
    {
        "id": 278,
        "task_id": 2124,
        "test_case_id": 7,
        "question": "Recently Vladik discovered a new entertainment — coding bots for social networks. He would like to use machine learning in his bots so now he want to prepare some learning data for them.\n\nAt first, he need to download t chats. Vladik coded a script which should have downloaded the chats, however, something went wrong. In particular, some of the messages have no information of their sender. It is known that if a person sends several messages in a row, they all are merged into a single message. It means that there could not be two or more messages in a row with the same sender. Moreover, a sender never mention himself in his messages.\n\nVladik wants to recover senders of all the messages so that each two neighboring messages will have different senders and no sender will mention himself in his messages.\n\nHe has no idea of how to do this, and asks you for help. Help Vladik to recover senders in each of the chats!\n\n\n-----Input-----\n\nThe first line contains single integer t (1 ≤ t ≤ 10) — the number of chats. The t chats follow. Each chat is given in the following format.\n\nThe first line of each chat description contains single integer n (1 ≤ n ≤ 100) — the number of users in the chat.\n\nThe next line contains n space-separated distinct usernames. Each username consists of lowercase and uppercase English letters and digits. The usernames can't start with a digit. Two usernames are different even if they differ only with letters' case. The length of username is positive and doesn't exceed 10 characters.\n\nThe next line contains single integer m (1 ≤ m ≤ 100) — the number of messages in the chat. The next m line contain the messages in the following formats, one per line:   <username>:<text> — the format of a message with known sender. The username should appear in the list of usernames of the chat.  <?>:<text> — the format of a message with unknown sender. \n\nThe text of a message can consist of lowercase and uppercase English letter, digits, characters '.' (dot), ',' (comma), '!' (exclamation mark), '?' (question mark) and ' ' (space). The text doesn't contain trailing spaces. The length of the text is positive and doesn't exceed 100 characters.\n\nWe say that a text mention a user if his username appears in the text as a word. In other words, the username appears in a such a position that the two characters before and after its appearance either do not exist or are not English letters or digits. For example, the text \"Vasya, masha13 and Kate!\" can mention users \"Vasya\", \"masha13\", \"and\" and \"Kate\", but not \"masha\".\n\nIt is guaranteed that in each chat no known sender mention himself in his messages and there are no two neighboring messages with the same known sender.\n\n\n-----Output-----\n\nPrint the information about the t chats in the following format:\n\nIf it is not possible to recover senders, print single line \"Impossible\" for this chat. Otherwise print m messages in the following format:\n\n<username>:<text>\n\nIf there are multiple answers, print any of them.\n\n\n-----Examples-----\nInput\n1\n2\nVladik netman\n2\n?: Hello, Vladik!\n?: Hi\n\nOutput\nnetman: Hello, Vladik!\nVladik: Hi\n\nInput\n1\n2\nnetman vladik\n3\nnetman:how are you?\n?:wrong message\nvladik:im fine\n\nOutput\nImpossible\n\nInput\n2\n3\nnetman vladik Fedosik\n2\n?: users are netman, vladik, Fedosik\nvladik: something wrong with this chat\n4\nnetman tigerrrrr banany2001 klinchuh\n4\n?: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\n?: yes, netman\nbanany2001: yes of course.\n\nOutput\nImpossible\nnetman: tigerrrrr, banany2001, klinchuh, my favourite team ever, are you ready?\nklinchuh: yes, coach!\ntigerrrrr: yes, netman\nbanany2001: yes of course.",
        "solutions": "[\"def pr( name , lvl , dp , u , tot ): \\n    if lvl == 0:\\n        print(name + ':' + tot[lvl])\\n        return\\n\\n    pr( u[lvl][name] , lvl - 1 , dp , u , tot )\\n    print(name + ':' + tot[lvl])\\n\\ndef solve(): \\n    n = int(input())\\n    users = input().split()\\n    m = int(input())\\n    dp = [] \\n    u = []\\n    tot = [] \\n    for i in range( m ) : \\n        dp.append( set() ) \\n        u.append( {} ) \\n        line = input().split(':')\\n        sender = line[0]\\n        tot.append( line[1] ) \\n        line[1] = line[1].replace( '?' , ' ' )\\n        line[1] = line[1].replace( '.' , ' ' )\\n        line[1] = line[1].replace( ',' , ' ' )\\n        line[1] = line[1].replace( '!' , ' ' )\\n        mess = line[1].split()\\n\\n        if sender == '?' : \\n            if i != 0:\\n                for name in users:\\n                    for x in dp[i-1]: \\n                        if x != name and name not in mess:\\n                            dp[i].add( name ) \\n                            u[i][name] = x\\n            else : \\n                for name in users: \\n                    if name not in mess:\\n                        dp[i].add( name ) \\n        else: \\n            if i != 0: \\n                for x in dp[i-1]: \\n                    if x != sender: \\n                        dp[i].add( sender ) \\n                        u[i][sender] = x\\n            else: \\n                dp[i].add( sender )\\n        \\n        \\n    if dp[m-1]: \\n        pr( list(dp[m-1])[0] , m-1 , dp , u , tot )\\n    else: \\n        print(\\\"Impossible\\\")\\n\\n\\nt = int(input())\\nfor i in range( t ) : \\n    solve()\\n\", \"#!/usr/bin/env python3\\n\\nimport re\\n\\ndef cont(line, user):\\n\\treturn re.search('[^a-zA-Z0-9]' + user + '[^a-zA-Z0-9]', '_' + line + '_')\\n\\ntests = int(input())\\nfor test in range(tests):\\n\\tinput()\\n\\tusers = input().rstrip('\\\\n').split(' ')\\n\\tlinecnt = int(input())\\n\\tlines = [input().rstrip('\\\\n') for _ in range(linecnt)]\\n\\tlines = [x.split(':', maxsplit=1) for x in lines]\\n\\tposs = [[]] * linecnt\\n\\tfor i, (user, line) in enumerate(lines):\\n\\t\\tif user != '?':\\n\\t\\t\\tposs[i] = [user]\\n\\t\\telse:\\n\\t\\t\\tposs[i] = [u for u in users if not cont(line, u)]\\n\\n\\tel = list(enumerate(lines))\\n\\trel = list(reversed(el))\\n\\n\\tchanged = False\\n\\twhile True:\\n\\t\\tfor i, (user, line) in el:\\n\\t\\t\\tif i > 0 and len(poss[i-1]) == 1:\\n\\t\\t\\t\\tif poss[i-1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i-1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tfor i, (user, line) in rel:\\n\\t\\t\\tif i < linecnt - 1 and len(poss[i+1]) == 1:\\n\\t\\t\\t\\tif poss[i+1][0] in poss[i]:\\n\\t\\t\\t\\t\\tposs[i].remove(poss[i+1][0])\\n\\t\\t\\t\\t\\tchanged = True\\n\\t\\tif not changed:\\n\\t\\t\\tbreak\\n\\t\\tchanged = False\\n\\n\\tif all(len(p) > 0 for p in poss):\\n\\t\\tfor i, p, (user, line) in zip(list(range(linecnt)), poss, lines):\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\tif poss[i-1][0] in p:\\n\\t\\t\\t\\t\\tp.remove(poss[i-1][0])\\n\\t\\t\\tprint(p[0] + ':' + line)\\n\\telse:\\n\\t\\tprint('Impossible')\\n\", \"import re\\nimport sys\\n\\n\\ndef g(cts, names):\\n  la = dict([(p, None) for p in names.difference(cts[0])])\\n  if len(la) == 0:\\n    return None\\n  d = [la]\\n  for i in range(1, len(cts)):\\n    prev = d[-1]\\n    la = dict()\\n    for p in names.difference(cts[i]):\\n      for n in prev.keys():\\n        if n != p:\\n          la[p] = n\\n          break\\n    if len(la) == 0:\\n      return None\\n    d.append(la)\\n  cur = list(d[-1].keys())[0]\\n  result = []\\n  for i in range(len(cts) - 1, -1, -1):\\n    result.append(cur)\\n    cur = d[i][cur]\\n  result.reverse()\\n  return result\\n\\n\\ndef solve():\\n  n = int(input())\\n  names = input().split()\\n  set_names = set(names)\\n\\n  def get_names(text):\\n    result = []\\n    for p in re.split(\\\"\\\\W+\\\", text):\\n      if p in set_names:\\n        result.append(p)\\n    return result\\n\\n  m = int(input())\\n  messages = []\\n  for i in range(m):\\n    s = input()\\n    colon = s.find(\\\":\\\")\\n    name = s[:colon]\\n    text = s[colon + 1:]\\n    if name == \\\"?\\\":\\n      name = None\\n    messages.append([name, text, get_names(text)])\\n\\n  for i in range(m):\\n    if messages[i][0]:\\n      continue\\n    j = i\\n    cts = []\\n    while j < m and not messages[j][0]:\\n      cts.append(set(messages[j][2]))\\n      j += 1\\n    if i > 0:\\n      cts[0].add(messages[i - 1][0])\\n    if j < m:\\n      cts[-1].add(messages[j][0])\\n    sb = g(cts, set_names)\\n    if not sb:\\n      return None\\n    for k in range(i, j):\\n      messages[k][0] = sb[k - i]\\n\\n  for p in messages:\\n    print(\\\"%s:%s\\\" % (p[0], p[1]))\\n  return True;\\n\\n\\ndef main():\\n  tests = int(input())\\n  for i in range(tests):\\n    if not solve():\\n      print(\\\"Impossible\\\")\\n  return 0\\n\\ndef __starting_point():\\n  return(main())\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"def dfs(ind):\\n    if ind > 0:\\n        if possible[ind][0] in possible[ind - 1]:\\n            possible[ind - 1].remove(possible[ind][0])\\n            if len(possible[ind - 1]) == 1:\\n                dfs(ind - 1)\\n    if ind < m - 1:\\n        if possible[ind][0] in possible[ind + 1]:\\n            possible[ind + 1].remove(possible[ind][0])\\n            if len(possible[ind + 1]) == 1:\\n                dfs(ind + 1)\\ndef Check(st, ms):\\n    for i in range(0, len(ms) - len(st) + 1):\\n        if st == ms[i:i+len(st)]:\\n            t = True\\n            if i > 0:\\n                if ms[i - 1] == ' ' or ms[i - 1] == '.' or ms[i - 1] == ',' or ms[i - 1] == '?' or ms[i - 1] == '!':\\n                    e = 0\\n                else:\\n                    t = False\\n            if i < len(ms) - len(st):\\n                if (ms[i + len(st)] == ' ' or ms[i + len(st)] == '.' or ms[i + len(st)] == ',' or ms[i + len(st)] == '?' or ms[i + len(st)] == '!'):\\n                    e = 0\\n                else:\\n                    t = False\\n            if t:\\n                return True\\n    return False\\nR = lambda:list(map(int, input().split(' ')))\\n#r, w = open(\\\"input.txt\\\", \\\"r\\\"), open(\\\"output.txt\\\", \\\"w\\\")\\nT = int(input())\\nwhile T:\\n    n = int(input())\\n    users = input().split(' ')\\n    m = int(input())\\n    possible = [[] for i in range(m)]\\n    sender, message = [], []\\n    for i in range(m):\\n        s = input().split(':')\\n        sender.append(s[0])\\n        message.append(s[1])\\n    for i in range(m):\\n        if sender[i] == '?':\\n            unallow = \\\" \\\"\\n            if i > 0 and len(possible[i - 1]) == 1:\\n                unallow = possible[i - 1][0]\\n            for j in users:\\n                if j == unallow:\\n                    continue\\n                done = Check(j, message[i])\\n                if not done:\\n                    possible[i].append(j)\\n        else:\\n            possible[i].append(sender[i])\\n    used = [0 for i in range(m)]\\n    for i in range(m):\\n        if len(possible[i]) == 1 and used[i] == 0:\\n            dfs(i)\\n    for i in range(m):\\n        if len(possible[i]) > 1:\\n            possible[i] = [possible[i][0]]\\n            dfs(i)\\n    done = False\\n    for i in possible:\\n        if len(i) == 0:\\n            print(\\\"Impossible\\\")\\n            done = True\\n            break\\n    if not done:\\n        for i in range(m):\\n            print(possible[i][0]+':'+message[i])\\n    T -= 1\\n\\n\\n    \\n\\n\", \"# Bartek Kostka\\n#  You are not prepared!\\n\\nimport re\\n\\n\\ndef go(i):\\n    nonlocal counter\\n    counter += 1\\n    if counter > 2000:\\n        return False\\n    if i == len(E):\\n        return True\\n    if E[i][0] != \\\"?\\\":\\n        prop = E[i][0]\\n        if i == 0 or W[i-1] != prop:\\n            W[i] = prop\\n            return go(i+1)\\n        else:\\n            return False\\n    for pos in E[i][1]:\\n        if i == 0 or W[i-1] != pos:\\n            W[i] = pos\\n            if go(i+1):\\n                return True\\n    return False\\n\\n\\nt = int(input())\\nfor ttt in range(t):\\n    n = int(input())\\n    users = input().split(\\\" \\\")\\n    users_set = set(users)\\n    m = int(input())\\n    E = []\\n    S = []\\n    counter = 0\\n    W = [\\\"\\\" for x in range(m)]\\n    for mmm in range(m):\\n        line = input()\\n        (user, sentence) = line.split(\\\":\\\")\\n        S.append((user, sentence))\\n        words = [x.strip() for x in re.split('\\\\W+', sentence)]\\n        mentions = list([x for x in words if x in users_set])\\n        E.append([user, set(mentions)])\\n    for i in range(len(E)-1):\\n        if E[i][0] != \\\"?\\\":\\n            E[i+1][1].add(E[i][0])\\n    for i in range(1, len(E)):\\n        if E[i][0] != \\\"?\\\":\\n            E[i-1][1].add(E[i][0])\\n    for i in range(len(E)):\\n        E[i][1] = E[i][1].symmetric_difference(users_set)\\n    if go(0):\\n        for i in range(len(E)):\\n            print(str(W[i])+\\\":\\\"+str(S[i][1]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import re\\nimport copy\\n\\nt = int(input())\\n\\nfor i in range(t):\\n\\tn = int(input())\\n\\tsenderList = []\\n\\tfor x in input().split():\\n\\t\\tsenderList.append(x)\\n\\n\\tm = int(input())\\n\\tmsg = [None] * m\\n\\tresSenders = [None] * m\\n\\tpossibleSenders = []\\n\\tfor i in range(m):\\n\\t\\tpossibleSenders.append(copy.copy(senderList))\\n\\n\\tfor i in range(m):\\n\\t\\tline = input()\\n\\t\\tblocks = re.findall(r\\\"[\\\\w]+\\\", line)\\n\\n\\t\\tfor x in blocks:\\n\\t\\t\\ttry:\\n\\t\\t\\t\\tpossibleSenders[i].remove(x)\\n\\t\\t\\texcept:\\n\\t\\t\\t\\tpass\\n\\n\\t\\tif line[0] != '?':\\n\\t\\t\\tresSenders[i] = blocks[0]\\n\\t\\t\\tif i > 0:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tmsg[i] = line[line.find(\\\":\\\")+1:]\\n\\n\\tresolved = False\\n\\twhile True:\\n\\t\\tdone = True\\n\\t\\tjustPick = True\\n\\t\\tfor i in range(m):\\n\\t\\t\\tif resSenders[i] != None:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tif len(possibleSenders[i]) == 0:\\n\\t\\t\\t\\tdone = True\\n\\t\\t\\t\\tresolved = True\\n\\t\\t\\t\\tprint(\\\"Impossible\\\")\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tif len(possibleSenders[i]) == 1:\\n\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\tif i > 0:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i-1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\t\\tjustPick = False\\n\\t\\t\\telse:\\n\\t\\t\\t\\tdone = False\\n\\t\\tif done:\\n\\t\\t\\tbreak\\n\\t\\tif justPick:\\n\\t\\t\\tfor i in range(m):\\n\\t\\t\\t\\tif resSenders[i] == None:\\n\\t\\t\\t\\t\\tresSenders[i] = possibleSenders[i][0]\\n\\t\\t\\t\\t\\tif i+1 < m:\\n\\t\\t\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\t\\t\\tpossibleSenders[i+1].remove(resSenders[i])\\n\\t\\t\\t\\t\\t\\texcept:\\n\\t\\t\\t\\t\\t\\t\\tpass\\n\\t\\t\\tbreak\\n\\n\\tif not resolved:\\n\\t\\tfor i in range(m):\\n\\t\\t\\tprint(resSenders[i] + \\\":\\\" + msg[i])\", \"import re\\n\\n\\ndef foo():\\n    n = int(input())\\n    who = input().split()\\n    m = int(input())\\n    msg = []\\n\\n    l = []\\n\\n    for num in range(m):\\n        a, t = input().split(':')\\n        msg.append(t)\\n\\n        st = set()\\n\\n        if a != '?':\\n            st.add(a)\\n        else:\\n\\n            names = re.split('\\\\W+', t)\\n\\n            for w in who:\\n                if not w in names:\\n                    st.add(w)\\n\\n        l.append(st)\\n\\n    d2 = []\\n\\n    for num in range(1, m):\\n        d = {}\\n        for w1 in l[num]:\\n            for w2 in l[num - 1]:\\n                if w1 != w2:\\n                    d[w1] = w2\\n                    break\\n\\n        l[num] = [x for x in d]\\n\\n        d2.append(d)\\n\\n    curr = None\\n\\n    for w in l[m - 1]:\\n        curr = w\\n\\n    res = [curr]\\n\\n    for num in list(reversed(list(range(1, m)))):\\n        curr = d2[num - 1].get(curr, None)  # d2[num - 1][curr]\\n        res.append(curr)\\n\\n    res = list(reversed(res))\\n\\n    if None in res:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for num in range(m):\\n            print(res[num] + ':' + msg[num])\\n\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    foo()\\n\", \"def main():\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    msg = [input().split(':') for _ in range(m)]\\n    texts = []\\n    for i in msg:\\n        texts.append(i[1])\\n        i[1] = i[1].replace(',', ' ').replace('.', ' ').replace('!', ' ').replace('?', ' ').split()\\n        if i[0] == '?':\\n            i.append([])\\n            for name in names:\\n                if name not in i[1]:\\n                    i[2].append(name)\\n            if len(i[2]) == 0:\\n                print('Impossible')\\n                return\\n    go_on = True\\n    while go_on:\\n        go_on = False\\n        for i in range(len(msg)):\\n            if msg[i][0] != '?':\\n                continue\\n            if i - 1 > -1 and msg[i - 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i - 1][0])\\n                go_on = True\\n            if i + 1 < m and msg[i + 1][0] in msg[i][2]:\\n                msg[i][2].remove(msg[i + 1][0])\\n                go_on = True\\n            if len(msg[i][2]) == 0:\\n                print('Impossible')\\n                return\\n            if len(msg[i][2]) == 1:\\n                msg[i][0] = msg[i][2][0]\\n                del msg[i][2]\\n                go_on = True\\n    for i in range(len(msg)):\\n        if msg[i][0] == '?':\\n            msg[i][0] = msg[i][2][0]\\n            if i < m - 1 and len(msg[i + 1]) == 3 and msg[i][0] in msg[i + 1][2]:\\n                msg[i + 1][2].remove(msg[i][0])\\n        print(msg[i][0], ':', texts[i], sep='')\\n\\nt = int(input())\\n\\nfor _ in range(t):\\n    main()\", \"import re\\n\\nt = int(input())\\n\\ndelimiters = \\\"?\\\", \\\".\\\", \\\" \\\", \\\",\\\", \\\"!\\\", \\\":\\\"\\nregexPattern = '|'.join(map(re.escape, delimiters))\\n\\nfor i in range(t):\\n  n = int(input())\\n  usernames = {x for x in str.split(input(), ' ')}\\n  m = int(input())\\n  possibilities = []\\n  for j in range(m):\\n    possibilities.append({x for x in usernames})\\n  messages = []\\n  for j in range(m):\\n    messages.append(input())\\n\\n  for j in range(m):\\n    if (messages[j][0] != '?'):\\n      messageSplit = re.split(':', messages[j])\\n      possibilities[j] = {messageSplit[0]}\\n    else:\\n      messageSplit = re.split(regexPattern, messages[j])\\n      for token in messageSplit:\\n        if token in usernames:\\n          possibilities[j] = possibilities[j] - {token}\\n  changed = True\\n  while changed:\\n    changed = False\\n    for j in range(m):\\n      if len(possibilities[j]) == 1:\\n        (poss,) = possibilities[j]\\n        if j < m-1 and poss in possibilities[j+1]:\\n          changed = True\\n          possibilities[j+1] = possibilities[j+1] - possibilities[j]\\n        if j > 0 and poss in possibilities[j-1]:\\n          changed = True\\n          possibilities[j-1] = possibilities[j-1] - possibilities[j]\\n  worked = True\\n  for j in range(m):\\n    if len(possibilities[j]) == 0:\\n      worked = False\\n      \\n  if not worked:\\n    print(\\\"Impossible\\\")\\n  else :\\n    for j in range(m):\\n      poss = next(iter(possibilities[j]))\\n      if (messages[j][0] == '?'):\\n        print(poss + messages[j][1:])\\n      else:\\n        print(messages[j])\\n      if (j < m-1):\\n        possibilities[j+1] = possibilities[j+1] - {poss}\\n\\n\", \"import re\\n\\ndef add_used(id, s, used):\\n    used[id-1].add(s)\\n    used[id].add(s)\\n    used[id+1].add(s)\\n\\ntcase = int(input())\\nfor cas in range(tcase):\\n    n = int(input())\\n    names = input().split()\\n    m = int(input())\\n    \\n    res = True\\n    ans = [''] * (m+1)\\n    msg = [''] * (m+1)\\n    used = [set() for i in range(m+1)]\\n    \\n    for i in range(m):\\n        ans[i], msg[i] = input().split(':')\\n        if ans[i] != '?':\\n            add_used(i, ans[i], used)\\n        \\n        mentioned = re.split('\\\\W+', msg[i])\\n        for s in mentioned:\\n            if s in names:\\n                used[i].add(s)\\n    \\n    for i in range(m):\\n        if ans[i] == '?' and len(used[i]) == n-1:\\n            ans[i] = list(set(names) - used[i])[0]\\n            add_used(i, ans[i], used)\\n    \\n    for i in range(m):\\n        if ans[i] == '?':\\n            if len(used[i]) == n:\\n                res = False\\n            else:\\n                ans[i] = list(set(names) - used[i])[0]\\n                add_used(i, ans[i], used)\\n    \\n    for i in range(m-1):\\n        if ans[i] == ans[i+1]:\\n            res = False\\n    \\n    if res:\\n        for i in range(m):\\n            print(ans[i] + ':' + msg[i])\\n    else:\\n        print('Impossible')\", \"from sys import stdin\\n\\nt = int(stdin.readline())\\n\\ndef c(l,r,msg):\\n    f = True\\n    if l > 0 and not msg[l-1].isdigit() and not msg[l-1].isalpha():\\n        pass\\n    elif l == 0:\\n        pass\\n    else:\\n        f = False\\n\\n    if r == len(msg) - 1:\\n        pass\\n    elif not msg[r+1].isdigit() and not msg[r+1].isalpha():\\n        pass\\n    else:\\n        f = False\\n\\n    return f\\n\\n\\ndef dd(i):\\n    if i - 1 in no_author_set:\\n        authors[i-1].discard(authors[i])\\n        if len(authors[i-1]) == 1:\\n            no_author_set.discard(i-1)\\n            for d in authors[i-1]:\\n                authors[i-1] = d\\n                break\\n            dd(i-1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    if i + 1 in no_author_set:\\n        authors[i+1].discard(authors[i])\\n        if len(authors[i+1]) == 1:\\n            no_author_set.discard(i+1)\\n            for d in authors[i+1]:\\n                authors[i+1] = d\\n                break\\n            dd(i+1)\\n        if len(authors[i-1]) == 0:\\n            return False\\n    return True\\n\\ntry:\\n    answers = []\\n    for i in range(t):\\n        ans = True\\n        n = int(stdin.readline())\\n        names = set(stdin.readline().strip().split())\\n        m = int(stdin.readline())\\n        authors = list()\\n        messages = list()\\n        no_author = list()\\n        no_author_set = set()\\n        for j in range(m):\\n            line = stdin.readline().strip()\\n            author, msg = line.split(':')\\n            messages.append(msg)\\n            if author == '?':\\n                no_author.append(j)\\n                no_author_set.add(j)\\n                a_set = set()\\n                for name in names:\\n                    l = msg.find(name)\\n                    while l != -1:\\n                        if c(l,l+len(name)-1, msg):\\n                            a_set.add(name)\\n                        l = msg.find(name,l+len(name))\\n                authors.append(names-a_set)\\n                if j-1 not in no_author_set:\\n                    authors[j].discard(authors[j-1])\\n            else:\\n                authors.append(author)\\n                if j - 1 in no_author_set:\\n                    authors[j-1].discard(author)\\n\\n        for j in no_author:\\n            if j in no_author_set:\\n                if len(authors[j]) == 1:\\n                    no_author_set.discard(j)\\n                    for d in authors[j]:\\n                        authors[j] = d\\n                        break\\n                    if not dd(j):\\n                        ans = False\\n                elif len(authors[j]) == 0:\\n                    ans = False\\n\\n        no_author = list()\\n        for j in no_author_set:\\n            no_author.append(j)\\n        no_author.sort()\\n        for i in no_author:\\n            for d in authors[i]:\\n                authors[i] = d\\n                break\\n            if i + 1 in no_author_set:\\n                authors[i+1].discard(authors[i])\\n        if not ans:\\n            answers.append('Impossible')\\n        else:\\n            for j in range(m):\\n                answers.append('%s:%s'%(authors[j],messages[j]))\\nexcept Exception as e:\\n    print(e)\\n\\nfor m in answers:\\n    print(m)\", \"import re\\nt = int(input())\\nfor _ in range(t):\\n    n = int(input())\\n    names = set(input().split())\\n    m = int(input())\\n    dp = []\\n    a = []\\n    for i in range(m): a.append(input())\\n    for i in range(m):\\n        sender, msg = a[i].split(':')\\n        ls = set([_f for _f in re.split('\\\\W+',msg) if _f])\\n        dp.append((names if sender == '?' else set([sender])) - ls)\\n        if i and len(dp[i-1]) == 1: dp[i] -= dp[i-1]\\n        # print(dp[i]);\\n    if any([len(p) == 0 for p in dp]): print(\\\"Impossible\\\")\\n    else:\\n        res = []\\n        for i in reversed(list(range(m))):\\n            res.append(dp[i].pop())\\n            if i > 0: dp[i-1].discard(res[-1])\\n        for i in range(m): \\n            sender, msg = a[i].split(':')\\n            sender = res[m-i-1]\\n            print(sender+':'+msg)\\n\", \"import re\\n\\nt = int( input() )\\nfor z in range(t):\\n\\tn = int( input() )\\n\\tusers = input().split();\\n\\t#print(users)\\n\\tm = int( input() )\\n\\tcan = [ [] for i in range(m) ]\\n\\tok = True\\n\\tL = []\\n\\tfor i in range(m):\\n\\t\\tl = input().split(':')\\n\\t\\t#print(l)\\n\\t\\tL += [l]\\n\\t\\tif l[0] != '?':\\n\\t\\t\\tcan[i] = [users.index(l[0])]\\n\\t\\telse:\\n\\t\\t\\tcan[i] = [x for x in range(n)]\\n\\t\\tll = re.sub(r'[^A-Za-z0-9]', ' ', l[1]).split()\\n\\t\\tfor j in ll:\\n\\t\\t\\t#print(j)\\n\\t\\t\\tif j in users:\\n\\t\\t\\t\\ttry:\\n\\t\\t\\t\\t\\tcan[i].remove( users.index(j) )\\n\\t\\t\\t\\texcept ValueError:\\n\\t\\t\\t\\t\\tpass\\n\\t\\tif len(can[i]) == 0:\\n\\t\\t\\tok = False\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\twhile 1:\\n\\t\\tflag = True\\n\\t\\tfor i in range(m - 1):\\n\\t\\t\\tif (len(can[i]) == 0): ok = False\\n\\t\\t\\tif (len(can[i]) == 1 and can[i][0] in can[i+1]):\\n\\t\\t\\t\\tcan[i+1].remove( can[i][0] )\\n\\t\\t\\t\\tL[i][0] = users[ can[i][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\t\\tif (len(can[i + 1]) == 1 and can[i + 1][0] in can[i]):\\n\\t\\t\\t\\tcan[i].remove( can[i + 1][0] )\\n\\t\\t\\t\\tL[i + 1][0] = users[ can[i + 1][0] ]\\n\\t\\t\\t\\tflag = False\\n\\t\\tif len(can[m - 1]) == 0: ok = False\\n\\t\\tif ok == False: break\\n\\t\\tif flag: break\\n\\tif ok == False:\\n\\t\\tprint('Impossible')\\n\\t\\tcontinue\\n\\tfor i in range(m):\\n\\t\\tif L[i][0] == '?':\\n\\t\\t\\tprint(users[ can[i][0] ], end='')\\n\\t\\t\\tif i < m - 1 and can[i][0] in can[i + 1]:\\n\\t\\t\\t\\tcan[i + 1].remove( can[i][0] )\\n\\t\\telse:\\n\\t\\t\\tprint(L[i][0], end='')\\n\\t\\tprint(':', L[i][1], sep = '');\\n\", \"3\\n\\nimport re\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    n = int(input())\\n    users = set(input().split(' '))\\n\\n    m = int(input())\\n    messages = []\\n\\n    guessed = [None for i in range(m)]\\n    denied = [set() for i in range(m)]\\n\\n    for i in range(m):\\n        mg = input()\\n        if not mg.startswith('?:'):\\n            guessed[i] = mg[:mg.find(':')]\\n\\n        mg = mg[mg.find(':'):]\\n        messages.append(mg)\\n\\n        m2 = mg + ' '\\n        for u in users:\\n            if re.search('[^a-zA-Z0-9]' + u + '[^a-zA-Z0-9]', m2):\\n                denied[i].add(u)\\n\\n    answer = True\\n\\n    for i in range(m):\\n        if guessed[i]:\\n            if i > 0: denied[i-1].add(guessed[i])\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n    for i in range(m):\\n        if guessed[i] in denied[i]:\\n            answer = False\\n\\n    #print(guessed)\\n    #print(denied)\\n\\n    changed = True\\n    while changed and answer:\\n        changed = False\\n        for i in range(m):\\n            if not guessed[i]:\\n                if len(users) - len(denied[i]) == 1:\\n                    changed = True\\n                    guessed[i] = (users - denied[i]).pop()\\n                    if i > 0: denied[i-1].add(guessed[i])\\n                    if i < m-1: denied[i+1].add(guessed[i])\\n                if len(users) == len(denied[i]):\\n                    answer = False\\n                    break\\n\\n    for i in range(m):\\n        if not guessed[i] and len(users) - len(denied[i]) >= 1:\\n            guessed[i] = (users - denied[i]).pop()\\n            if i < m-1: denied[i+1].add(guessed[i])\\n\\n\\n    for i in guessed:\\n        if not i:\\n            answer = False\\n\\n    if not answer:\\n        print(\\\"Impossible\\\")\\n    else:\\n        for i in range(m):\\n            print(guessed[i] + messages[i])\\n    \\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    is_fixed = [False] * m\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n            is_fixed = True\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n            is_fixed = True\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"import sys\\nimport re\\n\\ndef mentioned_usernames(line):\\n    return {x for x in re.split(r'[^A-Za-z0-9]+', line)}\\n\\nt = int(input())\\n\\nfor ti in range(t):\\n    possible_users = []\\n    messages = []\\n    n = int(input())\\n    usernames = set(input().split())\\n    # print(\\\"usernames =\\\", usernames, file=sys.stderr)\\n    m = int(input())\\n    for i in range(m):\\n        user, text = input().split(':')\\n        messages.append(text)\\n        if user == '?':\\n            mu = mentioned_usernames(text)\\n            possible_users.append(usernames - mentioned_usernames(text))\\n        else:\\n            possible_users.append({user})\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    for i in range(m-1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i+1].difference_update(possible_users[i])\\n    for i in range(m-1, 0, -1):\\n        if len(possible_users[i]) == 1:\\n            possible_users[i-1].difference_update(possible_users[i])\\n\\n    # print(possible_users, file=sys.stderr)\\n\\n    res = []\\n    is_possible = True\\n    prev_user = '$'\\n    for i in range(m):\\n        if possible_users[i]:\\n            pusers = possible_users[i] - {prev_user}\\n            resx = next(iter(pusers))\\n            res.append(resx)\\n            prev_user = resx\\n        else:\\n            is_possible = False\\n            break\\n\\n    if is_possible:\\n        for i in range(m):\\n            print('{}:{}'.format(res[i], messages[i]))\\n    else:\\n        print(\\\"Impossible\\\")\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  t = int(input())\\n  for t in range(t):\\n    # input\\n    n = int(input())\\n    users = set(input().split())\\n    m = int(input())\\n    msg = []\\n    for i in range(m):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        alts = users - {x for x in re.split(r'[^A-Za-z0-9]+',text)}\\n      msg.append(dict(user=user, text=text, users=alts))\\n    # remove before and after\\n    for i in range(m-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i+1]['users'].difference_update(msg[i]['users'])\\n    for i in range(m-1,0,-1):\\n      if len(msg[i]['users']) == 1:\\n        msg[i-1]['users'].difference_update(msg[i]['users'])\\n    # compute answer\\n    last = ''\\n    impo = False\\n    for i in range(m):\\n      msg[i]['users'].discard(last)\\n      if len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n      last = next(iter(msg[i]['users']))\\n      msg[i]['user'] = last\\n    if impo:\\n      print('Impossible')\\n      continue\\n    for i in range(m):\\n      print(msg[i]['user']+':'+msg[i]['text'])\\n    '''\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(1,n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]'''\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    impo = False\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?': user = users.index(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split(r'[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n      if msg[i]['user'] == '?' and len(msg[i]['users']) == 0:\\n        impo = True\\n        break\\n    if impo:\\n      print('Impossible')\\n      continue\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      u = msg[i]['user']\\n      for j in range(n+1):\\n        if u != '?':\\n          if u != j and dp[i+1][u]: dp[i][j] = u\\n        else:\\n          for alt in msg[i]['users']:\\n            if alt != j and dp[i+1][alt]:\\n              dp[i][j] = alt\\n              break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\", \"3\\n\\n# BEGIN template\\nimport sys\\nimport re\\nimport pprint\\n\\ndef dbg(x,y=''):\\n  if len(y) > 0: y += ' = '\\n  sys.stderr.write('\\\\n>>> '+y+pprint.pformat(x)+'\\\\n')\\n\\noo = 0x3f3f3f3f3f3f3f3f\\n# END template\\n\\ndef main():\\n  for t in range(int(input())):\\n    # input\\n    n = int(input())\\n    users = input().split()\\n    users_set = set(users)\\n    users.insert(0,'0')\\n    m = int(input())\\n    msg = [None]*(m+5)\\n    for i in range(1,m+1):\\n      user, text = input().split(':')\\n      alts = set()\\n      if user != '?':\\n        user = users.index(user)\\n        alts.add(user)\\n      else:\\n        # this shit is pretty fucked up, dude\\n        for alt in users_set - {x for x in re.split('[^A-Za-z0-9]+',text)}:\\n          alts.add(users.index(alt))\\n      msg[i] = dict(user=user, text=text, users=alts)\\n    # remove before and after\\n    for i in range(1,m+1):\\n      if 1 <= i-1:  msg[i]['users'].discard(msg[i-1]['user'])\\n      if i+1 <= m:  msg[i]['users'].discard(msg[i+1]['user'])\\n    # compute answer\\n    dp = [[0 for j in range(n+5)] for i in range(m+5)]\\n    for i in range(n+5):\\n      dp[m+1][i] = oo\\n    for i in range(m,0,-1):\\n      for j in range(n+1):\\n        for k in msg[i]['users']:\\n          if k != j and dp[i+1][k]:\\n            dp[i][j] = k\\n            break\\n    # output\\n    if not dp[1][0]:\\n      print('Impossible')\\n      continue\\n    j = 0\\n    for i in range(1,m+1):\\n      print(users[dp[i][j]]+':'+msg[i]['text'])\\n      j = dp[i][j]\\n\\nmain()\\n\"]",
        "difficulty": "interview",
        "input": "1\n3\nA B C\n3\nA: HI\n?: HI\nB: HI\n",
        "output": "A: HI\nC: HI\nB: HI\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/754/C"
    },
    {
        "id": 279,
        "task_id": 2273,
        "test_case_id": 3,
        "question": "You have a simple undirected graph consisting of $n$ vertices and $m$ edges. The graph doesn't contain self-loops, there is at most one edge between a pair of vertices. The given graph can be disconnected.\n\nLet's make a definition.\n\nLet $v_1$ and $v_2$ be two some nonempty subsets of vertices that do not intersect. Let $f(v_{1}, v_{2})$ be true if and only if all the conditions are satisfied:  There are no edges with both endpoints in vertex set $v_1$.  There are no edges with both endpoints in vertex set $v_2$.  For every two vertices $x$ and $y$ such that $x$ is in $v_1$ and $y$ is in $v_2$, there is an edge between $x$ and $y$. \n\nCreate three vertex sets ($v_{1}$, $v_{2}$, $v_{3}$) which satisfy the conditions below;  All vertex sets should not be empty.  Each vertex should be assigned to only one vertex set.  $f(v_{1}, v_{2})$, $f(v_{2}, v_{3})$, $f(v_{3}, v_{1})$ are all true. \n\nIs it possible to create such three vertex sets? If it's possible, print matching vertex set for each vertex.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($3 \\le n \\le 10^{5}$, $0 \\le m \\le \\text{min}(3 \\cdot 10^{5}, \\frac{n(n-1)}{2})$) — the number of vertices and edges in the graph.\n\nThe $i$-th of the next $m$ lines contains two integers $a_{i}$ and $b_{i}$ ($1 \\le a_{i} \\lt b_{i} \\le n$) — it means there is an edge between $a_{i}$ and $b_{i}$. The graph doesn't contain self-loops, there is at most one edge between a pair of vertices. The given graph can be disconnected.\n\n\n-----Output-----\n\nIf the answer exists, print $n$ integers. $i$-th integer means the vertex set number (from $1$ to $3$) of $i$-th vertex. Otherwise, print $-1$.\n\nIf there are multiple answers, print any.\n\n\n-----Examples-----\nInput\n6 11\n1 2\n1 3\n1 4\n1 5\n1 6\n2 4\n2 5\n2 6\n3 4\n3 5\n3 6\n\nOutput\n1 2 2 3 3 3 \nInput\n4 6\n1 2\n1 3\n1 4\n2 3\n2 4\n3 4\n\nOutput\n-1\n\n\n\n-----Note-----\n\nIn the first example, if $v_{1} = \\{ 1 \\}$, $v_{2} = \\{ 2, 3 \\}$, and $v_{3} = \\{ 4, 5, 6 \\}$ then vertex sets will satisfy all conditions. But you can assign vertices to vertex sets in a different way; Other answers like \"2 3 3 1 1 1\" will be accepted as well. [Image] \n\nIn the second example, it's impossible to make such vertex sets.",
        "solutions": "[\"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\n\\nN, M = list(map(int, input().split()))\\nE = [[] for _ in range(N)]\\nfor _ in range(M):\\n    a, b = list(map(int, input().split()))\\n    E[a-1].append(b-1)\\n    E[b-1].append(a-1)\\n\\ninf = 1 << 20\\nA, B, C = [], [], []\\nX = [0] * N\\nfor a in E[0]:\\n    X[a] = 1\\nA = [i for i in range(N) if X[i]==0]\\nb = min([i for i in range(N) if X[i] == 1] + [inf])\\nif b < inf:\\n    for a in E[b]:\\n        if X[a] == 1: X[a] = 2\\n    B = [i for i in range(N) if X[i]==1]\\nc = min([i for i in range(N) if X[i] == 2] + [inf])\\nif c < inf:\\n    for a in E[c]:\\n        if X[a] == 2: X[a] = 3\\n    C = [i for i in range(N) if X[i]==2]\\n\\nif max(X) == 2 and len(A) * len(B) * len(C) and (len(A) + len(B) + len(C) == N) and (len(A) * len(B) + len(B) * len(C) + len(A) * len(C) == M):\\n    f = 0\\n    for i in range(N):\\n        for j in E[i]:\\n            if X[i] == X[j]:\\n                f = 1\\n                break\\n        if f: break\\n    if f:\\n        print(-1)\\n    else:\\n        print(*[x+1 for x in X])\\nelse:\\n    print(-1)\\n\", \"n,m = list(map(int, input().split()))\\n\\nad = [[] for _ in range(n)]\\n\\nes = []\\nfor _ in range(m):\\n    v,u = list(map(int, input().split()))\\n    v-=1\\n    u-=1\\n    es.append((min(v,u),max(v,u)))\\nes = sorted(es, key=lambda x: (x[0], x[1]))\\n\\nfor e in es:\\n    v, u = e\\n    ad[v].append(str(u))\\n    ad[u].append(str(v))\\n\\n\\nad = [''.join(a) for a in ad]\\nd = {}\\nfor t in ad:\\n    if t not in d:\\n        d[t] = str(len(d) + 1)\\n    if len(d) > 3:\\n        print(-1)\\n        return\\nif len(d) != 3:\\n    print(-1)\\n    return\\n\\nans = []\\nfor t in ad:\\n    ans.append(d[t])\\nprint(' '.join(ans))\\n\", \"from collections import defaultdict, deque\\n\\n\\ndef threeSets(vs, es, d):\\n    sets = [3] * vs\\n    \\n    s1 = 0\\n    dist = [float('inf')] * vs\\n    sets[s1] = 1\\n    dist[s1] = 0\\n    queue = deque([s1])\\n    while queue:\\n        v1 = queue.pop()\\n        for v2 in d[v1]:\\n            if dist[v2] > dist[v1] + 1:\\n                dist[v2] = dist[v1] + 1\\n                queue.appendleft(v2)\\n    \\n    for i in range(vs):\\n        if dist[i] > 2:\\n            return [-1]\\n        elif dist[i] == 2:\\n            sets[i] = 1\\n        \\n    try:\\n        s2 = sets.index(3)\\n    except:\\n        return [-1]\\n    \\n    dist = [float('inf')] * vs\\n    sets[s2] = 2\\n    dist[s2] = 0\\n    queue = deque([s2])\\n    while queue:\\n        v1 = queue.pop()\\n        for v2 in d[v1]:\\n            if dist[v2] > dist[v1] + 1:\\n                dist[v2] = dist[v1] + 1\\n                queue.appendleft(v2)\\n    \\n    for i in range(vs):\\n        if dist[i] > 2:\\n            return [-1]\\n        elif dist[i] == 2:\\n            if sets[i] == 1:\\n                return [-1]\\n            else:\\n                sets[i] = 2\\n                \\n    VS = [set() for i in range(3)]\\n    for i in range(vs):\\n        g = sets[i] - 1\\n        VS[g].add(i)\\n    for V in VS:\\n        if not len(V):\\n            return [-1]\\n        \\n    for v1 in VS[0]:\\n        for v2 in VS[1]:\\n            if v2 not in d[v1]:\\n                return [-1]\\n        \\n        for v2 in VS[2]:\\n            if v2 not in d[v1]:\\n                return [-1]\\n    \\n    for v1 in VS[1]:\\n        for v2 in VS[2]:\\n            if v2 not in d[v1]:\\n                return [-1]\\n            \\n    valid_es = len(VS[0]) * len(VS[1]) + len(VS[0]) * len(VS[2]) + len(VS[1]) * len(VS[2])\\n    return sets if es == valid_es else [-1]       \\n        \\n    \\nd = defaultdict(set)\\n\\nvs, es = list(map(int, input().split()))\\nfor _ in range(es):\\n    v1, v2 = list(map(int, input().split()))\\n    v1 -= 1\\n    v2 -= 1\\n    d[v1].add(v2)\\n    d[v2].add(v1)\\n        \\nprint(*threeSets(vs, es, d))\\n\", \"# python template for atcoder1\\nimport sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\n\\nclass UnionFind:\\n    def __init__(self, N):\\n        self.parent = [i for i in range(N)]\\n        self.size = [1 for _ in range(N)]\\n\\n    def find(self, x):\\n        if self.parent[x] == x:\\n            return x\\n        else:\\n            return self.find(self.parent[x])\\n\\n    def union(self, x, y):\\n        px = self.find(x)\\n        py = self.find(y)\\n        if px == py:\\n            return\\n        if self.size[px] < self.size[py]:\\n            self.parent[px] = py\\n            self.size[py] += self.size[px]\\n        else:\\n            self.parent[py] = px\\n            self.size[px] += self.size[py]\\n\\n    def same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def connectedNum(self, x):\\n        return self.size[self.find(x)]\\n\\n    def component_NUM(self):\\n        par = set()\\n        for i in self.parent:\\n            par.add(self.find(i))\\n        return len(par)\\n\\n\\nN, M = list(map(int, input().split()))\\nadj = [set() for _ in range(N)]\\nUn = UnionFind(N)\\nfor _ in range(M):\\n    a, b = [int(x)-1 for x in input().split()]\\n    adj[a].add(b)\\n    adj[b].add(a)\\n\\nadded = set()\\nrepresentative = set()\\nfor i in range(N):\\n    if i in added:\\n        continue\\n    added.add(i)\\n    representative.add(i)\\n    for j in range(i+1, N):\\n        if j in added:\\n            continue\\n        if adj[i] == adj[j]:\\n            added.add(j)\\n            Un.union(i, j)\\n    if len(representative) > 3:\\n        print(-1)\\n        return\\nif Un.component_NUM() == 3:\\n    group = {}\\n    ans = []\\n    for p in range(N):\\n        par = Un.find(p)\\n        if par not in list(group.keys()):\\n            group[par] = len(group)+1\\n        ans.append(group[par])\\n    print(\\\" \\\".join(map(str, ans)))\\nelse:\\n    print(-1)\\n\", \"def main():\\n    from array import array\\n    from sys import stdin, stdout\\n    n, m = list(map(int, stdin.readline().split()))\\n    inp = {tuple(map(int, stdin.readline().split())) for _ in range(m)}\\n    ans = array('b', (0,)) * (n + 1)\\n    ans[1] = 1\\n    c1 = 1\\n    c2 = c3 = 0\\n    i = 1\\n    for j in range(2, n + 1):\\n        if (1, j) not in inp:\\n            ans[j] = 1\\n            c1 += 1\\n        else:\\n            if i == 1:\\n                i = j\\n            if (i, j) in inp:\\n                ans[j] = 2\\n                c2 += 1\\n            else:\\n                ans[j] = 3\\n                c3 += 1\\n    if m != c1 * (c2 + c3) + c2 * c3 or not c2:\\n        stdout.write('-1')\\n    else:\\n        for i, j in inp:\\n            if ans[i] == ans[j]:\\n                stdout.write('-1')\\n                break\\n        else:\\n            stdout.write(' '.join((str(ansi) for ansi in ans if ansi)))\\n\\n\\nmain()\\n\", \"\\\"\\\"\\\"\\nNTC here\\n\\\"\\\"\\\"\\nfrom sys import setcheckinterval, stdin, setrecursionlimit\\nsetcheckinterval(1000)\\nsetrecursionlimit(10**7)\\n \\n# print(\\\"Case #{}: {} {}\\\".format(i, n + m, n * m))\\n \\n \\ndef iin(): return int(stdin.readline())\\n \\n \\ndef lin(): return list(map(int, stdin.readline().split()))\\n\\ndef BFS(s, adj):\\n    parent = {s: None}\\n    color = [-1]*(len(adj))\\n    color[s]+=1\\n    u = [s]\\n    while u:  # runs till u is []\\n        nextu = []\\n        for i in u:\\n            for v in adj[i]:\\n                if v not in parent:\\n                    color[v]=(color[i]+1)%3\\n                    parent[v] = i\\n                    nextu.append(v)\\n                else:\\n                    if color[v]==color[i]:\\n                        color[v]=(color[v]+1)%3\\n        u = nextu.copy()\\n    for i in range(len(adj)):\\n        color[i]+=1\\n    return color\\n\\n\\n\\nn,m=lin()\\nadj=[[] for i in range(n)]\\nfor _ in range(m):\\n    i,j=lin()\\n    adj[i-1].append(j-1)\\n    adj[j-1].append(i-1)\\n\\nsol=BFS(0, adj)\\nc1=[0,0,0] #count of 1,2,3\\nfor i in sol:\\n    if i==0:\\n        print(-1)\\n        return\\n    c1[i-1]+=1\\nch=[[0 for i in range(3)] for j in range(3)] #12,13,23\\n\\nfor v in range(n):\\n    for u in adj[v]:\\n        ch[sol[v]-1][sol[u]-1]+=1\\n#        ch[sol[u]-1][sol[v]-1]+=1\\n        if sol[v]==sol[u]:\\n            print(-1)\\n            return\\na12=ch[0][1]+ch[1][0]\\na13=ch[0][2]+ch[2][0]\\na23=ch[1][2]+ch[2][1]\\n#print(a12,a13,a23,c1,sol)\\nif a12==2*c1[0]*c1[1] and a13==2*c1[0]*c1[2] and a23==2*c1[1]*c1[2]:\\n    print(*sol)\\nelse:\\n    print(-1)\", \"n, m = map(int, input().split())\\nG = [[] for _ in range(n)]\\nfor i in range(m):\\n\\ta, b = map(int, input().split())\\n\\ta -= 1\\n\\tb -= 1\\n\\tG[a].append(b)\\n\\tG[b].append(a)\\n\\nif len(G[0]) < 2:\\n\\tprint(-1)\\n\\treturn\\n\\nres = [1]*n\\nfor a in G[0]:\\n\\tres[a] = 2\\na2 = G[0][0]\\nfor b in G[a2]:\\n\\tif res[b] == 2: res[b] = 3\\nsizes = [n-len(G[0]), n-len(G[a2]), len(G[0])+len(G[a2])-n]\\nif 0 in sizes:\\n\\tprint(-1)\\n\\treturn\\n\\nfor i in range(n):\\n\\tg = res[i]\\n\\ts = sizes[g-1]\\n\\tif len(G[i]) != n-s:\\n\\t\\tprint(-1)\\n\\t\\treturn\\n\\tfor j in G[i]:\\n\\t\\tif res[j] == g:\\n\\t\\t\\tprint(-1)\\n\\t\\t\\treturn\\n\\nprint(*res)\", \"import sys\\ninput = sys.stdin.readline\\nfrom collections import deque\\nn, m = [int(item) for item in input().split()]\\nab = []\\nedges = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b = [int(item) for item in input().split()]\\n    a -= 1; b -= 1\\n    edges[a].append(b)\\n    edges[b].append(a)\\n    ab.append((a, b))\\n\\n    \\ngroupA = [1] * n\\nfor a, b in ab:\\n    if a == 0:\\n        groupA[b] = 0\\n    elif b == 0:\\n        groupA[a] = 0\\npar = None\\nfor i, item in enumerate(groupA):\\n    if item == 0:\\n        par = i\\n        break\\nif par == None:\\n    print(-1)\\n    return\\n\\ngroupB = [1] * n\\nfor a, b in ab:\\n    if a == par:\\n        groupB[b] = 0\\n    elif b == par:\\n        groupB[a] = 0\\n\\npar = None\\nfor i, (p, q) in enumerate(zip(groupA, groupB)):\\n    if p == 0 and q == 0:\\n        par = i\\n        break\\nif par == None:\\n    print(-1)\\n    return\\n\\ngroupC = [1] * n\\nfor a, b in ab:\\n    if a == par:\\n        groupC[b] = 0\\n    elif b == par:\\n        groupC[a] = 0\\n\\n# Check edge num\\nsumA = sum(groupA)\\nsumB = sum(groupB)\\nsumC = sum(groupC)\\ne_abc = [0, n - sumA, n - sumB, n - sumC]\\nedge_num = sumA * sumB + sumB * sumC + sumC * sumA \\nif edge_num != m:\\n    print(-1)\\n    return\\n\\n# Answer\\nsetA = set()\\nsetB = set()\\nsetC = set()\\ngroup = [groupA, groupB, groupC]\\nret = []\\nfor i, (ga, gb, gc) in enumerate(zip(groupA, groupB, groupC)):\\n    total = ga + gb + gc\\n    if total != 1:\\n        print(-1)\\n        return\\n    if ga:\\n        ret.append(1)\\n        setA.add(i)\\n    elif gb:\\n        ret.append(2)\\n        setB.add(i)\\n    else:\\n        ret.append(3)\\n        setC.add(i)\\ns_ABC = [set(), setA, setB, setC]\\nfor i, item in enumerate(ret):\\n    if e_abc[item] != len(edges[i]):\\n        print(-1)\\n        return\\n    for node in edges[i]:\\n        if node in s_ABC[item]:\\n            print(-1)\\n            return\\nprint(\\\" \\\".join([str(item) for item in ret]))\", \"import sys\\ninput = sys.stdin.readline\\n\\nn,m=list(map(int,input().split()))\\nE=[set() for i in range(n+1)]\\nECOUNT=[0]*(n+1)\\n\\nfor i in range(m):\\n    x,y=list(map(int,input().split()))\\n    E[x].add(y)\\n    E[y].add(x)\\n    ECOUNT[x]+=1\\n    ECOUNT[y]+=1\\n\\n\\nGroup=[i for i in range(n+1)]\\n\\ndef find(x):\\n    while Group[x] != x:\\n        x=Group[x]\\n    return x\\n\\ndef Union(x,y):\\n    if find(x) != find(y):\\n        Group[find(y)]=Group[find(x)]=min(find(y),find(x))\\n\\nSCORE=0\\nfor i in range(1,n+1):\\n    if find(i)==i:\\n        SCORE+=1\\n        if SCORE>=6:\\n            print(-1)\\n            return\\n            \\n        for j in range(i+1,n+1):\\n            if j in E[i]:\\n                continue\\n            else:\\n                Union(i,j)\\n\\nFD=[find(i) for i in range(n+1)]\\nif len(set(FD[1:]))!=3:\\n    print(-1)\\n    return\\n\\nfor i in range(n+1):\\n    for j in E[i]:\\n        if FD[i]==FD[j]:\\n            print(-1)\\n            return\\n\\n\\ncompression_dict={a: ind for ind, a in enumerate(sorted(set(FD)))}\\nANS=[compression_dict[a] for a in FD]\\n\\nVFD=[0,0,0]\\nEFD=[0,0,0]\\nfor i in range(1,n+1):\\n    VFD[ANS[i]-1]+=1\\n    EFD[ANS[i]-1]+=ECOUNT[i]\\n\\nfor i in range(3):\\n    VS=0\\n    for j in range(3):\\n        if i==j:\\n            continue\\n        VS+=VFD[i]*VFD[j]\\n\\n    if EFD[i]==VS:\\n        continue\\n    else:\\n        print(-1)\\n        return\\n\\n\\nprint(*ANS[1:])\\n\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn,m=list(map(int,input().split()))\\nE=[set() for i in range(n+1)]\\nECOUNT=[0]*(n+1)\\n\\nfor i in range(m):\\n    x,y=list(map(int,input().split()))\\n    E[x].add(y)\\n    E[y].add(x)\\n    ECOUNT[x]+=1\\n    ECOUNT[y]+=1\\n\\n\\nGroup=[i for i in range(n+1)]\\n\\ndef find(x):\\n    while Group[x] != x:\\n        x=Group[x]\\n    return x\\n\\ndef Union(x,y):\\n    if find(x) != find(y):\\n        Group[find(y)]=Group[find(x)]=min(find(y),find(x))\\n\\nSCORE=0\\nfor i in range(1,n+1):\\n    if find(i)==i:\\n        SCORE+=1\\n        if SCORE>=4:\\n            print(-1)\\n            return\\n            \\n        for j in range(i+1,n+1):\\n            if j in E[i]:\\n                continue\\n            else:\\n                Union(i,j)\\n\\nFD=[find(i) for i in range(n+1)]\\nif len(set(FD[1:]))!=3:\\n    print(-1)\\n    return\\n\\nfor i in range(n+1):\\n    for j in E[i]:\\n        if FD[i]==FD[j]:\\n            print(-1)\\n            return\\n\\n\\ncompression_dict={a: ind for ind, a in enumerate(sorted(set(FD)))}\\nANS=[compression_dict[a] for a in FD]\\n\\nVFD=[0,0,0]\\nEFD=[0,0,0]\\nfor i in range(1,n+1):\\n    VFD[ANS[i]-1]+=1\\n    EFD[ANS[i]-1]+=ECOUNT[i]\\n\\nfor i in range(3):\\n    VS=0\\n    for j in range(3):\\n        if i==j:\\n            continue\\n        VS+=VFD[i]*VFD[j]\\n\\n    if EFD[i]==VS:\\n        continue\\n    else:\\n        print(-1)\\n        return\\n\\n\\nprint(*ANS[1:])\\n\\n\\n\", \"from sys import stdin\\ninput = stdin.readline\\nv, e = map(int,input().split())\\nd = {}\\nfor i in range(1, v+1):\\n\\td[i] = []\\nfor ed in range(e):\\n\\ta,b = map(int,input().split())\\n\\td[a].append(b)\\n\\td[b].append(a)\\nfor i in d:\\n\\td[i].sort()\\nsas = {}\\nfor i in d:\\n\\tsas[tuple(d[i])] = 0\\nif len(sas) != 3:\\n\\tprint(-1)\\nelse:\\n\\tnbs = []\\n\\tfor i in sas:\\n\\t\\tnbs.append(set(i))\\n\\ta = nbs[0].intersection(nbs[1])\\n\\tb = nbs[0].intersection(nbs[2])\\n\\tc = nbs[1].intersection(nbs[2])\\n\\tkol = [0]*(v+1)\\n\\tfor i in a:\\n\\t\\tkol[i] = 1\\n\\tfor i in b:\\n\\t\\tkol[i] = 2\\n\\tfor i in c:\\n\\t\\tkol[i] = 3\\n\\tzer = 0\\n\\tfor i in range(1,len(kol)):\\n\\t\\tif kol[i] == 0:\\n\\t\\t\\tzer += 1\\n\\t\\t\\tbreak\\n\\tif zer > 0:\\n\\t\\tprint(-1)\\n\\telse:\\n\\t\\tif len(a)+len(b)+len(c) !=v:\\n\\t\\t\\tprint(-1)\\n\\t\\telse:\\n\\t\\t\\tprint(*kol[1:])\", \"# https://codeforces.com/contest/1228/problem/D\\n# all neightbor in group --> pass 1\\n# all neighbor not in group --> merge 0\\n# invalid 2\\n# WA\\ndef type_(list_v, group):\\n    cnt_0 = 0\\n    cnt_1 = 0\\n    \\n    for v in list_v:\\n        if v in group:\\n            cnt_1 += 1\\n        else:\\n            cnt_0 += 1\\n            \\n    if cnt_1 == len(group):\\n        return 1\\n    \\n    if cnt_0 == len(list_v):\\n        return 0\\n    \\n    return 2\\n    \\ndef is_all_type_1(ex_index, list_group, v):\\n    for i, group in list(list_group.items()):\\n        if i == ex_index:\\n            continue\\n            \\n        if type_(g[v], group) != 1: \\n            return False\\n        \\n    return True\\n    \\ndef check(v, list_group):\\n    t = None\\n    for i, group in list(list_group.items()):\\n        t  = type_(g[v], group)\\n        \\n        if t == 0 or t == 2:\\n            if t == 0:\\n                if is_all_type_1(i, list_group, v) == True:\\n                    group[v] = 1\\n                else:\\n                    return 2\\n            return t\\n        \\n    return t    \\n    \\ngroup = {}    \\ndef process(g):    \\n    for v in g:\\n        if len(group) == 0:\\n            group[0]    = {}\\n            group[0][v] = 1\\n            continue\\n    \\n        t = check(v, group)\\n        \\n        if t == 2:\\n            return -1\\n        \\n        if t == 1:\\n            if len(group) == 3:\\n                return -1\\n            \\n            group[len(group)]    = {}\\n            group[len(group)-1][v] = 1\\n    return group\\n\\ng = {}\\nn, m = list(map(int, input().split()))\\n\\nfor _ in range(m):\\n    u, v = list(map(int, input().split()))\\n    if u not in g:\\n        g[u] = []\\n    if v not in g:\\n        g[v] = []\\n        \\n    g[u].append(v)    \\n    g[v].append(u)\\n    \\nans = process(g)\\n\\nif ans == -1 or len(ans) < 3:\\n    print(-1)\\nelse:\\n    pr  = [0] * n\\n    \\n    cnt = 0\\n    for k, gr in list(group.items()):\\n        for v in gr:\\n            cnt += 1\\n            pr[v-1] = str(k+1)\\n    \\n    if cnt == n:\\n        print(' '.join(pr))\\n    else:\\n        print(-1)\\n# 1,2  3,4  5,6\\n#6 12\\n#1 3\\n#1 4\\n#2 3\\n#2 4\\n#1 5 \\n#1 6\\n#2 5\\n#2 6\\n#3 5\\n#3 6\\n#4 5\\n#4 6\\n\", \"from collections import defaultdict,deque\\nimport sys,heapq,bisect,math,itertools,string,queue,copy,time\\nsys.setrecursionlimit(10**8)\\nINF = float('inf')\\nmod = 10**9+7\\neps = 10**-7\\ndef inp(): return int(sys.stdin.readline())\\ndef inpl(): return list(map(int, sys.stdin.readline().split()))\\ndef inpl_str(): return list(sys.stdin.readline().split())\\n\\nN,M = inpl()\\n\\nif M == 0:\\n    print(-1)\\n    return\\n\\ncnts = [0]*N\\nlines = defaultdict(set)\\nfor _ in range(M):\\n    a,b = inpl()\\n    a-=1 ; b-=1\\n    cnts[a] += 1\\n    cnts[b] += 1\\n    lines[a].add(b)\\n    lines[b].add(a)\\n\\nsetc = set(cnts)\\nL = len(setc)\\nans = [-1]*N\\nif L == 1:\\n    ans[a] = 1\\n    ans[b] = 2\\n    for i in range(N):\\n        if ans[i] != 0:\\n            if i not in lines[a]:\\n                ans[i] = 1\\n            elif i not in lines[b]:\\n                ans[i] = 2\\n            else:\\n                ans[i] = 3\\nelif L == 2:\\n    c1,c2 = list(setc)\\n    if c1*2 + c2 == N*2:\\n        i1 = cnts.index(c1)\\n        i2 = cnts.index(c2)\\n        for i,c in enumerate(cnts):\\n            if c == c2:\\n                ans[i] = 2\\n            else:\\n                if i not in lines[i1]:\\n                    ans[i] = 1\\n                else:\\n                    ans[i] = 3\\n    elif c1 + c2*2 == N*2:\\n        i1 = cnts.index(c1)\\n        i2 = cnts.index(c2)\\n        for i,c in enumerate(cnts):\\n            if c == c1:\\n                ans[i] = 1\\n            else:\\n                if i not in lines[i2]:\\n                    ans[i] = 2\\n                else:\\n                    ans[i] = 3\\n    else:\\n        print(-1)\\n        return\\nelif L == 3:\\n    c1,c2,c3 = list(setc)\\n    for i,c in enumerate(cnts):\\n        if c == c1:\\n            ans[i] = 1\\n        elif c == c2:\\n            ans[i] = 2\\n        elif c == c3:\\n            ans[i] = 3\\nelse:\\n    print(-1)\\n    return\\n\\n\\nif len(set(ans)) != 3:\\n    print(-1)\\n    return\\n\\nfor s in range(N):\\n    for t in lines[s]:\\n        if ans[s] == ans[t]:\\n            print(-1)\\n            return\\n\\nprint(' '.join(map(str,ans)))\\n\", \"import sys\\ndef fill(graph,n):\\n    \\n    dp=[[True,True,True] for _ in range(n+1)]\\n    l=[-1 for i in range(n+1)]\\n    from collections import defaultdict\\n    vis=defaultdict(int)\\n    count1,count2,count3=0,0,0\\n    for i in graph:\\n        if dp[i][0]:\\n            #fill\\n            l[i],count1=1,count1+1\\n            vis[i]=1\\n            for j in graph[i]:\\n                dp[j][0]=False\\n        elif dp[i][1]:\\n            #fill\\n            l[i]=2\\n            count2+=1\\n            vis[i]=2\\n            for j in graph[i]:\\n                dp[j][1]=False\\n        elif dp[i][2]:\\n            #fill\\n            l[i]=3\\n            count3+=1\\n            vis[i]=3\\n            for j in graph[i]:\\n                dp[j][2]=False\\n        else:\\n            return [-1]\\n    \\n    if count1==0 or count2==0 or count3==0:\\n        return [-1]\\n    if count1+count2+count3!=n:\\n        return [-1]\\n    if count1*count2+count2*count3+count1*count3!=m:\\n        return [-1]\\n    l.pop(0)\\n    for i in l:\\n        if i==-1:\\n            return [-1]\\n    return l\\nn,m=list(map(int,sys.stdin.readline().split()))\\nfrom collections import defaultdict\\ngraph=defaultdict(list)\\nfor i in range(m):\\n    a,b=list(map(int,sys.stdin.readline().split()))\\n    graph[a].append(b)\\n    graph[b].append(a)\\nk=fill(graph,n)\\nprint(*k)\\n\", \"a=list(map(int,input().split()))\\nn=a[0]\\ne=a[1]\\ng={}\\nfor itr in range(1,n+1):\\n    g[itr]=[]\\nfor i in range(e):\\n    a=list(map(int,input().split()))\\n    g[a[0]].append(a[1])\\n    g[a[1]].append(a[0])\\nfor itr in range(1,n+1):\\n    g[itr]=frozenset(g[itr])\\na={}\\nk=1\\nres=[]\\nfor i in range(1,n+1):\\n    if len(g[i])==0:\\n        k=100\\n        break\\n    if g[i] in a: \\n        res.append(a[g[i]])\\n    else: \\n        a[g[i]]=k\\n        k+=1\\n        res.append(a[g[i]])\\n    if len(a)>3:break\\nif k!=4 : print(-1)\\nelse: print(*res)\", \"n,m=list(map(int,input().split()))\\nEE=[]\\nif m<3:\\n  print(-1)\\nelse:\\n  edge = [set() for i in range(n)]\\n  a,b=list(map(int,input().split()))\\n  edge[a-1].add(b-1)\\n  edge[b-1].add(a-1)\\n  EE.append([a-1,b-1])\\n\\n  for i in range(m-1):\\n    x,y=list(map(int,input().split()))\\n    edge[x-1].add(y-1)\\n    edge[y-1].add(x-1)\\n    EE.append([x-1,y-1])\\n  c=0  \\n  for i in range(n):\\n    if a-1 in edge[i] and b-1 in edge[i]:\\n      c=i+1\\n      break\\n  if c==0:\\n    print(-1)\\n  else:\\n    Ans=[0]*n\\n    Ans[a-1]=1\\n    Ans[b-1]=2\\n    Ans[c-1]=3\\n    flg=True\\n    C=[1]*3\\n    for i in range(n):\\n      if Ans[i]!=0:\\n        continue\\n      else:\\n        E=edge[i]\\n        if a-1 in E and b-1 in E and c-1 not in E:\\n          Ans[i]=3\\n          C[2]+=1\\n        elif a-1 in E and b-1 not in E and c-1 in E:\\n          Ans[i]=2\\n          C[1]+=1\\n        elif a-1 not in E and b-1 in E and c-1 in E:\\n          Ans[i]=1\\n          C[0]+=1\\n        else:\\n          print(-1)\\n          flg=False\\n          break\\n    if flg:\\n      T=[0]*3\\n      for x,y in EE:\\n        xx,yy=Ans[x],Ans[y]\\n        if xx==yy:\\n          print(-1)\\n          flg=False\\n          break\\n        elif xx==1 and yy==2:\\n          T[0]+=1\\n        elif xx==2 and yy==1:\\n          T[0]+=1\\n        elif xx==1 and yy==3:\\n          T[1]+=1\\n        elif xx==3 and yy==1:\\n          T[1]+=1\\n        else:\\n          T[2]+=1\\n    if flg:\\n      if T[0]==C[0]*C[1] and T[1]==C[0]*C[2] and T[2]==C[1]*C[2]:\\n        print(*Ans)\\n      else:\\n        print(-1)\\n\\n\\n\\n\\n\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\n \\nN, M = map(int, input().split())\\nE = [[] for _ in range(N)]\\nfor _ in range(M):\\n    a, b = map(int, input().split())\\n    E[a-1].append(b-1)\\n    E[b-1].append(a-1)\\n \\ninf = 1 << 20\\nA, B, C = [], [], []\\nX = [0] * N\\nfor a in E[0]:\\n    X[a] = 1\\nA = [i for i in range(N) if X[i]==0]\\nb = min([i for i in range(N) if X[i] == 1] + [inf])\\nif b < inf:\\n    for a in E[b]:\\n        if X[a] == 1: X[a] = 2\\n    B = [i for i in range(N) if X[i]==1]\\nc = min([i for i in range(N) if X[i] == 2] + [inf])\\nif c < inf:\\n    for a in E[c]:\\n        if X[a] == 2: X[a] = 3\\n    C = [i for i in range(N) if X[i]==2]\\n \\nif max(X) == 2 and len(A) * len(B) * len(C) and (len(A) + len(B) + len(C) == N) and (len(A) * len(B) + len(B) * len(C) + len(A) * len(C) == M):\\n    f = 0\\n    for i in range(N):\\n        for j in E[i]:\\n            if X[i] == X[j]:\\n                f = 1\\n                break\\n        if f: break\\n    if f:\\n        print(-1)\\n    else:\\n        print(*[x+1 for x in X])\\nelse:\\n    print(-1)\", \"n,m=list(map(int,input().split()))\\nEE=[]\\nif m<3:\\n  print(-1)\\nelse:\\n  edge = [[] for i in range(n)]\\n  a,b=list(map(int,input().split()))\\n  edge[a-1].append(b-1)\\n  edge[b-1].append(a-1)\\n  EE.append([a-1,b-1])\\n\\n  for i in range(m-1):\\n    x,y=list(map(int,input().split()))\\n    edge[x-1].append(y-1)\\n    edge[y-1].append(x-1)\\n    EE.append([x-1,y-1])\\n  c=0  \\n  for i in range(n):\\n    if a-1 in edge[i] and b-1 in edge[i]:\\n      c=i+1\\n      break\\n  if c==0:\\n    print(-1)\\n  else:\\n    Ans=[0]*n\\n    Ans[a-1]=1\\n    Ans[b-1]=2\\n    Ans[c-1]=3\\n    flg=True\\n    C=[1]*3\\n    for i in range(n):\\n      if Ans[i]!=0:\\n        continue\\n      else:\\n        E=edge[i]\\n        if a-1 in E and b-1 in E and c-1 not in E:\\n          Ans[i]=3\\n          C[2]+=1\\n        elif a-1 in E and b-1 not in E and c-1 in E:\\n          Ans[i]=2\\n          C[1]+=1\\n        elif a-1 not in E and b-1 in E and c-1 in E:\\n          Ans[i]=1\\n          C[0]+=1\\n        else:\\n          print(-1)\\n          flg=False\\n          break\\n    if flg:\\n      T=[0]*3\\n      for x,y in EE:\\n        xx,yy=Ans[x],Ans[y]\\n        if xx==yy:\\n          print(-1)\\n          flg=False\\n          break\\n        elif xx==1 and yy==2:\\n          T[0]+=1\\n        elif xx==2 and yy==1:\\n          T[0]+=1\\n        elif xx==1 and yy==3:\\n          T[1]+=1\\n        elif xx==3 and yy==1:\\n          T[1]+=1\\n        else:\\n          T[2]+=1\\n    if flg:\\n      if T[0]==C[0]*C[1] and T[1]==C[0]*C[2] and T[2]==C[1]*C[2]:\\n        print(*Ans)\\n      else:\\n        print(-1)\\n\\n\\n\\n\\n\\n\", \"3\\n\\nimport array\\nimport math\\nimport os\\nimport random\\nimport sys\\n\\n\\nDEBUG = 'DEBUG' in os.environ\\n\\n\\ndef inp():\\n    return sys.stdin.readline().rstrip()\\n\\n\\ndef dprint(*value, sep=' ', end='\\\\n'):\\n    if DEBUG:\\n        print(*value, sep=sep, end=end)\\n\\n\\ndef solve(N, M, G):\\n    A = {}\\n\\n    for i in range(N):\\n        t = frozenset(G[i])\\n        if t not in A:\\n            A[t] = set([i])\\n        else:\\n            A[t].add(i)\\n\\n    if len(A) != 3:\\n        return None\\n\\n    (a1, v1), (a2, v2), (a3, v3) = A.items()\\n    v1 = frozenset(v1)\\n    v2 = frozenset(v2)\\n    v3 = frozenset(v3)\\n\\n    if a1 != v2 | v3 or a2 != v3 | v1 or a3 != v1 | v2:\\n        return None\\n\\n    ans = [0] * N\\n    for v in v1:\\n        ans[v] = 1\\n    for v in v2:\\n        ans[v] = 2\\n    for v in v3:\\n        ans[v] = 3\\n    return ans\\n\\n\\ndef main():\\n    N, M = [int(e) for e in inp().split()]\\n    G = [[] for _ in range(N)]\\n    for _ in range(M):\\n        a, b = [int(e) - 1 for e in inp().split()]\\n        G[a].append(b)\\n        G[b].append(a)\\n\\n    ans = solve(N, M, G)\\n    if not ans:\\n        print('-1')\\n    else:\\n        print(*ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"3\\n\\nimport math\\nimport os\\nimport sys\\n\\n\\nDEBUG = 'DEBUG' in os.environ\\n\\n\\ndef inp():\\n    return sys.stdin.readline().rstrip()\\n\\n\\ndef dprint(*value, sep=' ', end='\\\\n'):\\n    if DEBUG:\\n        print(*value, sep=sep, end=end)\\n\\n\\ndef solve(N, M, G):\\n    A = {}\\n\\n    for i in range(N):\\n        t = frozenset(G[i])\\n        if t not in A:\\n            A[t] = set([i])\\n        else:\\n            A[t].add(i)\\n\\n    if len(A) != 3:\\n        return None\\n\\n    (a1, v1), (a2, v2), (a3, v3) = A.items()\\n    v1 = frozenset(v1)\\n    v2 = frozenset(v2)\\n    v3 = frozenset(v3)\\n\\n    if a1 != v2 | v3 or a2 != v3 | v1 or a3 != v1 | v2:\\n        return None\\n\\n    ans = [0] * N\\n    for v in v1:\\n        ans[v] = 1\\n    for v in v2:\\n        ans[v] = 2\\n    for v in v3:\\n        ans[v] = 3\\n    return ans\\n\\n\\ndef main():\\n    N, M = [int(e) for e in inp().split()]\\n    G = [[] for _ in range(N)]\\n    for _ in range(M):\\n        a, b = [int(e) - 1 for e in inp().split()]\\n        G[a].append(b)\\n        G[b].append(a)\\n\\n    ans = solve(N, M, G)\\n    if not ans:\\n        print('-1')\\n    else:\\n        print(*ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,m=map(int,input().split())\\nmaxi=1000009\\nd={}\\na=[['0'] for i in range(n+1)]\\n\\nfor i in range(m):\\n    u,v=map(int,input().split())\\n    a[u].append(str(v)+'*')\\n    a[v].append(str(u)+'*')\\n\\ncount=1\\nfor i in range(1,n+1):\\n    if len(a[i])==1:\\n        print(\\\"-1\\\")\\n        return\\n    a[i].sort()\\nfor i in range(1,n+1):    \\n    a[i]=\\\"\\\".join(a[i])\\n    #print(a[i])\\nfor i in range(1,n+1):\\n    if a[i] not in d:\\n        d[a[i]]=count\\n        count+=1\\n#print(d)  \\nif len(d)!=3:\\n    #print(s)\\n    print(\\\"-1\\\")\\n    return\\nfor i in range(1,n+1):\\n    print(d[a[i]],end=\\\" \\\")\\n  \", \"n,m=map(int,input().split())\\nd={}\\na=[['0'] for i in range(n+1)]\\nfor i in range(m):\\n    u,v=map(int,input().split())\\n    a[u].append(str(v)+'*')\\n    a[v].append(str(u)+'*')\\ncount=1\\nfor i in range(1,n+1):\\n    if len(a[i])==1:\\n        print(\\\"-1\\\")\\n        return\\n    a[i].sort()\\nfor i in range(1,n+1):    \\n    a[i]=\\\"\\\".join(a[i])\\nfor i in range(1,n+1):\\n    if a[i] not in d:\\n        d[a[i]]=count\\n        count+=1\\nif len(d)!=3:\\n    print(\\\"-1\\\")\\n    return\\nfor i in range(1,n+1):\\n    print(d[a[i]],end=\\\" \\\")\\n  \\n    \", \"n,m=map(int,input().split())\\nd={}\\na=[['0'] for i in range(n+1)]\\nfor i in range(m):\\n    u,v=map(int,input().split())\\n    a[u].append(str(v)+'*')\\n    a[v].append(str(u)+'*')\\ncount=1\\nfor i in range(1,n+1):\\n    if len(a[i])==1:\\n        print(\\\"-1\\\")\\n        return\\n    a[i].sort()\\nfor i in range(1,n+1):    \\n    a[i]=\\\"\\\".join(a[i])\\nfor i in range(1,n+1):\\n    if a[i] not in d:\\n        d[a[i]]=count\\n        count+=1\\nif len(d)!=3:\\n    print(\\\"-1\\\")\\n    return\\nfor i in range(1,n+1):\\n    print(d[a[i]],end=\\\" \\\")\\n  \", \"try:\\n    from string_source import string_source\\nexcept ImportError:\\n    source = input\\n\\nfrom collections import defaultdict\\n\\n\\ndef complete_bipartite(n, edges):\\n    if not edges:\\n        return False\\n\\n    neighbors = defaultdict(set, {})\\n\\n    for e1, e2 in edges:\\n        neighbors[e1].add(e2)\\n        neighbors[e2].add(e1)\\n\\n    s1, s2 = edges[0]\\n\\n    groups = [None, None, None]\\n    groups[2] = neighbors[s1].intersection(neighbors[s2])\\n    groups[1] = neighbors[s1].difference(neighbors[s2])\\n    groups[0] = set(neighbors).difference(groups[2], groups[1])\\n\\n    def get_group(node):\\n        return next(idx for idx, g in enumerate(groups) if node in g)\\n\\n    sizes = [len(g) for g in groups]\\n\\n    if sum(len(g) for g in groups) < n:\\n        return False\\n    if any(len(g) == 0 for g in groups):\\n        return False\\n\\n    for e1, e2 in edges:\\n        if get_group(e1) == get_group(e2):\\n            return False\\n\\n    answer = [0] * n\\n    for node, neigh in list(neighbors.items()):\\n        g = get_group(node)\\n        if sum(s for idx, s in enumerate(sizes) if idx != g) != len(neigh):\\n            return False\\n\\n        answer[node - 1] = g + 1\\n\\n    return answer\\n\\n\\ndef answer(source):\\n    n, m = list(map(int, source().strip().split()))\\n    edges = [[int(k) for k in source().strip().split()] for _ in range(m)]\\n\\n    a = complete_bipartite(n, edges)\\n    if not a:\\n        print(-1)\\n    else:\\n        print(\\\" \\\".join(map(str, a)))\\n\\n\\nanswer(input)\\n\\nif False:\\n    answer(\\n        string_source(\\n            \\\"\\\"\\\"6 11\\n    1 2\\n    1 3\\n    1 4\\n    1 5\\n    1 6\\n    2 4\\n    2 5\\n    2 6\\n    3 4\\n    3 5\\n    3 6\\\"\\\"\\\"\\n        )\\n    )\\n\\n    answer(\\n        string_source(\\n            \\\"\\\"\\\"4 6\\n    1 2\\n    1 3\\n    1 4\\n    2 3\\n    2 4\\n    3 4\\\"\\\"\\\"\\n        )\\n    )\\n\"]",
        "difficulty": "interview",
        "input": "3 0\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1228/D"
    },
    {
        "id": 280,
        "task_id": 2459,
        "test_case_id": 1,
        "question": "You are given an array a of size n, and q queries to it. There are queries of two types:   1 l_{i} r_{i} — perform a cyclic shift of the segment [l_{i}, r_{i}] to the right. That is, for every x such that l_{i} ≤ x < r_{i} new value of a_{x} + 1 becomes equal to old value of a_{x}, and new value of a_{l}_{i} becomes equal to old value of a_{r}_{i};  2 l_{i} r_{i} — reverse the segment [l_{i}, r_{i}].  \n\nThere are m important indices in the array b_1, b_2, ..., b_{m}. For each i such that 1 ≤ i ≤ m you have to output the number that will have index b_{i} in the array after all queries are performed.\n\n\n-----Input-----\n\nThe first line contains three integer numbers n, q and m (1 ≤ n, q ≤ 2·10^5, 1 ≤ m ≤ 100). \n\nThe second line contains n integer numbers a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ 10^9). \n\nThen q lines follow. i-th of them contains three integer numbers t_{i}, l_{i}, r_{i}, where t_{i} is the type of i-th query, and [l_{i}, r_{i}] is the segment where this query is performed (1 ≤ t_{i} ≤ 2, 1 ≤ l_{i} ≤ r_{i} ≤ n). \n\nThe last line contains m integer numbers b_1, b_2, ..., b_{m} (1 ≤ b_{i} ≤ n) — important indices of the array. \n\n\n-----Output-----\n\nPrint m numbers, i-th of which is equal to the number at index b_{i} after all queries are done.\n\n\n-----Example-----\nInput\n6 3 5\n1 2 3 4 5 6\n2 1 3\n2 3 6\n1 1 6\n2 2 1 5 3\n\nOutput\n3 3 1 5 2",
        "solutions": "[\"# https://codeforces.com/contest/863/problem/D\\n\\n\\nfrom sys import stdin, stdout\\ninput = stdin.readline\\nprint = stdout.write\\n# solve the reversed problem\\nn, q, m = map(int, input().split())\\na = list(map(int, input().split()))\\nops = [list(map(int, input().split())) for _ in range(q)]\\nb = list(map(int, input().split()))\\n\\n\\ndef solve(index, ops):\\n    def _solve(index, op):\\n        t, l, r = op\\n        if index < l or index > r:\\n            return index\\n        if t == 1:\\n            if index == l:\\n                return r\\n            else:\\n                return index - 1\\n        else:\\n            return l + r - index\\n\\n    for op in ops[::-1]:\\n        index = _solve(index, op)\\n    return index\\n\\n\\nb = list(map(lambda x: solve(x, ops), b))\\nfor i in b:\\n    print(str(a[i-1])+\\\" \\\")\\n\\n# Cartesian tree:\\n# https://codeforces.com/contest/863/submission/30693678\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict,deque\\ndef get(ind ,arr):\\n\\tn = len(arr)\\n\\tfor i in range(n):\\n\\t\\tt,l,r = arr[i]\\n\\t\\tif t == 1:\\n\\t\\t\\tif l <= ind <= r:\\n\\t\\t\\t\\tif ind == l:\\n\\t\\t\\t\\t\\tind = r\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\tind -= 1\\n\\t\\t\\tcontinue\\n\\t\\tif t == 2:\\n\\t\\t\\tif l <=ind <= r:\\n\\t\\t\\t\\tind = (r - ind + l)\\n\\t\\t\\tcontinue\\n\\treturn ind\\nn,q,m = map(int,sys.stdin.readline().split())\\narr = list(map(int,sys.stdin.readline().split()))\\nl = []\\nfor i in range(q):\\n\\ta,b,c = map(int,sys.stdin.readline().split())\\n\\tl.append([a,b,c])\\nl.reverse()\\nb = list(map(int,sys.stdin.readline().split()))\\nans = []\\nfor i in range(m):\\n\\tx = get(b[i],l)\\n\\tans.append(arr[x -1])\\nprint(*ans)\"]",
        "difficulty": "interview",
        "input": "6 3 5\n1 2 3 4 5 6\n2 1 3\n2 3 6\n1 1 6\n2 2 1 5 3\n",
        "output": "3 3 1 5 2 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/863/D"
    },
    {
        "id": 281,
        "task_id": 2459,
        "test_case_id": 2,
        "question": "You are given an array a of size n, and q queries to it. There are queries of two types:   1 l_{i} r_{i} — perform a cyclic shift of the segment [l_{i}, r_{i}] to the right. That is, for every x such that l_{i} ≤ x < r_{i} new value of a_{x} + 1 becomes equal to old value of a_{x}, and new value of a_{l}_{i} becomes equal to old value of a_{r}_{i};  2 l_{i} r_{i} — reverse the segment [l_{i}, r_{i}].  \n\nThere are m important indices in the array b_1, b_2, ..., b_{m}. For each i such that 1 ≤ i ≤ m you have to output the number that will have index b_{i} in the array after all queries are performed.\n\n\n-----Input-----\n\nThe first line contains three integer numbers n, q and m (1 ≤ n, q ≤ 2·10^5, 1 ≤ m ≤ 100). \n\nThe second line contains n integer numbers a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ 10^9). \n\nThen q lines follow. i-th of them contains three integer numbers t_{i}, l_{i}, r_{i}, where t_{i} is the type of i-th query, and [l_{i}, r_{i}] is the segment where this query is performed (1 ≤ t_{i} ≤ 2, 1 ≤ l_{i} ≤ r_{i} ≤ n). \n\nThe last line contains m integer numbers b_1, b_2, ..., b_{m} (1 ≤ b_{i} ≤ n) — important indices of the array. \n\n\n-----Output-----\n\nPrint m numbers, i-th of which is equal to the number at index b_{i} after all queries are done.\n\n\n-----Example-----\nInput\n6 3 5\n1 2 3 4 5 6\n2 1 3\n2 3 6\n1 1 6\n2 2 1 5 3\n\nOutput\n3 3 1 5 2",
        "solutions": "[\"# https://codeforces.com/contest/863/problem/D\\n\\n\\nfrom sys import stdin, stdout\\ninput = stdin.readline\\nprint = stdout.write\\n# solve the reversed problem\\nn, q, m = map(int, input().split())\\na = list(map(int, input().split()))\\nops = [list(map(int, input().split())) for _ in range(q)]\\nb = list(map(int, input().split()))\\n\\n\\ndef solve(index, ops):\\n    def _solve(index, op):\\n        t, l, r = op\\n        if index < l or index > r:\\n            return index\\n        if t == 1:\\n            if index == l:\\n                return r\\n            else:\\n                return index - 1\\n        else:\\n            return l + r - index\\n\\n    for op in ops[::-1]:\\n        index = _solve(index, op)\\n    return index\\n\\n\\nb = list(map(lambda x: solve(x, ops), b))\\nfor i in b:\\n    print(str(a[i-1])+\\\" \\\")\\n\\n# Cartesian tree:\\n# https://codeforces.com/contest/863/submission/30693678\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict,deque\\ndef get(ind ,arr):\\n\\tn = len(arr)\\n\\tfor i in range(n):\\n\\t\\tt,l,r = arr[i]\\n\\t\\tif t == 1:\\n\\t\\t\\tif l <= ind <= r:\\n\\t\\t\\t\\tif ind == l:\\n\\t\\t\\t\\t\\tind = r\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\tind -= 1\\n\\t\\t\\tcontinue\\n\\t\\tif t == 2:\\n\\t\\t\\tif l <=ind <= r:\\n\\t\\t\\t\\tind = (r - ind + l)\\n\\t\\t\\tcontinue\\n\\treturn ind\\nn,q,m = map(int,sys.stdin.readline().split())\\narr = list(map(int,sys.stdin.readline().split()))\\nl = []\\nfor i in range(q):\\n\\ta,b,c = map(int,sys.stdin.readline().split())\\n\\tl.append([a,b,c])\\nl.reverse()\\nb = list(map(int,sys.stdin.readline().split()))\\nans = []\\nfor i in range(m):\\n\\tx = get(b[i],l)\\n\\tans.append(arr[x -1])\\nprint(*ans)\"]",
        "difficulty": "interview",
        "input": "5 2 5\n64 3 4 665 2\n1 1 3\n2 1 5\n1 2 3 4 5\n",
        "output": "2 665 3 64 4 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/863/D"
    },
    {
        "id": 282,
        "task_id": 2744,
        "test_case_id": 1,
        "question": "Musicians of a popular band \"Flayer\" have announced that they are going to \"make their exit\" with a world tour. Of course, they will visit Berland as well.\n\nThere are n cities in Berland. People can travel between cities using two-directional train routes; there are exactly m routes, i-th route can be used to go from city v_{i} to city u_{i} (and from u_{i} to v_{i}), and it costs w_{i} coins to use this route.\n\nEach city will be visited by \"Flayer\", and the cost of the concert ticket in i-th city is a_{i} coins.\n\nYou have friends in every city of Berland, and they, knowing about your programming skills, asked you to calculate the minimum possible number of coins they have to pay to visit the concert. For every city i you have to compute the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).\n\nFormally, for every $i \\in [ 1, n ]$ you have to calculate $\\operatorname{min}_{j = 1} 2 d(i, j) + a_{j}$, where d(i, j) is the minimum number of coins you have to spend to travel from city i to city j. If there is no way to reach city j from city i, then we consider d(i, j) to be infinitely large.\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 2·10^5, 1 ≤ m ≤ 2·10^5).\n\nThen m lines follow, i-th contains three integers v_{i}, u_{i} and w_{i} (1 ≤ v_{i}, u_{i} ≤ n, v_{i} ≠ u_{i}, 1 ≤ w_{i} ≤ 10^12) denoting i-th train route. There are no multiple train routes connecting the same pair of cities, that is, for each (v, u) neither extra (v, u) nor (u, v) present in input.\n\nThe next line contains n integers a_1, a_2, ... a_{k} (1 ≤ a_{i} ≤ 10^12) — price to attend the concert in i-th city.\n\n\n-----Output-----\n\nPrint n integers. i-th of them must be equal to the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).\n\n\n-----Examples-----\nInput\n4 2\n1 2 4\n2 3 7\n6 20 1 25\n\nOutput\n6 14 1 25 \n\nInput\n3 3\n1 2 1\n2 3 1\n1 3 1\n30 10 20\n\nOutput\n12 10 12",
        "solutions": "[\"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(fd, stream.getvalue()) + stream.truncate(0) + stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 2*10**12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(x))\\n    sys.stdout.write('\\\\n')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(fd, stream.getvalue()) + stream.truncate(0) + stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write('\\\\n')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\n###### ACTUAL CODE\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nimport heapq\\nQ = [(best[i],i) for i in range(n)]\\nheapq.heapify(Q)\\n\\nwhile Q:\\n    c,node = heapq.heappop(Q)\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            heapq.heappush(Q,(C, nei))\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\n###### REWRITTEN HEAPQ\\ndef heappush(heap, item):\\n    heap.append(item)\\n    _siftdown(heap, len(heap)-1)\\n\\ndef heappop(heap):\\n    lastelt = heap.pop()\\n    if not heap:\\n        return lastelt\\n\\n    returnitem,heap[0] = heap[0],lastelt\\n    _siftup(heap)\\n    return returnitem\\n\\n# Does a pop and then a push\\ndef heapreplace(heap, item):\\n    returnitem,heap[0] = heap[0],item\\n    _siftup(heap)\\n    return returnitem\\n\\n# Does a push and then a pop\\ndef heappushpop(heap, item):\\n    if heap and heap[0] < item:\\n        item, heap[0] = heap[0], item\\n        _siftup(heap)\\n    return item\\n\\ndef heapify(x):\\n    for i in reversed(range(len(x)//2)):\\n        _siftup(x, i)\\n\\n\\ndef _siftdown(heap, pos):\\n    newitem = heap[pos]\\n    ppos = (pos-1) >> 1\\n    while pos and newitem < heap[ppos]:\\n        heap[pos] = heap[ppos]\\n        pos = ppos\\n        ppos = (pos-1) >> 1\\n    heap[pos] = newitem\\n\\ndef _siftup(heap, pos=0):\\n    # Move the item at pos to a leaf\\n    # by switching place with smallest child (bias to right)\\n    newitem = heap[pos]\\n    \\n    leftchild = 2*pos + 1\\n    rightchild = leftchild + 1\\n    while rightchild < len(heap):\\n        if heap[leftchild] < heap[rightchild]:\\n            heap[pos] = heap[leftchild]\\n            pos = leftchild\\n        else:\\n            heap[pos] = heap[rightchild]\\n            pos = rightchild\\n        leftchild = 2*pos + 1\\n        rightchild = leftchild + 1\\n    if leftchild < len(heap): # Special case of only one child\\n        heap[pos] = heap[leftchild]\\n        pos = leftchild\\n    # Now newitem has been moved to an leaf\\n    heap[pos] = newitem\\n    _siftdown(heap,pos)\\n\\n###### ACTUAL CODE\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\n\\nQ = [(best[i],i) for i in range(n)]\\nheapify(Q)\\n\\nwhile Q:\\n    c,node = heappop(Q)\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            heappush(Q,(C, nei))\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s[-1]>=b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readuint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readfloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0; sign = 1.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            elif s[i] == b'-'[0]: sign = -1.0\\n            else: A.append(sign*numb); numb = 0.0; sign = 1.0\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readufloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            else: A.append(numb); numb = 0.0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readufloat()\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readfloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0; sign = 1.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            elif s[i] == b'-'[0]: sign = -1.0\\n            else: A.append(sign*numb); numb = 0.0; sign = 1.0\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readufloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            else: A.append(numb); numb = 0.0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readfloat()\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize:break\\n            c = self.stream.read(1)[0]\\n            curpos += 1\\n            if c >= b'0'[0]: numb = 10 * numb + (conv(c) - 48)\\n            elif c == b'-'[0]: sign = -1\\n            elif c != b'\\\\r'[0]: A.append(sign*numb); numb = zero; sign = 1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod, (Note, currently experimenting, not fully working)\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize:break\\n            c = self.stream.read(1)[0]\\n            curpos += 1\\n            if c >= b'0'[0]: numb = 10 * numb + (conv(c) - 48)\\n            elif c == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = zero; sign = 1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(1000,buffsize-curpos))\\n            for i in range(len(s)):\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n                if len(A)==n: break\\n            curpos += i+1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10000,buffsize-curpos))\\n            for i in range(len(s)):\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n                if len(A)==n: break\\n            curpos += i+1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10000,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(256,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(32,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, zero=0):\\n        conv = ord if py2 else lambda x:x\\n        s = self.read()\\n        A = []; numb = zero; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n            elif s[i] != b'\\\\r'[0]:\\n                if s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\ninp = sys.stdin.readnumbers(0.0)\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(128,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\n# OLD VERSION\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            small_buff = min(64,buffsize-curpos)\\n            s = self.stream.read(small_buff)\\n            i = 0\\n            while i<small_buff and len(A)<n:\\n                if s[i] >= b'-'[0]:\\n                    if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                    else: sign = -1    \\n                elif s[i] != b'\\\\r'[0]: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\"]",
        "difficulty": "interview",
        "input": "4 2\n1 2 4\n2 3 7\n6 20 1 25\n",
        "output": "6 14 1 25 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/938/D"
    },
    {
        "id": 283,
        "task_id": 2744,
        "test_case_id": 2,
        "question": "Musicians of a popular band \"Flayer\" have announced that they are going to \"make their exit\" with a world tour. Of course, they will visit Berland as well.\n\nThere are n cities in Berland. People can travel between cities using two-directional train routes; there are exactly m routes, i-th route can be used to go from city v_{i} to city u_{i} (and from u_{i} to v_{i}), and it costs w_{i} coins to use this route.\n\nEach city will be visited by \"Flayer\", and the cost of the concert ticket in i-th city is a_{i} coins.\n\nYou have friends in every city of Berland, and they, knowing about your programming skills, asked you to calculate the minimum possible number of coins they have to pay to visit the concert. For every city i you have to compute the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).\n\nFormally, for every $i \\in [ 1, n ]$ you have to calculate $\\operatorname{min}_{j = 1} 2 d(i, j) + a_{j}$, where d(i, j) is the minimum number of coins you have to spend to travel from city i to city j. If there is no way to reach city j from city i, then we consider d(i, j) to be infinitely large.\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 2·10^5, 1 ≤ m ≤ 2·10^5).\n\nThen m lines follow, i-th contains three integers v_{i}, u_{i} and w_{i} (1 ≤ v_{i}, u_{i} ≤ n, v_{i} ≠ u_{i}, 1 ≤ w_{i} ≤ 10^12) denoting i-th train route. There are no multiple train routes connecting the same pair of cities, that is, for each (v, u) neither extra (v, u) nor (u, v) present in input.\n\nThe next line contains n integers a_1, a_2, ... a_{k} (1 ≤ a_{i} ≤ 10^12) — price to attend the concert in i-th city.\n\n\n-----Output-----\n\nPrint n integers. i-th of them must be equal to the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).\n\n\n-----Examples-----\nInput\n4 2\n1 2 4\n2 3 7\n6 20 1 25\n\nOutput\n6 14 1 25 \n\nInput\n3 3\n1 2 1\n2 3 1\n1 3 1\n30 10 20\n\nOutput\n12 10 12",
        "solutions": "[\"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(fd, stream.getvalue()) + stream.truncate(0) + stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 2*10**12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(x))\\n    sys.stdout.write('\\\\n')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(fd, stream.getvalue()) + stream.truncate(0) + stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write('\\\\n')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\n###### ACTUAL CODE\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nimport heapq\\nQ = [(best[i],i) for i in range(n)]\\nheapq.heapify(Q)\\n\\nwhile Q:\\n    c,node = heapq.heappop(Q)\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            heapq.heappush(Q,(C, nei))\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\n###### REWRITTEN HEAPQ\\ndef heappush(heap, item):\\n    heap.append(item)\\n    _siftdown(heap, len(heap)-1)\\n\\ndef heappop(heap):\\n    lastelt = heap.pop()\\n    if not heap:\\n        return lastelt\\n\\n    returnitem,heap[0] = heap[0],lastelt\\n    _siftup(heap)\\n    return returnitem\\n\\n# Does a pop and then a push\\ndef heapreplace(heap, item):\\n    returnitem,heap[0] = heap[0],item\\n    _siftup(heap)\\n    return returnitem\\n\\n# Does a push and then a pop\\ndef heappushpop(heap, item):\\n    if heap and heap[0] < item:\\n        item, heap[0] = heap[0], item\\n        _siftup(heap)\\n    return item\\n\\ndef heapify(x):\\n    for i in reversed(range(len(x)//2)):\\n        _siftup(x, i)\\n\\n\\ndef _siftdown(heap, pos):\\n    newitem = heap[pos]\\n    ppos = (pos-1) >> 1\\n    while pos and newitem < heap[ppos]:\\n        heap[pos] = heap[ppos]\\n        pos = ppos\\n        ppos = (pos-1) >> 1\\n    heap[pos] = newitem\\n\\ndef _siftup(heap, pos=0):\\n    # Move the item at pos to a leaf\\n    # by switching place with smallest child (bias to right)\\n    newitem = heap[pos]\\n    \\n    leftchild = 2*pos + 1\\n    rightchild = leftchild + 1\\n    while rightchild < len(heap):\\n        if heap[leftchild] < heap[rightchild]:\\n            heap[pos] = heap[leftchild]\\n            pos = leftchild\\n        else:\\n            heap[pos] = heap[rightchild]\\n            pos = rightchild\\n        leftchild = 2*pos + 1\\n        rightchild = leftchild + 1\\n    if leftchild < len(heap): # Special case of only one child\\n        heap[pos] = heap[leftchild]\\n        pos = leftchild\\n    # Now newitem has been moved to an leaf\\n    heap[pos] = newitem\\n    _siftdown(heap,pos)\\n\\n###### ACTUAL CODE\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\n\\nQ = [(best[i],i) for i in range(n)]\\nheapify(Q)\\n\\nwhile Q:\\n    c,node = heappop(Q)\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            heappush(Q,(C, nei))\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s[-1]>=b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readuint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readfloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0; sign = 1.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            elif s[i] == b'-'[0]: sign = -1.0\\n            else: A.append(sign*numb); numb = 0.0; sign = 1.0\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readufloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            else: A.append(numb); numb = 0.0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readufloat()\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readfloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0; sign = 1.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            elif s[i] == b'-'[0]: sign = -1.0\\n            else: A.append(sign*numb); numb = 0.0; sign = 1.0\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readufloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            else: A.append(numb); numb = 0.0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readfloat()\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize:break\\n            c = self.stream.read(1)[0]\\n            curpos += 1\\n            if c >= b'0'[0]: numb = 10 * numb + (conv(c) - 48)\\n            elif c == b'-'[0]: sign = -1\\n            elif c != b'\\\\r'[0]: A.append(sign*numb); numb = zero; sign = 1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod, (Note, currently experimenting, not fully working)\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize:break\\n            c = self.stream.read(1)[0]\\n            curpos += 1\\n            if c >= b'0'[0]: numb = 10 * numb + (conv(c) - 48)\\n            elif c == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = zero; sign = 1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(1000,buffsize-curpos))\\n            for i in range(len(s)):\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n                if len(A)==n: break\\n            curpos += i+1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10000,buffsize-curpos))\\n            for i in range(len(s)):\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n                if len(A)==n: break\\n            curpos += i+1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10000,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(256,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(32,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, zero=0):\\n        conv = ord if py2 else lambda x:x\\n        s = self.read()\\n        A = []; numb = zero; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n            elif s[i] != b'\\\\r'[0]:\\n                if s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\ninp = sys.stdin.readnumbers(0.0)\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(128,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\n# OLD VERSION\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            small_buff = min(64,buffsize-curpos)\\n            s = self.stream.read(small_buff)\\n            i = 0\\n            while i<small_buff and len(A)<n:\\n                if s[i] >= b'-'[0]:\\n                    if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                    else: sign = -1    \\n                elif s[i] != b'\\\\r'[0]: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\"]",
        "difficulty": "interview",
        "input": "3 3\n1 2 1\n2 3 1\n1 3 1\n30 10 20\n",
        "output": "12 10 12 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/938/D"
    },
    {
        "id": 284,
        "task_id": 2744,
        "test_case_id": 3,
        "question": "Musicians of a popular band \"Flayer\" have announced that they are going to \"make their exit\" with a world tour. Of course, they will visit Berland as well.\n\nThere are n cities in Berland. People can travel between cities using two-directional train routes; there are exactly m routes, i-th route can be used to go from city v_{i} to city u_{i} (and from u_{i} to v_{i}), and it costs w_{i} coins to use this route.\n\nEach city will be visited by \"Flayer\", and the cost of the concert ticket in i-th city is a_{i} coins.\n\nYou have friends in every city of Berland, and they, knowing about your programming skills, asked you to calculate the minimum possible number of coins they have to pay to visit the concert. For every city i you have to compute the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).\n\nFormally, for every $i \\in [ 1, n ]$ you have to calculate $\\operatorname{min}_{j = 1} 2 d(i, j) + a_{j}$, where d(i, j) is the minimum number of coins you have to spend to travel from city i to city j. If there is no way to reach city j from city i, then we consider d(i, j) to be infinitely large.\n\n\n-----Input-----\n\nThe first line contains two integers n and m (2 ≤ n ≤ 2·10^5, 1 ≤ m ≤ 2·10^5).\n\nThen m lines follow, i-th contains three integers v_{i}, u_{i} and w_{i} (1 ≤ v_{i}, u_{i} ≤ n, v_{i} ≠ u_{i}, 1 ≤ w_{i} ≤ 10^12) denoting i-th train route. There are no multiple train routes connecting the same pair of cities, that is, for each (v, u) neither extra (v, u) nor (u, v) present in input.\n\nThe next line contains n integers a_1, a_2, ... a_{k} (1 ≤ a_{i} ≤ 10^12) — price to attend the concert in i-th city.\n\n\n-----Output-----\n\nPrint n integers. i-th of them must be equal to the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).\n\n\n-----Examples-----\nInput\n4 2\n1 2 4\n2 3 7\n6 20 1 25\n\nOutput\n6 14 1 25 \n\nInput\n3 3\n1 2 1\n2 3 1\n1 3 1\n30 10 20\n\nOutput\n12 10 12",
        "solutions": "[\"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(fd, stream.getvalue()) + stream.truncate(0) + stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 2*10**12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(x))\\n    sys.stdout.write('\\\\n')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(fd, stream.getvalue()) + stream.truncate(0) + stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write('\\\\n')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\n###### ACTUAL CODE\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nimport heapq\\nQ = [(best[i],i) for i in range(n)]\\nheapq.heapify(Q)\\n\\nwhile Q:\\n    c,node = heapq.heappop(Q)\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            heapq.heappush(Q,(C, nei))\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\n###### REWRITTEN HEAPQ\\ndef heappush(heap, item):\\n    heap.append(item)\\n    _siftdown(heap, len(heap)-1)\\n\\ndef heappop(heap):\\n    lastelt = heap.pop()\\n    if not heap:\\n        return lastelt\\n\\n    returnitem,heap[0] = heap[0],lastelt\\n    _siftup(heap)\\n    return returnitem\\n\\n# Does a pop and then a push\\ndef heapreplace(heap, item):\\n    returnitem,heap[0] = heap[0],item\\n    _siftup(heap)\\n    return returnitem\\n\\n# Does a push and then a pop\\ndef heappushpop(heap, item):\\n    if heap and heap[0] < item:\\n        item, heap[0] = heap[0], item\\n        _siftup(heap)\\n    return item\\n\\ndef heapify(x):\\n    for i in reversed(range(len(x)//2)):\\n        _siftup(x, i)\\n\\n\\ndef _siftdown(heap, pos):\\n    newitem = heap[pos]\\n    ppos = (pos-1) >> 1\\n    while pos and newitem < heap[ppos]:\\n        heap[pos] = heap[ppos]\\n        pos = ppos\\n        ppos = (pos-1) >> 1\\n    heap[pos] = newitem\\n\\ndef _siftup(heap, pos=0):\\n    # Move the item at pos to a leaf\\n    # by switching place with smallest child (bias to right)\\n    newitem = heap[pos]\\n    \\n    leftchild = 2*pos + 1\\n    rightchild = leftchild + 1\\n    while rightchild < len(heap):\\n        if heap[leftchild] < heap[rightchild]:\\n            heap[pos] = heap[leftchild]\\n            pos = leftchild\\n        else:\\n            heap[pos] = heap[rightchild]\\n            pos = rightchild\\n        leftchild = 2*pos + 1\\n        rightchild = leftchild + 1\\n    if leftchild < len(heap): # Special case of only one child\\n        heap[pos] = heap[leftchild]\\n        pos = leftchild\\n    # Now newitem has been moved to an leaf\\n    heap[pos] = newitem\\n    _siftdown(heap,pos)\\n\\n###### ACTUAL CODE\\ns = sys.stdin.read().replace(b'\\\\r',b'')\\ninp = []\\nnumb = 0\\n \\nfor i in range(len(s)):\\n    if s[i]>=48:\\n        numb = 10*numb + s[i]-48\\n    elif s[i]!=13:\\n        inp.append(numb)\\n        numb = 0\\nif s[-1]>=48:\\n    inp.append(numb)\\n\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\n\\nQ = [(best[i],i) for i in range(n)]\\nheapify(Q)\\n\\nwhile Q:\\n    c,node = heappop(Q)\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            heappush(Q,(C, nei))\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s[-1]>=b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readuint()\\nind = 0\\n\\nn = inp[ind]\\nind += 1\\nm = inp[ind]\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = inp[ind+0]-1\\n    u = inp[ind+1]-1\\n    w = 1.0*inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [1.0*inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readfloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0; sign = 1.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            elif s[i] == b'-'[0]: sign = -1.0\\n            else: A.append(sign*numb); numb = 0.0; sign = 1.0\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readufloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            else: A.append(numb); numb = 0.0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readufloat()\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readuint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            else: A.append(numb); numb = 0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n    def readint(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10*numb + conv(s[i])-48\\n            elif s[i] == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = 0; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readfloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0; sign = 1.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            elif s[i] == b'-'[0]: sign = -1.0\\n            else: A.append(sign*numb); numb = 0.0; sign = 1.0\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n    def readufloat(self):\\n        conv = ord if py2 else lambda x:x\\n        s = sys.stdin.read().replace(b'\\\\r',b'')\\n        A = []; numb = 0.0\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10.0*numb + conv(s[i])-48.0\\n            else: A.append(numb); numb = 0.0\\n        if s and s[-1] >= b'0'[0]: A.append(numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\n\\n\\ninp = sys.stdin.readfloat()\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize:break\\n            c = self.stream.read(1)[0]\\n            curpos += 1\\n            if c >= b'0'[0]: numb = 10 * numb + (conv(c) - 48)\\n            elif c == b'-'[0]: sign = -1\\n            elif c != b'\\\\r'[0]: A.append(sign*numb); numb = zero; sign = 1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod, (Note, currently experimenting, not fully working)\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize:break\\n            c = self.stream.read(1)[0]\\n            curpos += 1\\n            if c >= b'0'[0]: numb = 10 * numb + (conv(c) - 48)\\n            elif c == b'-'[0]: sign = -1\\n            else: A.append(sign*numb); numb = zero; sign = 1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(1000,buffsize-curpos))\\n            for i in range(len(s)):\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n                if len(A)==n: break\\n            curpos += i+1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size).replace(b'\\\\r',b'')\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10000,buffsize-curpos))\\n            for i in range(len(s)):\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n                if len(A)==n: break\\n            curpos += i+1\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10000,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(10,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(256,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(32,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, zero=0):\\n        conv = ord if py2 else lambda x:x\\n        s = self.read()\\n        A = []; numb = zero; sign = 1\\n        for i in range(len(s)):\\n            if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n            elif s[i] != b'\\\\r'[0]:\\n                if s[i] == b'-'[0]: sign = -1\\n                else: A.append(sign*numb); numb = zero; sign = 1\\n        if s and s[-1] >= b'0'[0]: A.append(sign*numb)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\ninp = sys.stdin.readnumbers(0.0)\\nind = 0\\n\\nn = int(inp[ind])\\nind += 1\\nm = int(inp[ind])\\nind += 1\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[ind+0]-1)\\n    u = int(inp[ind+1]-1)\\n    w = inp[ind+2]\\n    ind += 3\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\nbest = [inp[ind+i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(128,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)<n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\n# OLD VERSION\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                self.read2buffer()\\n                self.stream.seek(0,2)\\n                buffsize = self.stream.tell()\\n                self.stream.seek(curpos)\\n                if curpos==buffsize: break\\n            s = self.stream.read(min(64,buffsize-curpos))\\n            i = 0\\n            while i<len(s) and len(A)!=n:\\n                if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                elif s[i] != b'\\\\r'[0]: \\n                    if s[i] == b'-'[0]: sign = -1\\n                    else: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\", \"from __future__ import division, print_function\\npy2 = round(0.5)\\n\\nif py2:\\n    from future_builtins import ascii, filter, hex, map, oct, zip\\n    range = xrange\\n\\nimport os, sys\\nfrom io import BytesIO, IOBase\\n\\n# FastIO for PyPy2 and PyPy3 by Pajenegod,\\nclass FastI(object):\\n    def __init__(self, fd=0, buffersize=2**14):\\n        self.stream = stream = BytesIO(); self.bufendl = 0\\n        def read2buffer():\\n            curpos = stream.tell(); s = os.read(fd, buffersize + os.fstat(fd).st_size)\\n            stream.seek(0,2); stream.write(s); stream.seek(curpos); return s\\n        self.read2buffer = read2buffer\\n    def read(self):\\n        while self.read2buffer(): pass\\n        return self.stream.read() if self.stream.tell() else self.stream.getvalue()\\n    def readline(self):\\n        while self.bufendl == 0: s = self.read2buffer(); self.bufendl += s.count(b'\\\\n') + (not s)\\n        self.bufendl -= 1; return self.stream.readline()\\n    def input(self): return self.readline().rstrip(b'\\\\r\\\\n')\\n    def readnumbers(self, n,zero=0):\\n        conv = ord if py2 else lambda x:x\\n        \\n        A = []; numb = zero; sign = 1\\n        \\n        curpos = self.stream.tell()\\n        self.stream.seek(0,2)\\n        buffsize = self.stream.tell()\\n        self.stream.seek(curpos)\\n        \\n        while len(A)<n:\\n            if curpos>=buffsize:\\n                buffsize += len(self.read2buffer())\\n                if curpos==buffsize: break\\n            small_buff = min(64,buffsize-curpos)\\n            s = self.stream.read(small_buff)\\n            i = 0\\n            while i<small_buff and len(A)<n:\\n                if s[i] >= b'-'[0]:\\n                    if s[i] >= b'0'[0]: numb = 10 * numb + (conv(s[i]) - 48)\\n                    else: sign = -1    \\n                elif s[i] != b'\\\\r'[0]: A.append(sign*numb); numb = zero; sign = 1\\n                i += 1\\n            curpos += i\\n        if curpos == buffsize and len(A)<n: A.append(sign*numb)\\n        assert(len(A)==n)\\n        if self.stream.tell()!=curpos: self.stream.seek(curpos)\\n        return A\\n\\nclass FastO(IOBase):\\n    def __init__(self, fd=1):\\n        stream = BytesIO()\\n        self.flush = lambda: os.write(1, stream.getvalue()) and not stream.truncate(0) and stream.seek(0)\\n        self.write = stream.write if py2 else lambda s: stream.write(s.encode())\\n\\nsys.stdin, sys.stdout = FastI(), FastO()\\ninput = sys.stdin.readline\\n\\nbig = 3E12\\n\\nclass segheap:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n:m*=2\\n        self.n = n\\n        self.m = m\\n\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i],self.data[2*i+1])\\n\\n    def mini(self):\\n        i = 1\\n        while i<self.m:\\n            if self.data[i]==self.data[2*i]:\\n                i = 2*i\\n            else:\\n                i = 2*i+1\\n        i -= self.m\\n        self.setter(i,big)\\n        return i\\n    def setter(self,ind,val):\\n        ind += self.m\\n        if val<self.data[ind]:\\n            while ind>0 and self.data[ind]>val:\\n                self.data[ind] = val\\n                ind //= 2\\n        elif val>self.data[ind]:\\n            old_val = self.data[ind]\\n            self.data[ind] = val\\n            ind //= 2\\n            while ind>0 and self.data[ind]==old_val:\\n                self.data[ind] = min(self.data[2*ind],self.data[2*ind+1])\\n                ind //= 2\\n\\nn, m = [int(x) for x in sys.stdin.readline().split()]\\n\\ninp = sys.stdin.readnumbers(3*m, 0.0)\\n\\ncoupl = [[] for _ in range(n)]\\ncost = [[] for _ in range(n)]\\nfor _ in range(m):\\n    v = int(inp[_*3+0]-1)\\n    u = int(inp[_*3+1]-1)\\n    w = inp[_*3+2]\\n    coupl[v].append(u)\\n    coupl[u].append(v)\\n    cost[u].append(w)\\n    cost[v].append(w)\\n\\ninp = sys.stdin.readnumbers(n, 0.0)\\n\\nbest = [inp[i] for i in range(n)]\\n\\nQ = segheap(best)\\n\\nwhile Q.data[1]!=big:\\n    c = Q.data[1]\\n    node = Q.mini()\\n    if best[node]!=c:\\n        continue\\n    for j in range(len(coupl[node])):\\n        nei = coupl[node][j]\\n        C = c+2*cost[node][j]\\n        if C<best[nei]:\\n            best[nei] = C\\n            Q.setter(nei,C)\\n\\nfor x in best:\\n    sys.stdout.write(str(int(x)))\\n    sys.stdout.write(' ')\"]",
        "difficulty": "interview",
        "input": "7 7\n1 6 745325\n2 3 3581176\n2 4 19\n3 6 71263060078\n5 4 141198\n7 4 163953\n5 6 15994\n1 297404206755 82096176217 14663411 187389745 21385 704393\n",
        "output": "1 335807 7498159 335769 53373 21385 663675 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/938/D"
    },
    {
        "id": 285,
        "task_id": 2811,
        "test_case_id": 1,
        "question": "You are given a rooted tree with $n$ nodes. The nodes are numbered $1..n$. The root is node $1$, and $m$ of the nodes are colored red, the rest are black.\n\nYou would like to choose a subset of nodes such that there is no node in your subset which is an ancestor of any other node in your subset. For example, if A is the parent of B and B is the parent of C, then you could have at most one of A, B or C in your subset. In addition, you would like exactly $k$ of your chosen nodes to be red.\n\nIf exactly $m$ of the nodes are red, then for all $k=0..m$, figure out how many ways you can choose subsets with $k$ red nodes, and no node is an ancestor of any other node.\n\n-----Input-----\nEach input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line with two integers $n$ ($1 \\le n \\le 2 \\times 10^5$) and $m$ ($0 \\le m \\le min(10^3,\\ n)$), where $n$ is the number of nodes in the tree, and $m$ is the number of nodes which are red. The nodes are numbered $1..n$.\n\nEach of the next $n-1$ lines will contain a single integer $p$ ($1 \\le p \\le n$), which is the number of the parent of this node. The nodes are listed in order, starting with node $2$, then node $3$, and so on. Node $1$ is skipped, since it is the root. It is guaranteed that the nodes form a single tree, with a single root at node $1$ and no cycles.\n\nEach of the next $m$ lines will contain single integer $r$ ($1 \\le r \\le n$). These are the numbers of the red nodes. No value of $r$ will be repeated.\n\n-----Output-----\nOutput $m+1$ lines, corresponding to the number of subsets satisfying the given criteria with a number of red nodes equal to $k=0..m$, in that order. Output this number modulo $10^9+7$.\n\n-----Examples-----\nSample Input 1:\n4 1\n1\n1\n1\n3\nSample Output 1:\n5\n4\n\nSample Input 2:\n4 4\n1\n1\n1\n1\n2\n3\n4\nSample Output 2:\n1\n4\n3\n1\n0",
        "solutions": "",
        "difficulty": "interview",
        "input": "4 1\n1\n1\n1\n3\n",
        "output": "5\n4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/redblacktree"
    },
    {
        "id": 286,
        "task_id": 2811,
        "test_case_id": 2,
        "question": "You are given a rooted tree with $n$ nodes. The nodes are numbered $1..n$. The root is node $1$, and $m$ of the nodes are colored red, the rest are black.\n\nYou would like to choose a subset of nodes such that there is no node in your subset which is an ancestor of any other node in your subset. For example, if A is the parent of B and B is the parent of C, then you could have at most one of A, B or C in your subset. In addition, you would like exactly $k$ of your chosen nodes to be red.\n\nIf exactly $m$ of the nodes are red, then for all $k=0..m$, figure out how many ways you can choose subsets with $k$ red nodes, and no node is an ancestor of any other node.\n\n-----Input-----\nEach input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line with two integers $n$ ($1 \\le n \\le 2 \\times 10^5$) and $m$ ($0 \\le m \\le min(10^3,\\ n)$), where $n$ is the number of nodes in the tree, and $m$ is the number of nodes which are red. The nodes are numbered $1..n$.\n\nEach of the next $n-1$ lines will contain a single integer $p$ ($1 \\le p \\le n$), which is the number of the parent of this node. The nodes are listed in order, starting with node $2$, then node $3$, and so on. Node $1$ is skipped, since it is the root. It is guaranteed that the nodes form a single tree, with a single root at node $1$ and no cycles.\n\nEach of the next $m$ lines will contain single integer $r$ ($1 \\le r \\le n$). These are the numbers of the red nodes. No value of $r$ will be repeated.\n\n-----Output-----\nOutput $m+1$ lines, corresponding to the number of subsets satisfying the given criteria with a number of red nodes equal to $k=0..m$, in that order. Output this number modulo $10^9+7$.\n\n-----Examples-----\nSample Input 1:\n4 1\n1\n1\n1\n3\nSample Output 1:\n5\n4\n\nSample Input 2:\n4 4\n1\n1\n1\n1\n2\n3\n4\nSample Output 2:\n1\n4\n3\n1\n0",
        "solutions": "",
        "difficulty": "interview",
        "input": "4 4\n1\n1\n1\n1\n2\n3\n4\n",
        "output": "1\n4\n3\n1\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/redblacktree"
    },
    {
        "id": 287,
        "task_id": 2811,
        "test_case_id": 3,
        "question": "You are given a rooted tree with $n$ nodes. The nodes are numbered $1..n$. The root is node $1$, and $m$ of the nodes are colored red, the rest are black.\n\nYou would like to choose a subset of nodes such that there is no node in your subset which is an ancestor of any other node in your subset. For example, if A is the parent of B and B is the parent of C, then you could have at most one of A, B or C in your subset. In addition, you would like exactly $k$ of your chosen nodes to be red.\n\nIf exactly $m$ of the nodes are red, then for all $k=0..m$, figure out how many ways you can choose subsets with $k$ red nodes, and no node is an ancestor of any other node.\n\n-----Input-----\nEach input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line with two integers $n$ ($1 \\le n \\le 2 \\times 10^5$) and $m$ ($0 \\le m \\le min(10^3,\\ n)$), where $n$ is the number of nodes in the tree, and $m$ is the number of nodes which are red. The nodes are numbered $1..n$.\n\nEach of the next $n-1$ lines will contain a single integer $p$ ($1 \\le p \\le n$), which is the number of the parent of this node. The nodes are listed in order, starting with node $2$, then node $3$, and so on. Node $1$ is skipped, since it is the root. It is guaranteed that the nodes form a single tree, with a single root at node $1$ and no cycles.\n\nEach of the next $m$ lines will contain single integer $r$ ($1 \\le r \\le n$). These are the numbers of the red nodes. No value of $r$ will be repeated.\n\n-----Output-----\nOutput $m+1$ lines, corresponding to the number of subsets satisfying the given criteria with a number of red nodes equal to $k=0..m$, in that order. Output this number modulo $10^9+7$.\n\n-----Examples-----\nSample Input 1:\n4 1\n1\n1\n1\n3\nSample Output 1:\n5\n4\n\nSample Input 2:\n4 4\n1\n1\n1\n1\n2\n3\n4\nSample Output 2:\n1\n4\n3\n1\n0",
        "solutions": "",
        "difficulty": "interview",
        "input": "14 4\n1\n2\n1\n2\n3\n4\n5\n5\n13\n8\n10\n4\n4\n8\n3\n12\n13\n",
        "output": "100\n169\n90\n16\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/redblacktree"
    },
    {
        "id": 288,
        "task_id": 2813,
        "test_case_id": 1,
        "question": "A haiku is a Japanese form of poetry. It consists of three phrases of $5$, $7$ and $5$ syllables each.\n\nOnce a year, HiQ has a haiku contest, where employees submit their best poems. The poems are then judged based on a wide range of aspects, such as\n - creativity\n - simplicity\n - beauty\n - whether the poem is actually a haiku\n\nThis last point turned out to be quite a challenge for the judges (many problems arose when half the judges indexed syllables starting at $0$ and the other half at $1$).\n\nCan you help the judges to determine whether a submitted poem is a haiku, given a set of syllables? Note that there may exist multiple decompositions of a single word in the poem into syllables. In this case, you should determine whether some decomposition is a haiku.\n\n-----Input-----\nThe first line of input contains a single integer $1 \\le S \\le 100$, the number of syllables. The next line contains $S$ words separated by spaces (the syllables). Each syllable contains at most $7$ lowercase letters a-z.\n\nThen, three lines containing the poem follow. Each line contains a (non-empty) list of words separated by spaces, representing a phrase. The words contain only lowercase letters a-z. The length of each line is at most $100$ characters (including spaces).\n\nIt is guaranteed that there exists at least one decomposition of the poem into the given syllables.\n\n-----Output-----\nOutput “haiku” if the given poem is a haiku, and “come back next year” otherwise (without the quotes).\n\n-----Explanation of Sample Input 1-----\nOne possible decomposition into a haiku is:\n\nspe-lling ve-ry hard\near-ly in mor-ning ti-red\ni need cov-fe-fe\n\n\n-----Explanation of Sample Input 3-----\nNo matter how the third line is decomposed, it contains $8$ syllables instead of $5$, so it can not be a haiku.\n\n-----Examples-----\nSample Input 1:\n20\nva\nfi\nve\nmor\nlling\nspe\nin\ni\nsh\nti\nhard\ncov\near\nfe\nred\nja\nry\nneed\nly\nning\nspelling very hard\nearly in morning tired\ni need covfefe\nSample Output 1:\nhaiku\n\nSample Input 2:\n22\nq\nc\nda\nplus\nto\nthe\ne\nthee\nun\nlate\nci\na\nshall\nby\ncom\ni\nru\npare\ntemp\nble\nhi\nde\nshall i compare thee\nto a c plus plus template\nundecidable\nSample Output 2:\nhaiku",
        "solutions": "",
        "difficulty": "interview",
        "input": "20\nva\nfi\nve\nmor\nlling\nspe\nin\ni\nsh\nti\nhard\ncov\near\nfe\nred\nja\nry\nneed\nly\nning\nspelling very hard\nearly in morning tired\ni need covfefe\n",
        "output": "haiku\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/haiku"
    },
    {
        "id": 289,
        "task_id": 2813,
        "test_case_id": 2,
        "question": "A haiku is a Japanese form of poetry. It consists of three phrases of $5$, $7$ and $5$ syllables each.\n\nOnce a year, HiQ has a haiku contest, where employees submit their best poems. The poems are then judged based on a wide range of aspects, such as\n - creativity\n - simplicity\n - beauty\n - whether the poem is actually a haiku\n\nThis last point turned out to be quite a challenge for the judges (many problems arose when half the judges indexed syllables starting at $0$ and the other half at $1$).\n\nCan you help the judges to determine whether a submitted poem is a haiku, given a set of syllables? Note that there may exist multiple decompositions of a single word in the poem into syllables. In this case, you should determine whether some decomposition is a haiku.\n\n-----Input-----\nThe first line of input contains a single integer $1 \\le S \\le 100$, the number of syllables. The next line contains $S$ words separated by spaces (the syllables). Each syllable contains at most $7$ lowercase letters a-z.\n\nThen, three lines containing the poem follow. Each line contains a (non-empty) list of words separated by spaces, representing a phrase. The words contain only lowercase letters a-z. The length of each line is at most $100$ characters (including spaces).\n\nIt is guaranteed that there exists at least one decomposition of the poem into the given syllables.\n\n-----Output-----\nOutput “haiku” if the given poem is a haiku, and “come back next year” otherwise (without the quotes).\n\n-----Explanation of Sample Input 1-----\nOne possible decomposition into a haiku is:\n\nspe-lling ve-ry hard\near-ly in mor-ning ti-red\ni need cov-fe-fe\n\n\n-----Explanation of Sample Input 3-----\nNo matter how the third line is decomposed, it contains $8$ syllables instead of $5$, so it can not be a haiku.\n\n-----Examples-----\nSample Input 1:\n20\nva\nfi\nve\nmor\nlling\nspe\nin\ni\nsh\nti\nhard\ncov\near\nfe\nred\nja\nry\nneed\nly\nning\nspelling very hard\nearly in morning tired\ni need covfefe\nSample Output 1:\nhaiku\n\nSample Input 2:\n22\nq\nc\nda\nplus\nto\nthe\ne\nthee\nun\nlate\nci\na\nshall\nby\ncom\ni\nru\npare\ntemp\nble\nhi\nde\nshall i compare thee\nto a c plus plus template\nundecidable\nSample Output 2:\nhaiku",
        "solutions": "",
        "difficulty": "interview",
        "input": "22\nq\nc\nda\nplus\nto\nthe\ne\nthee\nun\nlate\nci\na\nshall\nby\ncom\ni\nru\npare\ntemp\nble\nhi\nde\nshall i compare thee\nto a c plus plus template\nundecidable\n",
        "output": "haiku\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/haiku"
    },
    {
        "id": 290,
        "task_id": 2813,
        "test_case_id": 3,
        "question": "A haiku is a Japanese form of poetry. It consists of three phrases of $5$, $7$ and $5$ syllables each.\n\nOnce a year, HiQ has a haiku contest, where employees submit their best poems. The poems are then judged based on a wide range of aspects, such as\n - creativity\n - simplicity\n - beauty\n - whether the poem is actually a haiku\n\nThis last point turned out to be quite a challenge for the judges (many problems arose when half the judges indexed syllables starting at $0$ and the other half at $1$).\n\nCan you help the judges to determine whether a submitted poem is a haiku, given a set of syllables? Note that there may exist multiple decompositions of a single word in the poem into syllables. In this case, you should determine whether some decomposition is a haiku.\n\n-----Input-----\nThe first line of input contains a single integer $1 \\le S \\le 100$, the number of syllables. The next line contains $S$ words separated by spaces (the syllables). Each syllable contains at most $7$ lowercase letters a-z.\n\nThen, three lines containing the poem follow. Each line contains a (non-empty) list of words separated by spaces, representing a phrase. The words contain only lowercase letters a-z. The length of each line is at most $100$ characters (including spaces).\n\nIt is guaranteed that there exists at least one decomposition of the poem into the given syllables.\n\n-----Output-----\nOutput “haiku” if the given poem is a haiku, and “come back next year” otherwise (without the quotes).\n\n-----Explanation of Sample Input 1-----\nOne possible decomposition into a haiku is:\n\nspe-lling ve-ry hard\near-ly in mor-ning ti-red\ni need cov-fe-fe\n\n\n-----Explanation of Sample Input 3-----\nNo matter how the third line is decomposed, it contains $8$ syllables instead of $5$, so it can not be a haiku.\n\n-----Examples-----\nSample Input 1:\n20\nva\nfi\nve\nmor\nlling\nspe\nin\ni\nsh\nti\nhard\ncov\near\nfe\nred\nja\nry\nneed\nly\nning\nspelling very hard\nearly in morning tired\ni need covfefe\nSample Output 1:\nhaiku\n\nSample Input 2:\n22\nq\nc\nda\nplus\nto\nthe\ne\nthee\nun\nlate\nci\na\nshall\nby\ncom\ni\nru\npare\ntemp\nble\nhi\nde\nshall i compare thee\nto a c plus plus template\nundecidable\nSample Output 2:\nhaiku",
        "solutions": "",
        "difficulty": "interview",
        "input": "19\nju\nwin\nsoft\nit\neast\nsun\nand\nlight\nder\nthe\nis\nthrough\nli\nbut\nyon\nwhat\nbreaks\net\ndow\nbut soft what light through\nyonder window breaks it is\nthe east and juliet is the sun\n",
        "output": "come back next year\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/haiku"
    },
    {
        "id": 291,
        "task_id": 2901,
        "test_case_id": 1,
        "question": "Johnny has been roped into a fantasy soccer league and needs your help to set up the best team. \n\nHe has a list of $10$ players that are on his fantasy team (no goalkeepers in this league). He also knows what country they’re from, what league they play in, and what team they play for.\n\nHe doesn’t know much about soccer, but he does know these things:\n - If two players are from the same country, a link between the two players will have a synergy score of $1$.\n - If two players are in the same league, a link between the two players will have a synergy score of $1$.\n - If two players are on the same team, a link between the two players will have a synergy score of $2$.\n - If two players are from the same country and in the same league, a link between the two players will have a synergy score of $2$.\n - If two players are from the same country and on the same team, a link between the two players will have a synergy score of $3$.\n\nA team can only be in one league and no two teams will have the same name unless they are the same team.\n\nHe has to place the players on his team into a formation of $10$ nodes which can be represented as an undirected graph. The illustration shows the first sample. Therefore, Johnny has to place a player in each node of the graph. Given a particular formation and the members of Johnny’s team, output whether it is possible for Johnny to organize his team to get a perfect team.\n\nA team is a perfect team if and only if every player is placed on a node with a synergy score that is greater or equal to the node’s degree. A node’s degree is the number of links to which it is linked in the formation. A player placed on a node in the formation derives synergy from all players placed on nodes to which the player is linked in the formation. Thus, a node’s synergy score is the sum of the synergy scores of all links to which the node is connected.\n\n-----Input-----\nThe input will contain a single test case. The first line of the input will have one integer $c$ ($0 \\leq c \\leq 45$). $c$ represents the number of edges in the formation. The next $c$ lines represent the connections between nodes represented as two integers $a$ ($0 \\leq a < 10$) and $b$ ($0 \\leq b < 10$), where $a$ is not equal to $b$. Then, the next $10$ lines will be the players on Johnny’s team. Each line will contain four strings in the following order: player name, nation, league, team, which are delimited by a single space. These strings are guaranteed to be non-empty and consist of up to $15$ alphanumeric characters, which includes lower- and uppercase English letters, digits, and hyphens.\n\n-----Output-----\nIf a perfect team can be organized by Johnny, print yes. Otherwise, print no.\n\n-----Examples-----\nSample Input:\n15\n0 1\n1 2\n2 3\n0 4\n1 5\n2 6\n3 7\n4 5\n5 6\n6 7\n4 8\n5 8\n6 9\n7 9\n8 9\nGriezmann France LaLiga AtleticoMadrid\nBenzema France LaLiga RealMadrid\nNtep France Ligue1 StadeRennais\nSissoko France PremierLeague Spurs\nTolisso France Ligue1 Lyon\nDiarra France Ligue1 OM\nEvra France CalcioA Juventus\nKoscielny France PremierLeague Arsenal\nVarane France LaLiga RealMadrid\nSagna France PremierLeague ManCity\nSample Output:\nyes",
        "solutions": "",
        "difficulty": "interview",
        "input": "15\n0 1\n1 2\n2 3\n0 4\n1 5\n2 6\n3 7\n4 5\n5 6\n6 7\n4 8\n5 8\n6 9\n7 9\n8 9\nGriezmann France LaLiga AtleticoMadrid\nBenzema France LaLiga RealMadrid\nNtep France Ligue1 StadeRennais\nSissoko France PremierLeague Spurs\nTolisso France Ligue1 Lyon\nDiarra France Ligue1 OM\nEvra France CalcioA Juventus\nKoscielny France PremierLeague Arsenal\nVarane France LaLiga RealMadrid\nSagna France PremierLeague ManCity\n",
        "output": "yes\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/victorythroughsynergy"
    },
    {
        "id": 292,
        "task_id": 2901,
        "test_case_id": 2,
        "question": "Johnny has been roped into a fantasy soccer league and needs your help to set up the best team. \n\nHe has a list of $10$ players that are on his fantasy team (no goalkeepers in this league). He also knows what country they’re from, what league they play in, and what team they play for.\n\nHe doesn’t know much about soccer, but he does know these things:\n - If two players are from the same country, a link between the two players will have a synergy score of $1$.\n - If two players are in the same league, a link between the two players will have a synergy score of $1$.\n - If two players are on the same team, a link between the two players will have a synergy score of $2$.\n - If two players are from the same country and in the same league, a link between the two players will have a synergy score of $2$.\n - If two players are from the same country and on the same team, a link between the two players will have a synergy score of $3$.\n\nA team can only be in one league and no two teams will have the same name unless they are the same team.\n\nHe has to place the players on his team into a formation of $10$ nodes which can be represented as an undirected graph. The illustration shows the first sample. Therefore, Johnny has to place a player in each node of the graph. Given a particular formation and the members of Johnny’s team, output whether it is possible for Johnny to organize his team to get a perfect team.\n\nA team is a perfect team if and only if every player is placed on a node with a synergy score that is greater or equal to the node’s degree. A node’s degree is the number of links to which it is linked in the formation. A player placed on a node in the formation derives synergy from all players placed on nodes to which the player is linked in the formation. Thus, a node’s synergy score is the sum of the synergy scores of all links to which the node is connected.\n\n-----Input-----\nThe input will contain a single test case. The first line of the input will have one integer $c$ ($0 \\leq c \\leq 45$). $c$ represents the number of edges in the formation. The next $c$ lines represent the connections between nodes represented as two integers $a$ ($0 \\leq a < 10$) and $b$ ($0 \\leq b < 10$), where $a$ is not equal to $b$. Then, the next $10$ lines will be the players on Johnny’s team. Each line will contain four strings in the following order: player name, nation, league, team, which are delimited by a single space. These strings are guaranteed to be non-empty and consist of up to $15$ alphanumeric characters, which includes lower- and uppercase English letters, digits, and hyphens.\n\n-----Output-----\nIf a perfect team can be organized by Johnny, print yes. Otherwise, print no.\n\n-----Examples-----\nSample Input:\n15\n0 1\n1 2\n2 3\n0 4\n1 5\n2 6\n3 7\n4 5\n5 6\n6 7\n4 8\n5 8\n6 9\n7 9\n8 9\nGriezmann France LaLiga AtleticoMadrid\nBenzema France LaLiga RealMadrid\nNtep France Ligue1 StadeRennais\nSissoko France PremierLeague Spurs\nTolisso France Ligue1 Lyon\nDiarra France Ligue1 OM\nEvra France CalcioA Juventus\nKoscielny France PremierLeague Arsenal\nVarane France LaLiga RealMadrid\nSagna France PremierLeague ManCity\nSample Output:\nyes",
        "solutions": "",
        "difficulty": "interview",
        "input": "15\n0 1\n1 2\n2 3\n0 4\n1 5\n2 6\n3 7\n4 5\n5 6\n6 7\n4 8\n5 8\n6 9\n7 9\n8 9\nPlayerA France A1 A1-1\nPlayerB France B1 B1-1\nPlayerC France C1 C1-1\nPlayerD France D1 D1-1\nPlayerE France E1 E1-1\nPlayerF France F1 F1-1\nPlayerG France G1 G1-1\nPlayerH France H1 H1-1\nPlayerI France I1 I1-1\nPlayerJ Germany J1 J1-1\n",
        "output": "no\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/victorythroughsynergy"
    },
    {
        "id": 293,
        "task_id": 2948,
        "test_case_id": 1,
        "question": "A land far, far away has N Members of Parliament (MP). They had a turbulent and passionate debate on the law on amendments to the law on a new referendum on referendums. From Monday to Friday, all MPs joyfully came to work and argued all day.\n\nA diligent news reporter photographed MPs at their workplace in the heat of the argument every working day of the week. What she captured on the photos are pairs of MPs fighting and scowling at each other. The five photographs have been forwarded to you for thorough analysis.\n\nIt is a fact that each MP belongs to one of the two political parties. Let’s denote them with letters A and B. Your task is to estimate which MP belongs to which party, so that the following holds for your estimation: each MP argued with at most two distinct members of their own party.\n\n-----Input-----\nThe first line of input contains an integer $N$ ($2 \\leq N \\leq 200000$), the number of MPs. MPs are denoted with numbers from $1$ to $N$.\n\nThe following five lines describe the photographs taken from Monday to Friday. Each of the five lines contains the list of pairs of MPs that are arguing on the photograph that day (scowling at each other). Stated first is the number of pairs $P$ ($1 \\leq P \\leq N/2$), followed by $P$ pairs in the form “$K$ $L$”, where $K$ and $L$ are labels of MPs scowling at each other. Before each pair there is a double space.\n\nOf course, each MP is stated at most once per line, and no MP ever argues with herself.\n\n-----Output-----\nThe first and only line of output must contain an array consisting of only characters A and B, so that the $K$-th character denotes the party of $K$-th MP in a division that satisfies the given conditions.\n\nSince the solution isn’t going to be unique, output any.\n\n-----Examples-----\nSample Input:\n7\n2  1 2  7 3\n2  1 3  7 4\n2  1 4  7 5\n2  1 5  7 6\n2  1 6  7 2\nSample Output:\nABBBBBA",
        "solutions": "",
        "difficulty": "interview",
        "input": "7\n2  1 2  7 3\n2  1 3  7 4\n2  1 4  7 5\n2  1 5  7 6\n2  1 6  7 2\n",
        "output": "ABBBBBA\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/sabor"
    },
    {
        "id": 294,
        "task_id": 2948,
        "test_case_id": 2,
        "question": "A land far, far away has N Members of Parliament (MP). They had a turbulent and passionate debate on the law on amendments to the law on a new referendum on referendums. From Monday to Friday, all MPs joyfully came to work and argued all day.\n\nA diligent news reporter photographed MPs at their workplace in the heat of the argument every working day of the week. What she captured on the photos are pairs of MPs fighting and scowling at each other. The five photographs have been forwarded to you for thorough analysis.\n\nIt is a fact that each MP belongs to one of the two political parties. Let’s denote them with letters A and B. Your task is to estimate which MP belongs to which party, so that the following holds for your estimation: each MP argued with at most two distinct members of their own party.\n\n-----Input-----\nThe first line of input contains an integer $N$ ($2 \\leq N \\leq 200000$), the number of MPs. MPs are denoted with numbers from $1$ to $N$.\n\nThe following five lines describe the photographs taken from Monday to Friday. Each of the five lines contains the list of pairs of MPs that are arguing on the photograph that day (scowling at each other). Stated first is the number of pairs $P$ ($1 \\leq P \\leq N/2$), followed by $P$ pairs in the form “$K$ $L$”, where $K$ and $L$ are labels of MPs scowling at each other. Before each pair there is a double space.\n\nOf course, each MP is stated at most once per line, and no MP ever argues with herself.\n\n-----Output-----\nThe first and only line of output must contain an array consisting of only characters A and B, so that the $K$-th character denotes the party of $K$-th MP in a division that satisfies the given conditions.\n\nSince the solution isn’t going to be unique, output any.\n\n-----Examples-----\nSample Input:\n7\n2  1 2  7 3\n2  1 3  7 4\n2  1 4  7 5\n2  1 5  7 6\n2  1 6  7 2\nSample Output:\nABBBBBA",
        "solutions": "",
        "difficulty": "interview",
        "input": "10\n3  1 2  7 3  9 4\n3  1 3  7 4  9 5\n3  1 4  7 5  9 6\n3  1 5  7 6  9 8\n3  1 6  7 8  9 10\n",
        "output": "ABBBBBAAAA\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/sabor"
    },
    {
        "id": 295,
        "task_id": 2965,
        "test_case_id": 1,
        "question": "Pumpkin Pete is trying out a new type of “rapid-growth” pumpkin seed that he bought from the farmer’s market. Without looking at the directions, Pumpkin Pete tears through the packaging and plants the seeds he has into his pumpkin patch. Unbeknownst to Pumpkin Pete, his rival, Gourd Gary, is watching him plant the new seeds from a secret vantage point. After Pumpkin Pete leaves the pumpkin patch, Gourd Gary approaches the patch and picks up the packaging that Pumpkin Pete left on the ground. The packaging says the following:\n - A pumpkin starts with four roots of length zero.\n - Each of the pumpkin’s four roots grow a single unit in a different cardinal direction each day.\n - When a pumpkin dies, its remains will not disappear.\n - If any of the roots grow into another pumpkin or its roots – dead or alive – the pumpkin will die at the end of that day.\n - Roots cannot grow outside of the bounds of a plot. In other words, a pumpkin will die if one of its roots tries to go outside the bounds of the pumpkin patch.\n - If the roots of multiple pumpkins reach the same spot on the same day, each one of the affected roots stops growing (i.e. fight for nutrients) and in turn, the pumpkins will die at the end of the day.\n - When a pumpkin dies, its roots do not grow on subsequent days.\n\nWith this information and the knowledge of where each of the pumpkin seeds were planted, Gourd Gary starts to think about which pumpkins would still be alive if they were left to grow for $D$ days$\\ldots $Input\n\nThe first line of input contains three integers: the number of pumpkins $P$, the number of days $D$ that will pass ($1 \\leq D \\leq 10$), and $N$ ($1 \\leq N \\leq 30, 1 \\leq P \\leq N^2$) the dimension of the $N\\times N$ grid. The next $P$ lines of input contain two integers, $R$ and $C$ ($0 \\leq R,C < N$), representing the row and column position of each pumpkin. No two pumpkins will be at the same position. Position $(0,0)$ is the top left corner of the grid.\n\n-----Output-----\nThe output will consist of a single line per pumpkin in the same relative order as the input. If the pumpkin is alive after $D$ days have passed, print “ALIVE”. Otherwise, print the day the pumpkin died as a single integer.\n\n-----Examples-----\nSample Input:\n4 2 8\n3 2\n5 5\n4 3\n1 1\nSample Output:\n1\n2\n1\n2",
        "solutions": "",
        "difficulty": "interview",
        "input": "4 2 8\n3 2\n5 5\n4 3\n1 1\n",
        "output": "1\n2\n1\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/pumpkinpatch"
    },
    {
        "id": 296,
        "task_id": 2965,
        "test_case_id": 2,
        "question": "Pumpkin Pete is trying out a new type of “rapid-growth” pumpkin seed that he bought from the farmer’s market. Without looking at the directions, Pumpkin Pete tears through the packaging and plants the seeds he has into his pumpkin patch. Unbeknownst to Pumpkin Pete, his rival, Gourd Gary, is watching him plant the new seeds from a secret vantage point. After Pumpkin Pete leaves the pumpkin patch, Gourd Gary approaches the patch and picks up the packaging that Pumpkin Pete left on the ground. The packaging says the following:\n - A pumpkin starts with four roots of length zero.\n - Each of the pumpkin’s four roots grow a single unit in a different cardinal direction each day.\n - When a pumpkin dies, its remains will not disappear.\n - If any of the roots grow into another pumpkin or its roots – dead or alive – the pumpkin will die at the end of that day.\n - Roots cannot grow outside of the bounds of a plot. In other words, a pumpkin will die if one of its roots tries to go outside the bounds of the pumpkin patch.\n - If the roots of multiple pumpkins reach the same spot on the same day, each one of the affected roots stops growing (i.e. fight for nutrients) and in turn, the pumpkins will die at the end of the day.\n - When a pumpkin dies, its roots do not grow on subsequent days.\n\nWith this information and the knowledge of where each of the pumpkin seeds were planted, Gourd Gary starts to think about which pumpkins would still be alive if they were left to grow for $D$ days$\\ldots $Input\n\nThe first line of input contains three integers: the number of pumpkins $P$, the number of days $D$ that will pass ($1 \\leq D \\leq 10$), and $N$ ($1 \\leq N \\leq 30, 1 \\leq P \\leq N^2$) the dimension of the $N\\times N$ grid. The next $P$ lines of input contain two integers, $R$ and $C$ ($0 \\leq R,C < N$), representing the row and column position of each pumpkin. No two pumpkins will be at the same position. Position $(0,0)$ is the top left corner of the grid.\n\n-----Output-----\nThe output will consist of a single line per pumpkin in the same relative order as the input. If the pumpkin is alive after $D$ days have passed, print “ALIVE”. Otherwise, print the day the pumpkin died as a single integer.\n\n-----Examples-----\nSample Input:\n4 2 8\n3 2\n5 5\n4 3\n1 1\nSample Output:\n1\n2\n1\n2",
        "solutions": "",
        "difficulty": "interview",
        "input": "5 3 10\n0 0\n6 3\n6 4\n3 6\n2 1\n",
        "output": "1\n1\n1\nALIVE\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/pumpkinpatch"
    },
    {
        "id": 297,
        "task_id": 3013,
        "test_case_id": 1,
        "question": "Archimedes is the name of a new 2D video game, which is very simple: the player hits start, at which point their avatar moves along an Archimedean spiral starting from the origin. When they hit a button, the avatar detaches from the spiral in the direction it is currently moving. The goal is to hit a target. The avatar’s path after it detaches must not intersect with any part of the spiral.\n\nTo help study this game, you’re asked to write a program that given the spiral’s angular velocity computes the point at which the player would need to press the button to release the avatar from the spiral in order to hit the target.\n\nAn Archimedean spiral is created when a point moves with constant velocity along a line that rotates with constant angular velocity around the origin. When expressed in polar coordinates $(r, \\phi )$, an Archimedean spiral has a simple formulation: $r = b \\ \\phi $ where $r$ is the distance from the origin and $\\phi $ is the angle between the point, origin, and the unit vector $(1, 0)$. $b$ is a constant that determines the distance between successive rings of the spiral.\n\n-----Input-----\nThe input consists of a single line with $3$ real numbers $b$, $t_ x$, and $t_ y$, denoting the parameter $b$ ($0.01 \\le b \\le 10$) of the spiral described by $r = b \\ \\phi $ and the $x, y$ coordinates of the target $T = (t_ x, t_ y)$, restricted by $-10000 \\le t_ x, t_ y \\le 10000$. It is guaranteed that $\\sqrt{t_ x^2+t_ y^2} > 2 \\pi b$, i.e., the avatar will stay on the spiral for at least one full $360$ degree turn. It is also guaranteed that the distance from point $T$ to the closest point on the spiral will be greater than $10^{-3}$. There may be up to $12$ significant digits in $b$, and up to $3$ digits after the decimal point in $t_ x$ and $t_ y$.\n\n-----Output-----\nOutput the $x, y$ coordinates of the point on the spiral where the avatar should leave the spiral, continue in the direction it is moving, and hit the target without intersecting the spiral.\n\nYour answer will be considered correct if the absolute or relative error of both $x$ and $y$ does not exceed $10^{-5}$.\n\n-----Examples-----\nSample Input 1:\n0.5 -5.301 3.098\nSample Output 1:\n-1.26167861 3.88425357\n\nSample Input 2:\n0.5 8 8\nSample Output 2:\n9.21068947 2.56226688\n\nSample Input 3:\n1 8 8\nSample Output 3:\n6.22375968 -0.31921472\n\nSample Input 4:\n0.5 -8 8\nSample Output 4:\n-4.36385220 9.46891588",
        "solutions": "",
        "difficulty": "competition",
        "input": "0.5 -5.301 3.098\n",
        "output": "-1.26167861 3.88425357\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/archimedes"
    },
    {
        "id": 298,
        "task_id": 3013,
        "test_case_id": 2,
        "question": "Archimedes is the name of a new 2D video game, which is very simple: the player hits start, at which point their avatar moves along an Archimedean spiral starting from the origin. When they hit a button, the avatar detaches from the spiral in the direction it is currently moving. The goal is to hit a target. The avatar’s path after it detaches must not intersect with any part of the spiral.\n\nTo help study this game, you’re asked to write a program that given the spiral’s angular velocity computes the point at which the player would need to press the button to release the avatar from the spiral in order to hit the target.\n\nAn Archimedean spiral is created when a point moves with constant velocity along a line that rotates with constant angular velocity around the origin. When expressed in polar coordinates $(r, \\phi )$, an Archimedean spiral has a simple formulation: $r = b \\ \\phi $ where $r$ is the distance from the origin and $\\phi $ is the angle between the point, origin, and the unit vector $(1, 0)$. $b$ is a constant that determines the distance between successive rings of the spiral.\n\n-----Input-----\nThe input consists of a single line with $3$ real numbers $b$, $t_ x$, and $t_ y$, denoting the parameter $b$ ($0.01 \\le b \\le 10$) of the spiral described by $r = b \\ \\phi $ and the $x, y$ coordinates of the target $T = (t_ x, t_ y)$, restricted by $-10000 \\le t_ x, t_ y \\le 10000$. It is guaranteed that $\\sqrt{t_ x^2+t_ y^2} > 2 \\pi b$, i.e., the avatar will stay on the spiral for at least one full $360$ degree turn. It is also guaranteed that the distance from point $T$ to the closest point on the spiral will be greater than $10^{-3}$. There may be up to $12$ significant digits in $b$, and up to $3$ digits after the decimal point in $t_ x$ and $t_ y$.\n\n-----Output-----\nOutput the $x, y$ coordinates of the point on the spiral where the avatar should leave the spiral, continue in the direction it is moving, and hit the target without intersecting the spiral.\n\nYour answer will be considered correct if the absolute or relative error of both $x$ and $y$ does not exceed $10^{-5}$.\n\n-----Examples-----\nSample Input 1:\n0.5 -5.301 3.098\nSample Output 1:\n-1.26167861 3.88425357\n\nSample Input 2:\n0.5 8 8\nSample Output 2:\n9.21068947 2.56226688\n\nSample Input 3:\n1 8 8\nSample Output 3:\n6.22375968 -0.31921472\n\nSample Input 4:\n0.5 -8 8\nSample Output 4:\n-4.36385220 9.46891588",
        "solutions": "",
        "difficulty": "competition",
        "input": "0.5 8 8\n",
        "output": "9.21068947 2.56226688\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/archimedes"
    },
    {
        "id": 299,
        "task_id": 3013,
        "test_case_id": 3,
        "question": "Archimedes is the name of a new 2D video game, which is very simple: the player hits start, at which point their avatar moves along an Archimedean spiral starting from the origin. When they hit a button, the avatar detaches from the spiral in the direction it is currently moving. The goal is to hit a target. The avatar’s path after it detaches must not intersect with any part of the spiral.\n\nTo help study this game, you’re asked to write a program that given the spiral’s angular velocity computes the point at which the player would need to press the button to release the avatar from the spiral in order to hit the target.\n\nAn Archimedean spiral is created when a point moves with constant velocity along a line that rotates with constant angular velocity around the origin. When expressed in polar coordinates $(r, \\phi )$, an Archimedean spiral has a simple formulation: $r = b \\ \\phi $ where $r$ is the distance from the origin and $\\phi $ is the angle between the point, origin, and the unit vector $(1, 0)$. $b$ is a constant that determines the distance between successive rings of the spiral.\n\n-----Input-----\nThe input consists of a single line with $3$ real numbers $b$, $t_ x$, and $t_ y$, denoting the parameter $b$ ($0.01 \\le b \\le 10$) of the spiral described by $r = b \\ \\phi $ and the $x, y$ coordinates of the target $T = (t_ x, t_ y)$, restricted by $-10000 \\le t_ x, t_ y \\le 10000$. It is guaranteed that $\\sqrt{t_ x^2+t_ y^2} > 2 \\pi b$, i.e., the avatar will stay on the spiral for at least one full $360$ degree turn. It is also guaranteed that the distance from point $T$ to the closest point on the spiral will be greater than $10^{-3}$. There may be up to $12$ significant digits in $b$, and up to $3$ digits after the decimal point in $t_ x$ and $t_ y$.\n\n-----Output-----\nOutput the $x, y$ coordinates of the point on the spiral where the avatar should leave the spiral, continue in the direction it is moving, and hit the target without intersecting the spiral.\n\nYour answer will be considered correct if the absolute or relative error of both $x$ and $y$ does not exceed $10^{-5}$.\n\n-----Examples-----\nSample Input 1:\n0.5 -5.301 3.098\nSample Output 1:\n-1.26167861 3.88425357\n\nSample Input 2:\n0.5 8 8\nSample Output 2:\n9.21068947 2.56226688\n\nSample Input 3:\n1 8 8\nSample Output 3:\n6.22375968 -0.31921472\n\nSample Input 4:\n0.5 -8 8\nSample Output 4:\n-4.36385220 9.46891588",
        "solutions": "",
        "difficulty": "competition",
        "input": "1 8 8\n",
        "output": "6.22375968 -0.31921472\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/archimedes"
    },
    {
        "id": 300,
        "task_id": 3013,
        "test_case_id": 4,
        "question": "Archimedes is the name of a new 2D video game, which is very simple: the player hits start, at which point their avatar moves along an Archimedean spiral starting from the origin. When they hit a button, the avatar detaches from the spiral in the direction it is currently moving. The goal is to hit a target. The avatar’s path after it detaches must not intersect with any part of the spiral.\n\nTo help study this game, you’re asked to write a program that given the spiral’s angular velocity computes the point at which the player would need to press the button to release the avatar from the spiral in order to hit the target.\n\nAn Archimedean spiral is created when a point moves with constant velocity along a line that rotates with constant angular velocity around the origin. When expressed in polar coordinates $(r, \\phi )$, an Archimedean spiral has a simple formulation: $r = b \\ \\phi $ where $r$ is the distance from the origin and $\\phi $ is the angle between the point, origin, and the unit vector $(1, 0)$. $b$ is a constant that determines the distance between successive rings of the spiral.\n\n-----Input-----\nThe input consists of a single line with $3$ real numbers $b$, $t_ x$, and $t_ y$, denoting the parameter $b$ ($0.01 \\le b \\le 10$) of the spiral described by $r = b \\ \\phi $ and the $x, y$ coordinates of the target $T = (t_ x, t_ y)$, restricted by $-10000 \\le t_ x, t_ y \\le 10000$. It is guaranteed that $\\sqrt{t_ x^2+t_ y^2} > 2 \\pi b$, i.e., the avatar will stay on the spiral for at least one full $360$ degree turn. It is also guaranteed that the distance from point $T$ to the closest point on the spiral will be greater than $10^{-3}$. There may be up to $12$ significant digits in $b$, and up to $3$ digits after the decimal point in $t_ x$ and $t_ y$.\n\n-----Output-----\nOutput the $x, y$ coordinates of the point on the spiral where the avatar should leave the spiral, continue in the direction it is moving, and hit the target without intersecting the spiral.\n\nYour answer will be considered correct if the absolute or relative error of both $x$ and $y$ does not exceed $10^{-5}$.\n\n-----Examples-----\nSample Input 1:\n0.5 -5.301 3.098\nSample Output 1:\n-1.26167861 3.88425357\n\nSample Input 2:\n0.5 8 8\nSample Output 2:\n9.21068947 2.56226688\n\nSample Input 3:\n1 8 8\nSample Output 3:\n6.22375968 -0.31921472\n\nSample Input 4:\n0.5 -8 8\nSample Output 4:\n-4.36385220 9.46891588",
        "solutions": "",
        "difficulty": "competition",
        "input": "0.5 -8 8\n",
        "output": "-4.36385220 9.46891588\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/archimedes"
    },
    {
        "id": 301,
        "task_id": 3013,
        "test_case_id": 5,
        "question": "Archimedes is the name of a new 2D video game, which is very simple: the player hits start, at which point their avatar moves along an Archimedean spiral starting from the origin. When they hit a button, the avatar detaches from the spiral in the direction it is currently moving. The goal is to hit a target. The avatar’s path after it detaches must not intersect with any part of the spiral.\n\nTo help study this game, you’re asked to write a program that given the spiral’s angular velocity computes the point at which the player would need to press the button to release the avatar from the spiral in order to hit the target.\n\nAn Archimedean spiral is created when a point moves with constant velocity along a line that rotates with constant angular velocity around the origin. When expressed in polar coordinates $(r, \\phi )$, an Archimedean spiral has a simple formulation: $r = b \\ \\phi $ where $r$ is the distance from the origin and $\\phi $ is the angle between the point, origin, and the unit vector $(1, 0)$. $b$ is a constant that determines the distance between successive rings of the spiral.\n\n-----Input-----\nThe input consists of a single line with $3$ real numbers $b$, $t_ x$, and $t_ y$, denoting the parameter $b$ ($0.01 \\le b \\le 10$) of the spiral described by $r = b \\ \\phi $ and the $x, y$ coordinates of the target $T = (t_ x, t_ y)$, restricted by $-10000 \\le t_ x, t_ y \\le 10000$. It is guaranteed that $\\sqrt{t_ x^2+t_ y^2} > 2 \\pi b$, i.e., the avatar will stay on the spiral for at least one full $360$ degree turn. It is also guaranteed that the distance from point $T$ to the closest point on the spiral will be greater than $10^{-3}$. There may be up to $12$ significant digits in $b$, and up to $3$ digits after the decimal point in $t_ x$ and $t_ y$.\n\n-----Output-----\nOutput the $x, y$ coordinates of the point on the spiral where the avatar should leave the spiral, continue in the direction it is moving, and hit the target without intersecting the spiral.\n\nYour answer will be considered correct if the absolute or relative error of both $x$ and $y$ does not exceed $10^{-5}$.\n\n-----Examples-----\nSample Input 1:\n0.5 -5.301 3.098\nSample Output 1:\n-1.26167861 3.88425357\n\nSample Input 2:\n0.5 8 8\nSample Output 2:\n9.21068947 2.56226688\n\nSample Input 3:\n1 8 8\nSample Output 3:\n6.22375968 -0.31921472\n\nSample Input 4:\n0.5 -8 8\nSample Output 4:\n-4.36385220 9.46891588",
        "solutions": "",
        "difficulty": "competition",
        "input": "0.5 0 -8\n",
        "output": "-3.60855706 -3.61140618\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/archimedes"
    },
    {
        "id": 302,
        "task_id": 3120,
        "test_case_id": 1,
        "question": "In programming contest circles, one of the most important roles is that of the Chief Equality Officer (CEO). This person is responsible for making sure that every team has an equal chance of winning the contest. Since last year’s NWERC the current CEO, Gregor, has thought at length about how to make the contest even more fair and equal.\n\nHis answer is to introduce a new programming language as the only one allowed for submissions. This way, no team will be disadvantaged by not having mastered any of the allowed languages. This language is called Balloon, short for Building A Long List Of Ordinary Numbers. Its only data type is the list of integers. To keep the language fast, it contains only four instructions:\n - [x$_1$,…,x$_ n$] is the constructor for lists. It returns the integers inside the brackets in their given order.\n - concat(<Expr$_1$>,<Expr$_2$>) returns a list of all the integers returned when evaluating the expression <Expr$_1$> followed by all of the integers returned when evaluating <Expr$_2$>.\n - shuffle(<Expr>) returns a list of all the integers returned by <Expr>, rearranged according to a uniformly random permutation, i.e., each permutation of the elements is used with equal probability.\n - sorted(<Expr>) returns a list of all the integers returned by <Expr>, rearranged into non-decreasing order.\n\nAs an example, consider the first expression of Sample Input 1. The two shuffle exressions both take the list [1,2] as input and return one of the lists [1,2] and [2,1], each with probability $0.5$ (independently of each other). The outer concat operator takes the two returned lists as its input and returns their concatenation. I.e., it returns one of the lists [1,2,1,2], [1,2,2,1], [2,1,1,2], and [2,1,2,1], each with probability $0.25$.\n\nNaturally, we cannot use byte-by-byte output comparison any more when teams submit their solutions in Balloon, as its output is probabilistic. The judge server instead has to check whether a submitted program is equivalent to the sample solution created by the judges. Two programs are equivalent if for any list $L$ of integers, both programs have an equal probability of returning $L$.\n\nIt is your task to determine whether two given Balloon programs are equivalent.\n\n-----Input-----\nThe input consists of:\n - One line containing a string A, the first program.\n - One line containing a string B, the second program.\n\nEach program is a syntactically valid Balloon program with between $3$ and $10^6$ characters, and contains neither spacing nor empty lists (i.e., the strings “ ” or “[]” do not occur in the input).\n\nEach integer in each program is greater than $0$ and less than $10^{9}$.\n\n-----Output-----\nIf the two programs are equivalent, output “equal”, otherwise output “not equal”.\n\n-----Examples-----\nSample Input 1:\nconcat(shuffle([1,2]),shuffle([1,2]))\nshuffle([1,2,1,2])\nSample Output 1:\nnot equal\n\nSample Input 2:\nsorted(concat([3,2,1],[4,5,6]))\n[1,2,3,4,5,6]\nSample Output 2:\nequal",
        "solutions": "",
        "difficulty": "competition",
        "input": "concat(shuffle([1,2]),shuffle([1,2]))\nshuffle([1,2,1,2])\n",
        "output": "not equal\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/equality"
    },
    {
        "id": 303,
        "task_id": 3120,
        "test_case_id": 3,
        "question": "In programming contest circles, one of the most important roles is that of the Chief Equality Officer (CEO). This person is responsible for making sure that every team has an equal chance of winning the contest. Since last year’s NWERC the current CEO, Gregor, has thought at length about how to make the contest even more fair and equal.\n\nHis answer is to introduce a new programming language as the only one allowed for submissions. This way, no team will be disadvantaged by not having mastered any of the allowed languages. This language is called Balloon, short for Building A Long List Of Ordinary Numbers. Its only data type is the list of integers. To keep the language fast, it contains only four instructions:\n - [x$_1$,…,x$_ n$] is the constructor for lists. It returns the integers inside the brackets in their given order.\n - concat(<Expr$_1$>,<Expr$_2$>) returns a list of all the integers returned when evaluating the expression <Expr$_1$> followed by all of the integers returned when evaluating <Expr$_2$>.\n - shuffle(<Expr>) returns a list of all the integers returned by <Expr>, rearranged according to a uniformly random permutation, i.e., each permutation of the elements is used with equal probability.\n - sorted(<Expr>) returns a list of all the integers returned by <Expr>, rearranged into non-decreasing order.\n\nAs an example, consider the first expression of Sample Input 1. The two shuffle exressions both take the list [1,2] as input and return one of the lists [1,2] and [2,1], each with probability $0.5$ (independently of each other). The outer concat operator takes the two returned lists as its input and returns their concatenation. I.e., it returns one of the lists [1,2,1,2], [1,2,2,1], [2,1,1,2], and [2,1,2,1], each with probability $0.25$.\n\nNaturally, we cannot use byte-by-byte output comparison any more when teams submit their solutions in Balloon, as its output is probabilistic. The judge server instead has to check whether a submitted program is equivalent to the sample solution created by the judges. Two programs are equivalent if for any list $L$ of integers, both programs have an equal probability of returning $L$.\n\nIt is your task to determine whether two given Balloon programs are equivalent.\n\n-----Input-----\nThe input consists of:\n - One line containing a string A, the first program.\n - One line containing a string B, the second program.\n\nEach program is a syntactically valid Balloon program with between $3$ and $10^6$ characters, and contains neither spacing nor empty lists (i.e., the strings “ ” or “[]” do not occur in the input).\n\nEach integer in each program is greater than $0$ and less than $10^{9}$.\n\n-----Output-----\nIf the two programs are equivalent, output “equal”, otherwise output “not equal”.\n\n-----Examples-----\nSample Input 1:\nconcat(shuffle([1,2]),shuffle([1,2]))\nshuffle([1,2,1,2])\nSample Output 1:\nnot equal\n\nSample Input 2:\nsorted(concat([3,2,1],[4,5,6]))\n[1,2,3,4,5,6]\nSample Output 2:\nequal",
        "solutions": "",
        "difficulty": "competition",
        "input": "concat(sorted([4,3,2,1]),shuffle([1]))\nconcat(concat([1,2,3],shuffle([4])),sorted([1]))\n",
        "output": "equal\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/equality"
    },
    {
        "id": 304,
        "task_id": 3268,
        "test_case_id": 1,
        "question": "One day, a flock of hungry birds find a huge tree of elderberries. They want to eat all berries of this tree. These birds quickly alight on some leaves of the tree and eat all berries at where they stand. Now they are thinking of who will eat other berries …\n\nSince the flock has some giant birds and some tiny birds, and they observe that the tree has some big branches and some small branches, fairly distributing these berries among all birds turns out to be a tough problem.\n\nFormally, the tree of elderberries can be presented as a tree with the following properties:\n - The tree has $n$ vertices numbered from $1$ to $n$. The root of the tree is vertex $1$.\n - Each non-leaf vertex (vertex having at least one child) is called a branch. A branch is classified as either big or small. The root is always a big branch.\n - On each leaf of the tree, there is either one giant bird, one tiny bird or one elderberry.\n\nThe process of determining which bird eats which berry is as below:\n - First, every bird and every berry has a label. A label is a non-empty string of at most five lowercase English characters.\n - Second, every bird has a controlled area. Consider the bird at vertex $v$:\n - If it is a tiny bird, its controlled area is the subtree rooted at vertex $p_ v$, where $p_ v$ is the parent of $v$.\n - If it is a giant bird, its controlled area is the subtree rooted at vertex $b_ v$, where $b_ v$ is the closest vertex to $v$ among all ancestors of $v$ which are big branches.\n - Finally, for each berry, the bird who eats it is determined using the following rules:\n - A bird can only eat berries having the same label with it.\n - A bird can only eat berries inside its controlled area.\n - If there is more than one bird satisfying the two above conditions, only one bird with smallest controlled area eats it.\n\nAll birds also set some rules for labeling birds and berries:\n - A bird or a berry can have same label with other birds and berries. However, no groups of eight or more birds have the same label.\n - Each berry is inside the controlled area of at least one bird with same label.\n - No two birds with same label have the same controlled area.\n - From all of the above, it can be proven that the bird eating every berry can be uniquely determined.\n\nSadly, tiny birds think that these rules give advantages to giant birds over them. They claim that, by the rules of setting controlled areas, giant birds usually control larger areas, which means having more berries. (Although their thought is not always true.) They want their controlled areas to be determined using the rules of giant birds. (In other words, they want all tiny birds become giant birds).\n\nGiant birds accept the tiny birds’ claim. However, they try to change the labels of some birds and berries, so that if we assume all tiny birds become giant, the new set of labels still satisfy all the above conditions. Moreover, after changing labels, no berries have changed owner (i.e, every berry is still eaten by the same bird).\n\nThey want to change as few labels as possible. Please help them!\n\n-----Input-----\nThe first line contains one integer $n$ $(3 \\leq n \\leq 150000)$ — the number of vertices in the elderberry tree. The $i$-th of the next $n$ lines describes the $i$-th vertex in either of the following formats:\n - ‘$p_ i$$t_ i$’, where $t_ i$ is either ‘B’ or ‘S’: meaning that $i$ is a branch. $t_ i = B$ and $t_ i = S$ mean that $i$ is a big or small branch, respectively.\n - ‘$p_ i$$T_ i$ $L$’, where $T_ i$ is either ‘G’, ‘T’ or ‘E’ and $L$ is a non-empty string of at most five English characters: meaning that $i$ is a leaf. $t_ i = G, t_ i = T$ and $t_ i = E$ mean that in vertex $i$ there is either a giant bird, a tiny bird or an elderberry. $L$ is the label of this bird or berry.\n\nIn both formats, $p_ i$ satisfies $1 \\leq p_ i < i$ and denotes the parent of vertex $i$. We assume that $p_1 = 0$.\n\nIt is guaranteed that the input describes a valid tree, there is at least one bird and one berry, and all labels satisfy the rules presented above. Remember that no eight birds have the same label.\n\n-----Output-----\n - The first line contains an integer $k$ — the minimum number of labels need changing.\n - Each of the next $k$ lines contains an integer $x$ $(1 \\leq x \\leq n)$ and a label $S$ — meaning that the label of the bird or berry at vertex $x$ is assigned to $S$. $x$ should be a leaf of the tree. $S$ should be a non-empty string of at most five lowercase English characters.\n\nIf there are multiple optimal solutions, you can output any one of them.\n\n-----Explanation for examples-----\nBelow are figures depicting these three examples. Red vertices are big branches, green vertices are small branches, violet vertices have giant birds, yellow vertices have tiny birds and white vertices have berries.\n - In the first example:\n - Initially:\n - There are $3$ birds with label ‘a’ at vertices $6$, $7$ and $13$. Their controlled areas are the subtrees rooted at vertices $2$, $5$ and $1$, respectively.\n - There is $1$ bird with label ‘b’ at vertex $12$. Its controlled area is the subtree rooted at vertex $1$.\n - There are $3$ elderberries with label ‘a’ at vertices $3$, $8$ and $11$. They are eaten by the birds at vertices $6$, $7$ and $13$, respectively.\n - There are $2$ elderberries with label ‘b’ at vertices $4$ and $9$. They are both eaten by the bird at vertex $12$.\n - If all tiny birds become giant birds, both the controlled area of the birds at vertices $6$ and $7$ become the subtree rooted at vertex $2$, violating the rules (No two birds with same label having same controlled area). Hence, at least one bird’s label needs changing. We can change the label of the bird at vertex $6$ to ‘c’. As the bird at vertex $6$ eats the berry at vertex $3$, we need to change the label of the bird at vertex $3$ as well. The solution in which the bird/berry at vertices $7$ and $8$ get changed is also accepted.\n - In the second example:\n - Initially:\n - The controlled areas of the birds at vertices $3$ and $6$ are subtrees rooted at vertices $1$ and $5$, respectively.\n - The bird at vertex $3$ eats the berry at vertex $4$.\n - If all tiny birds become giant birds, their controlled areas are subtrees rooted at vertices $1$ and $2$, respectively. As a result, the bird at vertex $6$ eats the berry at vertex $4$, which is invalid, (The owner of every berry must not change). Hence the label of the bird at vertex $6$ must be changed.\n - In the third example, no changes are necessary.\n\n-----Examples-----\nSample Input 1:\n13\n0 B\n1 B\n2 E a\n2 E b\n2 S\n5 G a\n5 T a\n5 E a\n5 E b\n1 S\n10 E a\n10 G b\n1 T a\nSample Output 1:\n2\n3 c\n6 c\n\nSample Input 2:\n6\n0 B\n1 B\n1 T a\n2 E a\n2 S\n5 T a\nSample Output 2:\n1\n6 b",
        "solutions": "",
        "difficulty": "competition",
        "input": "13\n0 B\n1 B\n2 E a\n2 E b\n2 S\n5 G a\n5 T a\n5 E a\n5 E b\n1 S\n10 E a\n10 G b\n1 T a\n",
        "output": "2\n3 c\n6 c\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/enjoyingelderberries"
    },
    {
        "id": 305,
        "task_id": 3268,
        "test_case_id": 2,
        "question": "One day, a flock of hungry birds find a huge tree of elderberries. They want to eat all berries of this tree. These birds quickly alight on some leaves of the tree and eat all berries at where they stand. Now they are thinking of who will eat other berries …\n\nSince the flock has some giant birds and some tiny birds, and they observe that the tree has some big branches and some small branches, fairly distributing these berries among all birds turns out to be a tough problem.\n\nFormally, the tree of elderberries can be presented as a tree with the following properties:\n - The tree has $n$ vertices numbered from $1$ to $n$. The root of the tree is vertex $1$.\n - Each non-leaf vertex (vertex having at least one child) is called a branch. A branch is classified as either big or small. The root is always a big branch.\n - On each leaf of the tree, there is either one giant bird, one tiny bird or one elderberry.\n\nThe process of determining which bird eats which berry is as below:\n - First, every bird and every berry has a label. A label is a non-empty string of at most five lowercase English characters.\n - Second, every bird has a controlled area. Consider the bird at vertex $v$:\n - If it is a tiny bird, its controlled area is the subtree rooted at vertex $p_ v$, where $p_ v$ is the parent of $v$.\n - If it is a giant bird, its controlled area is the subtree rooted at vertex $b_ v$, where $b_ v$ is the closest vertex to $v$ among all ancestors of $v$ which are big branches.\n - Finally, for each berry, the bird who eats it is determined using the following rules:\n - A bird can only eat berries having the same label with it.\n - A bird can only eat berries inside its controlled area.\n - If there is more than one bird satisfying the two above conditions, only one bird with smallest controlled area eats it.\n\nAll birds also set some rules for labeling birds and berries:\n - A bird or a berry can have same label with other birds and berries. However, no groups of eight or more birds have the same label.\n - Each berry is inside the controlled area of at least one bird with same label.\n - No two birds with same label have the same controlled area.\n - From all of the above, it can be proven that the bird eating every berry can be uniquely determined.\n\nSadly, tiny birds think that these rules give advantages to giant birds over them. They claim that, by the rules of setting controlled areas, giant birds usually control larger areas, which means having more berries. (Although their thought is not always true.) They want their controlled areas to be determined using the rules of giant birds. (In other words, they want all tiny birds become giant birds).\n\nGiant birds accept the tiny birds’ claim. However, they try to change the labels of some birds and berries, so that if we assume all tiny birds become giant, the new set of labels still satisfy all the above conditions. Moreover, after changing labels, no berries have changed owner (i.e, every berry is still eaten by the same bird).\n\nThey want to change as few labels as possible. Please help them!\n\n-----Input-----\nThe first line contains one integer $n$ $(3 \\leq n \\leq 150000)$ — the number of vertices in the elderberry tree. The $i$-th of the next $n$ lines describes the $i$-th vertex in either of the following formats:\n - ‘$p_ i$$t_ i$’, where $t_ i$ is either ‘B’ or ‘S’: meaning that $i$ is a branch. $t_ i = B$ and $t_ i = S$ mean that $i$ is a big or small branch, respectively.\n - ‘$p_ i$$T_ i$ $L$’, where $T_ i$ is either ‘G’, ‘T’ or ‘E’ and $L$ is a non-empty string of at most five English characters: meaning that $i$ is a leaf. $t_ i = G, t_ i = T$ and $t_ i = E$ mean that in vertex $i$ there is either a giant bird, a tiny bird or an elderberry. $L$ is the label of this bird or berry.\n\nIn both formats, $p_ i$ satisfies $1 \\leq p_ i < i$ and denotes the parent of vertex $i$. We assume that $p_1 = 0$.\n\nIt is guaranteed that the input describes a valid tree, there is at least one bird and one berry, and all labels satisfy the rules presented above. Remember that no eight birds have the same label.\n\n-----Output-----\n - The first line contains an integer $k$ — the minimum number of labels need changing.\n - Each of the next $k$ lines contains an integer $x$ $(1 \\leq x \\leq n)$ and a label $S$ — meaning that the label of the bird or berry at vertex $x$ is assigned to $S$. $x$ should be a leaf of the tree. $S$ should be a non-empty string of at most five lowercase English characters.\n\nIf there are multiple optimal solutions, you can output any one of them.\n\n-----Explanation for examples-----\nBelow are figures depicting these three examples. Red vertices are big branches, green vertices are small branches, violet vertices have giant birds, yellow vertices have tiny birds and white vertices have berries.\n - In the first example:\n - Initially:\n - There are $3$ birds with label ‘a’ at vertices $6$, $7$ and $13$. Their controlled areas are the subtrees rooted at vertices $2$, $5$ and $1$, respectively.\n - There is $1$ bird with label ‘b’ at vertex $12$. Its controlled area is the subtree rooted at vertex $1$.\n - There are $3$ elderberries with label ‘a’ at vertices $3$, $8$ and $11$. They are eaten by the birds at vertices $6$, $7$ and $13$, respectively.\n - There are $2$ elderberries with label ‘b’ at vertices $4$ and $9$. They are both eaten by the bird at vertex $12$.\n - If all tiny birds become giant birds, both the controlled area of the birds at vertices $6$ and $7$ become the subtree rooted at vertex $2$, violating the rules (No two birds with same label having same controlled area). Hence, at least one bird’s label needs changing. We can change the label of the bird at vertex $6$ to ‘c’. As the bird at vertex $6$ eats the berry at vertex $3$, we need to change the label of the bird at vertex $3$ as well. The solution in which the bird/berry at vertices $7$ and $8$ get changed is also accepted.\n - In the second example:\n - Initially:\n - The controlled areas of the birds at vertices $3$ and $6$ are subtrees rooted at vertices $1$ and $5$, respectively.\n - The bird at vertex $3$ eats the berry at vertex $4$.\n - If all tiny birds become giant birds, their controlled areas are subtrees rooted at vertices $1$ and $2$, respectively. As a result, the bird at vertex $6$ eats the berry at vertex $4$, which is invalid, (The owner of every berry must not change). Hence the label of the bird at vertex $6$ must be changed.\n - In the third example, no changes are necessary.\n\n-----Examples-----\nSample Input 1:\n13\n0 B\n1 B\n2 E a\n2 E b\n2 S\n5 G a\n5 T a\n5 E a\n5 E b\n1 S\n10 E a\n10 G b\n1 T a\nSample Output 1:\n2\n3 c\n6 c\n\nSample Input 2:\n6\n0 B\n1 B\n1 T a\n2 E a\n2 S\n5 T a\nSample Output 2:\n1\n6 b",
        "solutions": "",
        "difficulty": "competition",
        "input": "6\n0 B\n1 B\n1 T a\n2 E a\n2 S\n5 T a\n",
        "output": "1\n6 b\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/enjoyingelderberries"
    },
    {
        "id": 306,
        "task_id": 3268,
        "test_case_id": 3,
        "question": "One day, a flock of hungry birds find a huge tree of elderberries. They want to eat all berries of this tree. These birds quickly alight on some leaves of the tree and eat all berries at where they stand. Now they are thinking of who will eat other berries …\n\nSince the flock has some giant birds and some tiny birds, and they observe that the tree has some big branches and some small branches, fairly distributing these berries among all birds turns out to be a tough problem.\n\nFormally, the tree of elderberries can be presented as a tree with the following properties:\n - The tree has $n$ vertices numbered from $1$ to $n$. The root of the tree is vertex $1$.\n - Each non-leaf vertex (vertex having at least one child) is called a branch. A branch is classified as either big or small. The root is always a big branch.\n - On each leaf of the tree, there is either one giant bird, one tiny bird or one elderberry.\n\nThe process of determining which bird eats which berry is as below:\n - First, every bird and every berry has a label. A label is a non-empty string of at most five lowercase English characters.\n - Second, every bird has a controlled area. Consider the bird at vertex $v$:\n - If it is a tiny bird, its controlled area is the subtree rooted at vertex $p_ v$, where $p_ v$ is the parent of $v$.\n - If it is a giant bird, its controlled area is the subtree rooted at vertex $b_ v$, where $b_ v$ is the closest vertex to $v$ among all ancestors of $v$ which are big branches.\n - Finally, for each berry, the bird who eats it is determined using the following rules:\n - A bird can only eat berries having the same label with it.\n - A bird can only eat berries inside its controlled area.\n - If there is more than one bird satisfying the two above conditions, only one bird with smallest controlled area eats it.\n\nAll birds also set some rules for labeling birds and berries:\n - A bird or a berry can have same label with other birds and berries. However, no groups of eight or more birds have the same label.\n - Each berry is inside the controlled area of at least one bird with same label.\n - No two birds with same label have the same controlled area.\n - From all of the above, it can be proven that the bird eating every berry can be uniquely determined.\n\nSadly, tiny birds think that these rules give advantages to giant birds over them. They claim that, by the rules of setting controlled areas, giant birds usually control larger areas, which means having more berries. (Although their thought is not always true.) They want their controlled areas to be determined using the rules of giant birds. (In other words, they want all tiny birds become giant birds).\n\nGiant birds accept the tiny birds’ claim. However, they try to change the labels of some birds and berries, so that if we assume all tiny birds become giant, the new set of labels still satisfy all the above conditions. Moreover, after changing labels, no berries have changed owner (i.e, every berry is still eaten by the same bird).\n\nThey want to change as few labels as possible. Please help them!\n\n-----Input-----\nThe first line contains one integer $n$ $(3 \\leq n \\leq 150000)$ — the number of vertices in the elderberry tree. The $i$-th of the next $n$ lines describes the $i$-th vertex in either of the following formats:\n - ‘$p_ i$$t_ i$’, where $t_ i$ is either ‘B’ or ‘S’: meaning that $i$ is a branch. $t_ i = B$ and $t_ i = S$ mean that $i$ is a big or small branch, respectively.\n - ‘$p_ i$$T_ i$ $L$’, where $T_ i$ is either ‘G’, ‘T’ or ‘E’ and $L$ is a non-empty string of at most five English characters: meaning that $i$ is a leaf. $t_ i = G, t_ i = T$ and $t_ i = E$ mean that in vertex $i$ there is either a giant bird, a tiny bird or an elderberry. $L$ is the label of this bird or berry.\n\nIn both formats, $p_ i$ satisfies $1 \\leq p_ i < i$ and denotes the parent of vertex $i$. We assume that $p_1 = 0$.\n\nIt is guaranteed that the input describes a valid tree, there is at least one bird and one berry, and all labels satisfy the rules presented above. Remember that no eight birds have the same label.\n\n-----Output-----\n - The first line contains an integer $k$ — the minimum number of labels need changing.\n - Each of the next $k$ lines contains an integer $x$ $(1 \\leq x \\leq n)$ and a label $S$ — meaning that the label of the bird or berry at vertex $x$ is assigned to $S$. $x$ should be a leaf of the tree. $S$ should be a non-empty string of at most five lowercase English characters.\n\nIf there are multiple optimal solutions, you can output any one of them.\n\n-----Explanation for examples-----\nBelow are figures depicting these three examples. Red vertices are big branches, green vertices are small branches, violet vertices have giant birds, yellow vertices have tiny birds and white vertices have berries.\n - In the first example:\n - Initially:\n - There are $3$ birds with label ‘a’ at vertices $6$, $7$ and $13$. Their controlled areas are the subtrees rooted at vertices $2$, $5$ and $1$, respectively.\n - There is $1$ bird with label ‘b’ at vertex $12$. Its controlled area is the subtree rooted at vertex $1$.\n - There are $3$ elderberries with label ‘a’ at vertices $3$, $8$ and $11$. They are eaten by the birds at vertices $6$, $7$ and $13$, respectively.\n - There are $2$ elderberries with label ‘b’ at vertices $4$ and $9$. They are both eaten by the bird at vertex $12$.\n - If all tiny birds become giant birds, both the controlled area of the birds at vertices $6$ and $7$ become the subtree rooted at vertex $2$, violating the rules (No two birds with same label having same controlled area). Hence, at least one bird’s label needs changing. We can change the label of the bird at vertex $6$ to ‘c’. As the bird at vertex $6$ eats the berry at vertex $3$, we need to change the label of the bird at vertex $3$ as well. The solution in which the bird/berry at vertices $7$ and $8$ get changed is also accepted.\n - In the second example:\n - Initially:\n - The controlled areas of the birds at vertices $3$ and $6$ are subtrees rooted at vertices $1$ and $5$, respectively.\n - The bird at vertex $3$ eats the berry at vertex $4$.\n - If all tiny birds become giant birds, their controlled areas are subtrees rooted at vertices $1$ and $2$, respectively. As a result, the bird at vertex $6$ eats the berry at vertex $4$, which is invalid, (The owner of every berry must not change). Hence the label of the bird at vertex $6$ must be changed.\n - In the third example, no changes are necessary.\n\n-----Examples-----\nSample Input 1:\n13\n0 B\n1 B\n2 E a\n2 E b\n2 S\n5 G a\n5 T a\n5 E a\n5 E b\n1 S\n10 E a\n10 G b\n1 T a\nSample Output 1:\n2\n3 c\n6 c\n\nSample Input 2:\n6\n0 B\n1 B\n1 T a\n2 E a\n2 S\n5 T a\nSample Output 2:\n1\n6 b",
        "solutions": "",
        "difficulty": "competition",
        "input": "6\n0 B\n1 G y\n1 E y\n1 E z\n1 T z\n1 E z\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/enjoyingelderberries"
    },
    {
        "id": 307,
        "task_id": 3353,
        "test_case_id": 2,
        "question": "Sally and her friends are trying to cross safely from one bank of a raging river to another. Boulders are scattered within the river, with log planks connecting the banks of the river to some of the boulders, and some pairs of boulders to each other.\n\nSally begins by trying to cross the river first. She starts at the left bank and crosses one plank at a time, with the goal of reaching the right bank. Walking across a plank takes Sally one second. Each time she crosses a plank, that plank becomes unstable and collapses into the river (so that neither Sally nor her friends can use that plank again). After Sally has safely reached the right bank, another friend tries to cross the river, and so on, until everyone who is able to make it has crossed the river.\n\nGiven the graph of banks/boulders and planks and the number of people who need the cross the river, what is the smallest amount of total time (in seconds) required for everyone to cross the river safely? If it is impossible for all people to cross, compute the minimum number of people $n$ who must be left behind and print n people left behind.\n\n-----Input-----\nThe first line of the input contains three integers $P$, $R$, and $L$: the number of people $P$ who must cross the river, the number of boulders $R$ in the river, and the number of logs $L$ spanning boulders/river banks. These integers satisfy $1 \\leq P \\leq 10$ and $0 \\leq R \\leq 1000$ and $0 \\leq L \\leq 1000$.\n\nThen follows $L$ lines, each of which contains two integers $E_1$ and $E_2$ specifying the endpoints on one log. The values for $E_1$ and $E_2$ are in the range $[-2,R-1]$, where $-2$ signifies the left river bank, $-1$ signifies the right river bank, and all other values indicate one of the boulders in the river.\n\nYou may assume that every log has two distinct endpoints, and that no two logs span the same pair of boulders/river banks. There is no guarantee that every boulder, or even the right river bank, is reachable from the left river bank.\n\n-----Output-----\nIf it is possible for all $P$ people to reach the right bank, print a single integer, the minimum total time (in seconds) required for all people to cross.\n\nIf some people must be left behind at the left bank, instead print n people left behind, where the integer $n$ is the least number of people who must be left behind.\n\n-----Examples-----\nSample Input:\n2 4 7\n-2 0\n0 -1\n-2 1\n1 0\n2 1\n2 3\n3 -1\nSample Output:\n6",
        "solutions": "",
        "difficulty": "competition",
        "input": "3 2 5\n-2 0\n-2 1\n0 1\n1 -1\n0 -1\n",
        "output": "1 people left behind\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/ragingriver"
    },
    {
        "id": 308,
        "task_id": 3367,
        "test_case_id": 1,
        "question": "In the TV quiz Monstermind, a contestant chooses a topic and is then asked questions about it during a fixed period of time. The contestant earns one point for each correct answer. When the time runs out, the contestant must be silent.\n\nTeresa has figured out such a niche topic that she knows all possible questions that may be asked about it, as well as all the answers. Since the competition is fierce, she has decided to sometimes answer a question before the host finishes reading it. The host picks each question uniformly at random from the pool of possible questions, and each question may be asked multiple times. When reading a question, the host reads at a pace of one word per second.\n\nTeresa can interrupt the host mid-question—between words, or even before hearing the first word—but not mid-word—that would be extremely impolite. Answering also takes one second, and the host will start reading another question immediately after an answer—unless Teresa interrupts again.\n\nShe wrote a program to help her choose the best moment to answer, and now there is only one question left for you. How many points does she expect to score?\n\nFor example, in the first sample test case the answer is completely determined after hearing one word, so it is optimal to answer after hearing it, and Teresa answers 2 questions correctly in 4 seconds. In the second sample test case, if the first word is What, then it takes too much time to wait for the question to finish. Therefore Teresa says Now! 4 times and expects to get $1/3$ of the answers right.\n\n-----Input-----\nThe first line contains two integers $t$ and $n$ ($1 \\leq t \\leq 100$, $1 \\leq n \\leq 100\\ 000$), the duration of the quiz and the number of questions. Each of the following $n$ lines contains a question, which is a space-separated list of words terminated by a question mark; and an answer, which is a single word.\n\nEach word is a sequence of non-space ASCII printable characters, between the ASCII values of ‘!’ and ‘$\\sim $’. Only the last word of a question has a question mark (‘?’). You can assume that no question is a prefix of another and that punctuation marks are part of a word. Words spelled with different upper/lower case are assumed to be different.\n\nIt is guaranteed that the total number of word characters is at most $100\\ 000$.\n\n-----Output-----\nOutput the expected score of an optimal strategy. Answers within a relative or absolute error of $10^{-6}$ will be accepted.\n\n-----Examples-----\nSample Input:\n4 4\nHow much is 6 times 9? 42\nHow much is 9 times 6? 42\nIs there intelligent life on Earth? Probably\nWhat is the air speed velocity of an unladen swallow? African?\nSample Output:\n2.0000000000",
        "solutions": "",
        "difficulty": "competition",
        "input": "4 4\nHow much is 6 times 9? 42\nHow much is 9 times 6? 42\nIs there intelligent life on Earth? Probably\nWhat is the air speed velocity of an unladen swallow? African?\n",
        "output": "2.0000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/justaquiz"
    },
    {
        "id": 309,
        "task_id": 3367,
        "test_case_id": 2,
        "question": "In the TV quiz Monstermind, a contestant chooses a topic and is then asked questions about it during a fixed period of time. The contestant earns one point for each correct answer. When the time runs out, the contestant must be silent.\n\nTeresa has figured out such a niche topic that she knows all possible questions that may be asked about it, as well as all the answers. Since the competition is fierce, she has decided to sometimes answer a question before the host finishes reading it. The host picks each question uniformly at random from the pool of possible questions, and each question may be asked multiple times. When reading a question, the host reads at a pace of one word per second.\n\nTeresa can interrupt the host mid-question—between words, or even before hearing the first word—but not mid-word—that would be extremely impolite. Answering also takes one second, and the host will start reading another question immediately after an answer—unless Teresa interrupts again.\n\nShe wrote a program to help her choose the best moment to answer, and now there is only one question left for you. How many points does she expect to score?\n\nFor example, in the first sample test case the answer is completely determined after hearing one word, so it is optimal to answer after hearing it, and Teresa answers 2 questions correctly in 4 seconds. In the second sample test case, if the first word is What, then it takes too much time to wait for the question to finish. Therefore Teresa says Now! 4 times and expects to get $1/3$ of the answers right.\n\n-----Input-----\nThe first line contains two integers $t$ and $n$ ($1 \\leq t \\leq 100$, $1 \\leq n \\leq 100\\ 000$), the duration of the quiz and the number of questions. Each of the following $n$ lines contains a question, which is a space-separated list of words terminated by a question mark; and an answer, which is a single word.\n\nEach word is a sequence of non-space ASCII printable characters, between the ASCII values of ‘!’ and ‘$\\sim $’. Only the last word of a question has a question mark (‘?’). You can assume that no question is a prefix of another and that punctuation marks are part of a word. Words spelled with different upper/lower case are assumed to be different.\n\nIt is guaranteed that the total number of word characters is at most $100\\ 000$.\n\n-----Output-----\nOutput the expected score of an optimal strategy. Answers within a relative or absolute error of $10^{-6}$ will be accepted.\n\n-----Examples-----\nSample Input:\n4 4\nHow much is 6 times 9? 42\nHow much is 9 times 6? 42\nIs there intelligent life on Earth? Probably\nWhat is the air speed velocity of an unladen swallow? African?\nSample Output:\n2.0000000000",
        "solutions": "",
        "difficulty": "competition",
        "input": "4 3\nWhat do we send? Code\nWhat do we want? Accepted\nWhen do we want it? Now!\n",
        "output": "1.333333333\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/justaquiz"
    },
    {
        "id": 310,
        "task_id": 3397,
        "test_case_id": 1,
        "question": "Yraglac is running a daycare service for dogs. He has $N$ dogs and he would like to take them on a walk. However, before he can walk the dogs, he needs to feed them first.\n\nYraglac has $M$ feeding bowls. Because the dogs are very well trained, he can instruct a dog to eat from a specific bowl, but he can’t instruct multiple dogs to eat from the same bowl. A dog is considered to be fed after eating from a single bowl. However, different dogs may have different food preferences, so they may eat from the same food bowl at different speeds.\n\nAs you know, dogs get impatient if they have to wait for long periods of time. Therefore, Yraglac wants to minimize the total amount of time that dogs spend waiting. More formally, suppose $t$ is the longest amount of time in seconds that any dog spent eating. Yraglac wants to minimize $T=\\sum _{i=1}^{N} t-t_ i$, where $t_ i$ is the time in seconds that the $i^\\textrm {th}$ dog spends eating.\n\n-----Input-----\nThe first line contains integers $N$ and $M$, where $2\\leq N\\leq M\\leq 50$.\n\nEach of the next $N$ lines of the input contains $M$ integers each. The $j^\\textrm {th}$ integers on the $i^\\textrm {th}$ line denotes the $t_{ij}$, the amount of time that the $i^\\textrm {th}$ dog will spend on eating food from the $j^\\textrm {th}$ bowl. Note that $1\\leq t_{ij}\\leq 200$.\n\n-----Output-----\nOutput $T$, the minimum total waiting time in seconds.\n\n-----Examples-----\nSample Input:\n2 3\n2 100 10\n100 1 10\nSample Output:\n0",
        "solutions": "",
        "difficulty": "competition",
        "input": "2 3\n2 100 10\n100 1 10\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/dogtrouble"
    },
    {
        "id": 311,
        "task_id": 3397,
        "test_case_id": 2,
        "question": "Yraglac is running a daycare service for dogs. He has $N$ dogs and he would like to take them on a walk. However, before he can walk the dogs, he needs to feed them first.\n\nYraglac has $M$ feeding bowls. Because the dogs are very well trained, he can instruct a dog to eat from a specific bowl, but he can’t instruct multiple dogs to eat from the same bowl. A dog is considered to be fed after eating from a single bowl. However, different dogs may have different food preferences, so they may eat from the same food bowl at different speeds.\n\nAs you know, dogs get impatient if they have to wait for long periods of time. Therefore, Yraglac wants to minimize the total amount of time that dogs spend waiting. More formally, suppose $t$ is the longest amount of time in seconds that any dog spent eating. Yraglac wants to minimize $T=\\sum _{i=1}^{N} t-t_ i$, where $t_ i$ is the time in seconds that the $i^\\textrm {th}$ dog spends eating.\n\n-----Input-----\nThe first line contains integers $N$ and $M$, where $2\\leq N\\leq M\\leq 50$.\n\nEach of the next $N$ lines of the input contains $M$ integers each. The $j^\\textrm {th}$ integers on the $i^\\textrm {th}$ line denotes the $t_{ij}$, the amount of time that the $i^\\textrm {th}$ dog will spend on eating food from the $j^\\textrm {th}$ bowl. Note that $1\\leq t_{ij}\\leq 200$.\n\n-----Output-----\nOutput $T$, the minimum total waiting time in seconds.\n\n-----Examples-----\nSample Input:\n2 3\n2 100 10\n100 1 10\nSample Output:\n0",
        "solutions": "",
        "difficulty": "competition",
        "input": "3 3\n100 20 30\n10 90 80\n99 90 98\n",
        "output": "12\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/dogtrouble"
    },
    {
        "id": 312,
        "task_id": 3480,
        "test_case_id": 1,
        "question": "A round is a musical arrangement in which two or more voices repeat the same melodic line, at times offset from one another. One of the more famous rounds is the nursery rhyme \"Row, Row, Your Boat\", shown here. \n\n \n\nIn western music, an integer count of time units can be assigned to each syllable to indicate how long that syllable is sung. A rest of one or more time units can be treated as a syllable that simply isn’t sung. If you know the time allocated to each syllable, and the time offset at which a second voice begins singing the song, you can determine which words will overlap in the round.\n\nYou are to write a program to display a two-voice round so that syllables that are sung simultaneously in the two voices appear vertically aligned. For each line of the original input, there are to be two lines of output: one for the original line sung by the first voice and one to display syllables (if any) that are started by the second voice during the time period where the first voice is singing the indicated line. Syllables for each voice in the output must be separated by one or more underscores (’_’), with each syllable displayed as far to the left as possible, subject to the following constraints:\n\nConsecutive syllables on a line are separated by at least one ’_’ character.\n\nTwo syllables that begin at the same time in their respective voices are displayed with their leftmost characters in the same column.\n\nConsider syllables S1 and S2, either sung by the same or different voices, that are displayed within the same pair of output lines. If S2 is sung beginning $k$ time units after S1 begins, for $k \\geq 1$, then the first character of S2 must be displayed at least $k$ columns to the right of the first character of S1.\n\nIn some cases there will be a first-voice line when no second-voice syllables are started. Print ’/’ for the second voice instead of an empty line.\n\nIt is possible (in fact, likely), that not all syllables of the second voice will be printed, as only those syllables that start while the first voice is active are to be displayed.\n\n-----Input-----\nThe first line contains two integers, $L$ and $D$, such that $1 \\leq L \\leq 10$ indicates the number of lines in the song and $0 \\leq D \\leq 128$ indicates the delay, in time units, between the time when the first voice signs the first syllable and the time when the second voice begins singing the first syllable.\n\nThe remainder of the input consists of $L$ pairs of lines. The first line in each pair contains the syllables of that line of the song. Adjacent syllables in the input will be separated by a single space The syllables are strings of any non-whitespace characters other than underscores or ’/’. This line contains at most 80 characters.\n\nThe second line in each pair will consist of positive integers, one per syllable from the first line of the pair, indicating the time allocated to the corresponding syllables. Each such integer $t$ will satisfy $1 \\leq t \\leq 128$.\n\n-----Output-----\nFor each dataset, display $2L$ lines corresponding to the two voices in the round, as described above.\n\n-----Examples-----\nSample Input:\n2 16\nHot cross buns! = Hot cross buns! =\n4 4 4 4 4 4 4 4\nOne a pen- ny, Two a pen- ny, Hot cross buns! =\n2 2 2 2 2 2 2 2 4 4 4 4\nSample Output:\nHot_cross_buns!_=___Hot_cross_buns!_=\n____________________Hot_cross_buns!_=\nOne_a_pen-_ny,_Two_a_pen-_ny,_Hot___cross____buns!_=\nHot___cross____buns!_=________One_a_pen-_ny,_Two_a_pen-_ny,",
        "solutions": "",
        "difficulty": "competition",
        "input": "2 16\nHot cross buns! = Hot cross buns! =\n4 4 4 4 4 4 4 4\nOne a pen- ny, Two a pen- ny, Hot cross buns! =\n2 2 2 2 2 2 2 2 4 4 4 4\n",
        "output": "Hot_cross_buns!_=___Hot_cross_buns!_=\n____________________Hot_cross_buns!_=\nOne_a_pen-_ny,_Two_a_pen-_ny,_Hot___cross____buns!_=\nHot___cross____buns!_=________One_a_pen-_ny,_Two_a_pen-_ny,\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/around"
    },
    {
        "id": 313,
        "task_id": 3480,
        "test_case_id": 2,
        "question": "A round is a musical arrangement in which two or more voices repeat the same melodic line, at times offset from one another. One of the more famous rounds is the nursery rhyme \"Row, Row, Your Boat\", shown here. \n\n \n\nIn western music, an integer count of time units can be assigned to each syllable to indicate how long that syllable is sung. A rest of one or more time units can be treated as a syllable that simply isn’t sung. If you know the time allocated to each syllable, and the time offset at which a second voice begins singing the song, you can determine which words will overlap in the round.\n\nYou are to write a program to display a two-voice round so that syllables that are sung simultaneously in the two voices appear vertically aligned. For each line of the original input, there are to be two lines of output: one for the original line sung by the first voice and one to display syllables (if any) that are started by the second voice during the time period where the first voice is singing the indicated line. Syllables for each voice in the output must be separated by one or more underscores (’_’), with each syllable displayed as far to the left as possible, subject to the following constraints:\n\nConsecutive syllables on a line are separated by at least one ’_’ character.\n\nTwo syllables that begin at the same time in their respective voices are displayed with their leftmost characters in the same column.\n\nConsider syllables S1 and S2, either sung by the same or different voices, that are displayed within the same pair of output lines. If S2 is sung beginning $k$ time units after S1 begins, for $k \\geq 1$, then the first character of S2 must be displayed at least $k$ columns to the right of the first character of S1.\n\nIn some cases there will be a first-voice line when no second-voice syllables are started. Print ’/’ for the second voice instead of an empty line.\n\nIt is possible (in fact, likely), that not all syllables of the second voice will be printed, as only those syllables that start while the first voice is active are to be displayed.\n\n-----Input-----\nThe first line contains two integers, $L$ and $D$, such that $1 \\leq L \\leq 10$ indicates the number of lines in the song and $0 \\leq D \\leq 128$ indicates the delay, in time units, between the time when the first voice signs the first syllable and the time when the second voice begins singing the first syllable.\n\nThe remainder of the input consists of $L$ pairs of lines. The first line in each pair contains the syllables of that line of the song. Adjacent syllables in the input will be separated by a single space The syllables are strings of any non-whitespace characters other than underscores or ’/’. This line contains at most 80 characters.\n\nThe second line in each pair will consist of positive integers, one per syllable from the first line of the pair, indicating the time allocated to the corresponding syllables. Each such integer $t$ will satisfy $1 \\leq t \\leq 128$.\n\n-----Output-----\nFor each dataset, display $2L$ lines corresponding to the two voices in the round, as described above.\n\n-----Examples-----\nSample Input:\n2 16\nHot cross buns! = Hot cross buns! =\n4 4 4 4 4 4 4 4\nOne a pen- ny, Two a pen- ny, Hot cross buns! =\n2 2 2 2 2 2 2 2 4 4 4 4\nSample Output:\nHot_cross_buns!_=___Hot_cross_buns!_=\n____________________Hot_cross_buns!_=\nOne_a_pen-_ny,_Two_a_pen-_ny,_Hot___cross____buns!_=\nHot___cross____buns!_=________One_a_pen-_ny,_Two_a_pen-_ny,",
        "solutions": "",
        "difficulty": "competition",
        "input": "5 32\nAnd ev- ry-\n1 1 1\none neath a vine and fig tree, = shall live in\n2 1 1 2 2 2 2 1 1 1 1\npeace and un- a- fraid. =\n2 2 2 2 6 2\nAnd in- to plow shares beat their swords.\n2 1 1 2 2 2 2 4\nNa- tions shall learn war no more =\n2 2 2 2 2 2 1 3\n",
        "output": "And_ev-_ry-\n/\none_neath_a_vine_and_fig_tree,_=_shall_live_in\n/\npeace_and_un-_a-_fraid.___=\n______________________And_ev-_ry-\nAnd_in-___to_plow_shares_beat_their_swords.\none_neath_a__vine_and____fig__tree,_=_shall_live_in\nNa-___tions_shall_learn_war_no_more_=\npeace_and___un-___a-____fraid._______=\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/around"
    },
    {
        "id": 314,
        "task_id": 3556,
        "test_case_id": 1,
        "question": "Several surveys indicate that the taller you are, the higher you can climb the corporate ladder. At TALL Enterprises Inc., this “de facto standard” has been properly formalized: your boss is always at least as tall as you are. Furthermore, you can safely assume that your boss earns a bit more than you do. In fact, you can be absolutely sure that your immediate boss is the person who earns the least among all the employees that earn more than you and are at least as tall as you are. Furthermore, if you are the immediate boss of someone, that person is your subordinate, and all of his or her subordinates are your subordinates as well. If you are nobody’s boss, then you have no subordinates. As simple as these rules are, many people working for TALL are unsure of to whom they should be turning in their weekly progress report and how many subordinates they have. Write a program that will help in determining for any employee who the immediate boss of that employee is and how many subordinates they have. Quality Assurance at TALL have devised a series of tests to ensure that your program is correct. These test are described below.\n\n-----Input-----\nOn the first line of input are two positive integers $m$ and $q$, where $m$ (at most $100000$) is the number of employees and $q$ (at most $10000$) is the number of queries. The following $m$ lines each list an employee by three integers on the same line: employee ID number (six decimal digits, the first one of which is not zero), yearly salary in Euros and finally height in $\\mu $m ($1 \\mu \\text {m} = 10^{-6}$ meters – accuracy is important at TALL). The chairperson is the employee that earns more than anyone else and is also the tallest person in the company. Then there are $q$ lines listing queries. Each query is a single legal employee ID.\n\nThe salary is a positive integer which is at most $10000000$. No two employees have the same ID, and no two employees have the same salary. The height of an employee is at least $1000000$ $\\mu $m and at most $2500000$ $\\mu $m.\n\n-----Output-----\nFor each employee ID $x$ in a query output a single line with two integers $y$ and $k$, where $y$ is the ID of $x$’s boss, and $k$ is the number of subordinates of $x$. If the query is the ID of the chairperson, then you should output $0$ as the ID of his or her boss (since the chairperson has no immediate boss except, possibly, God).\n\n-----Examples-----\nSample Input:\n3 3\n123456 14323 1700000\n123458 41412 1900000\n123457 15221 1800000\n123456\n123458\n123457\nSample Output:\n123457 0\n0 2\n123458 1",
        "solutions": "",
        "difficulty": "competition",
        "input": "3 3\n123456 14323 1700000\n123458 41412 1900000\n123457 15221 1800000\n123456\n123458\n123457\n",
        "output": "123457 0\n0 2\n123458 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/whostheboss"
    },
    {
        "id": 315,
        "task_id": 3556,
        "test_case_id": 2,
        "question": "Several surveys indicate that the taller you are, the higher you can climb the corporate ladder. At TALL Enterprises Inc., this “de facto standard” has been properly formalized: your boss is always at least as tall as you are. Furthermore, you can safely assume that your boss earns a bit more than you do. In fact, you can be absolutely sure that your immediate boss is the person who earns the least among all the employees that earn more than you and are at least as tall as you are. Furthermore, if you are the immediate boss of someone, that person is your subordinate, and all of his or her subordinates are your subordinates as well. If you are nobody’s boss, then you have no subordinates. As simple as these rules are, many people working for TALL are unsure of to whom they should be turning in their weekly progress report and how many subordinates they have. Write a program that will help in determining for any employee who the immediate boss of that employee is and how many subordinates they have. Quality Assurance at TALL have devised a series of tests to ensure that your program is correct. These test are described below.\n\n-----Input-----\nOn the first line of input are two positive integers $m$ and $q$, where $m$ (at most $100000$) is the number of employees and $q$ (at most $10000$) is the number of queries. The following $m$ lines each list an employee by three integers on the same line: employee ID number (six decimal digits, the first one of which is not zero), yearly salary in Euros and finally height in $\\mu $m ($1 \\mu \\text {m} = 10^{-6}$ meters – accuracy is important at TALL). The chairperson is the employee that earns more than anyone else and is also the tallest person in the company. Then there are $q$ lines listing queries. Each query is a single legal employee ID.\n\nThe salary is a positive integer which is at most $10000000$. No two employees have the same ID, and no two employees have the same salary. The height of an employee is at least $1000000$ $\\mu $m and at most $2500000$ $\\mu $m.\n\n-----Output-----\nFor each employee ID $x$ in a query output a single line with two integers $y$ and $k$, where $y$ is the ID of $x$’s boss, and $k$ is the number of subordinates of $x$. If the query is the ID of the chairperson, then you should output $0$ as the ID of his or her boss (since the chairperson has no immediate boss except, possibly, God).\n\n-----Examples-----\nSample Input:\n3 3\n123456 14323 1700000\n123458 41412 1900000\n123457 15221 1800000\n123456\n123458\n123457\nSample Output:\n123457 0\n0 2\n123458 1",
        "solutions": "",
        "difficulty": "competition",
        "input": "4 4\n200002 12234 1832001\n200003 15002 1745201\n200004 18745 1883410\n200001 24834 1921313\n200004\n200002\n200003\n200001\n",
        "output": "200001 2\n200004 0\n200004 0\n0 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/whostheboss"
    },
    {
        "id": 316,
        "task_id": 3570,
        "test_case_id": 1,
        "question": "A common plot device in story-telling is the “All Just A Dream” trope. Typical symptoms of this trope being used are talking lions, main characters dying, yodeling aliens on monocycles, and a general plethora of weird events. Then, of course, someone wakes up and it is revealed that everything that happened during the entire season did in fact not happen at all. It was All Just A Dream (or some kind of hallucination), and the days of our lives spent watching all those episodes are lost forever. In order to cause further confusion and uncertainty, this can also be done in layers, with characters having dreams within dreams within dreams, and so on. \n\nWhen the All Just A Dream trick is taken too far and gets used too often, it can get difficult to keep track of what has actually happened. This is where you enter the picture. You will be given a list of events, dreams, and scenarios. Each scenario specifies some events that have happened and some others that have not happened. Your job is to determine for each scenario whether that scenario is possible (possibly using the All Just A Dream trick).\n\n-----Input-----\nThe first line of input consists of an integer $0 \\le n \\le 50000$, the number of events, dreams and scenarios. Then follow $n$ lines, giving the events, dreams, and scenarios in chronological order. Each line is in one of the following forms:\n - An event line is of the form “E $e$”, indicating that event $e$ happens (see below for format of $e$).\n - A dream line is of the form “D $r$”, indicating that the last $r$ events that happened were All Just A Dream. Note that these events are now considered to not have happened, so they should not be counted when processing subsequent D lines.\n - A scenario line is of the form “S $k$ $e_1$ $\\ldots $ $e_ k$”, where $1 \\le k \\le 30$ is an integer giving the number of events and $e_1, \\ldots , e_ k$ is the list of events of the scenario. In a scenario, each event may be prefixed with a ‘!’, indicating that the event did not happen in this scenario.\n\nEvents are strings containing at most $20$ characters and using only the characters ‘a’-‘z’ and underscores (‘_’). For ‘D’ lines, you can assume that $r$ is an integer between $1$ and $R$, where $R$ is the total number of events that have happened so far (and that have not turned out to be a dream). For ‘E’ lines, you can assume that $e$ is not an event that has already happened, except if the previous occurence of the event turned out to be a dream, in which case it can happen again.Warning\n\nThis problem has somewhat large amounts of input and output. We recommend you to make sure that your input and output are properly buffered in order to make the most of the few seconds of execution time that we give you.\n\n-----Output-----\nFor each scenario in the input, output a line as follows:\n - “Yes” if the given scenario is consistent with what has happened so far.\n - “$r$ Just A Dream” if the given scenario would be consistent with what has happened so far, provided a “D $r$” line had occurred just before the scenario. If there are many possible values of $r$, choose the smallest value. Note that you should not consider this hypothetical “D $r$” line to have occurred (as illustrated by sample input 2 below).\n - “Plot Error” otherwise.\n\n-----Examples-----\nSample Input:\n10\nE business_as_usual\nE bobby_dies\nS 1 bobby_died\nE stuff_happens\nE jr_does_bad_things\nS 2 !bobby_dies business_as_usual\nE it_goes_on_and_on\nD 4\nS 1 !bobby_dies\nS 2 !bobby_dies it_goes_on_and_on\nSample Output:\nPlot Error\n3 Just A Dream\nYes\nPlot Error",
        "solutions": "",
        "difficulty": "competition",
        "input": "10\nE business_as_usual\nE bobby_dies\nS 1 bobby_died\nE stuff_happens\nE jr_does_bad_things\nS 2 !bobby_dies business_as_usual\nE it_goes_on_and_on\nD 4\nS 1 !bobby_dies\nS 2 !bobby_dies it_goes_on_and_on\n",
        "output": "Plot Error\n3 Just A Dream\nYes\nPlot Error\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/dream"
    },
    {
        "id": 317,
        "task_id": 3570,
        "test_case_id": 2,
        "question": "A common plot device in story-telling is the “All Just A Dream” trope. Typical symptoms of this trope being used are talking lions, main characters dying, yodeling aliens on monocycles, and a general plethora of weird events. Then, of course, someone wakes up and it is revealed that everything that happened during the entire season did in fact not happen at all. It was All Just A Dream (or some kind of hallucination), and the days of our lives spent watching all those episodes are lost forever. In order to cause further confusion and uncertainty, this can also be done in layers, with characters having dreams within dreams within dreams, and so on. \n\nWhen the All Just A Dream trick is taken too far and gets used too often, it can get difficult to keep track of what has actually happened. This is where you enter the picture. You will be given a list of events, dreams, and scenarios. Each scenario specifies some events that have happened and some others that have not happened. Your job is to determine for each scenario whether that scenario is possible (possibly using the All Just A Dream trick).\n\n-----Input-----\nThe first line of input consists of an integer $0 \\le n \\le 50000$, the number of events, dreams and scenarios. Then follow $n$ lines, giving the events, dreams, and scenarios in chronological order. Each line is in one of the following forms:\n - An event line is of the form “E $e$”, indicating that event $e$ happens (see below for format of $e$).\n - A dream line is of the form “D $r$”, indicating that the last $r$ events that happened were All Just A Dream. Note that these events are now considered to not have happened, so they should not be counted when processing subsequent D lines.\n - A scenario line is of the form “S $k$ $e_1$ $\\ldots $ $e_ k$”, where $1 \\le k \\le 30$ is an integer giving the number of events and $e_1, \\ldots , e_ k$ is the list of events of the scenario. In a scenario, each event may be prefixed with a ‘!’, indicating that the event did not happen in this scenario.\n\nEvents are strings containing at most $20$ characters and using only the characters ‘a’-‘z’ and underscores (‘_’). For ‘D’ lines, you can assume that $r$ is an integer between $1$ and $R$, where $R$ is the total number of events that have happened so far (and that have not turned out to be a dream). For ‘E’ lines, you can assume that $e$ is not an event that has already happened, except if the previous occurence of the event turned out to be a dream, in which case it can happen again.Warning\n\nThis problem has somewhat large amounts of input and output. We recommend you to make sure that your input and output are properly buffered in order to make the most of the few seconds of execution time that we give you.\n\n-----Output-----\nFor each scenario in the input, output a line as follows:\n - “Yes” if the given scenario is consistent with what has happened so far.\n - “$r$ Just A Dream” if the given scenario would be consistent with what has happened so far, provided a “D $r$” line had occurred just before the scenario. If there are many possible values of $r$, choose the smallest value. Note that you should not consider this hypothetical “D $r$” line to have occurred (as illustrated by sample input 2 below).\n - “Plot Error” otherwise.\n\n-----Examples-----\nSample Input:\n10\nE business_as_usual\nE bobby_dies\nS 1 bobby_died\nE stuff_happens\nE jr_does_bad_things\nS 2 !bobby_dies business_as_usual\nE it_goes_on_and_on\nD 4\nS 1 !bobby_dies\nS 2 !bobby_dies it_goes_on_and_on\nSample Output:\nPlot Error\n3 Just A Dream\nYes\nPlot Error",
        "solutions": "",
        "difficulty": "competition",
        "input": "11\nS 1 !something\nE one\nE two\nE three\nE four\nE five\nS 3 three !four one\nD 1\nS 3 three !four one\nD 1\nS 3 three !four one\n",
        "output": "Yes\n2 Just A Dream\n1 Just A Dream\nYes\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/dream"
    },
    {
        "id": 318,
        "task_id": 3596,
        "test_case_id": 1,
        "question": "Arnar is playing his favourite video game Association of Myths. He’s playing a character by the name of Lumen. Lumen has a special ability where she can shoot a laser beam across the entire map killing all enemies it hits. The only drawback is that the beam is rather narrow and Arnar isn’t always sure where the enemies are located. He now needs your help to determine where the enemies are hiding so he can hit them (and hopefully win the game).\n\nArnar is very serious about this game and is therefore willing to meticulously research how the opponents move. He knows they only travel in a straight path called the ’medial path’. We can therefore describe their location as a function $f$ of a single variable. After much thought Arnar has concluded that $f$ can be described by\\[ c\\int _ a^b \\left(t_1\\Gamma (x) + \\sqrt [t_2]{\\log (\\operatorname {erf}(t_3 x))} - J_ k(x)^{t_4}\\right)dx \\]\n\nwhere $\\log $ is the natural logarithm,\\[ \\Gamma (z) = \\int _0^{\\infty } x^{z - 1} e^{-x} dx, \\]\\[ \\operatorname {erf}(x) = \\frac{2}{\\sqrt {\\pi }} \\int _0^x e^{-t^2} dt, \\]\n\nand\\[ J_ k(x) = \\frac{1}{\\pi } \\int _0^{\\pi } \\cos (k \\tau - x \\sin \\tau ) d \\tau . \\]\n\nArnar thinks that it maybe a bit tough for you to compute $f$ as previously described so he tells you it is enough to calculate the $r$-th degree Taylor polynomial around $0$, i.e.\\[ P(x) = \\sum _{i = 0}^r \\frac{f^{(i)}(0)}{i!}x^i. \\]\n\nArnar is a afraid he was too aggressive when approximating $f$ with $P$ so he would like to modify $P$ a little bit further. He knows that as the game progresses his opponent will have more in-game currency and will therefore buy better, more agile shoes. To account for this Arnar recursively defines a sequence of polynomials by\\[ P_0(x) = P(x), \\quad P_ n(x) = \\sum _{i = 0}^{r + n} P_{n - 1}(i) x^i. \\]\n\nArnar finally notes that during the endgame phase the opponent will probably have to go to the bathroom and to account for that he wants reduce the size of the final polynomial in his sequence. He therefore takes $P_ s$ and differentiates it $\\operatorname {deg}(P_ s) + 1$ times and calls the outcome $g$. Arnar is now satisfied that\\[ \\frac{(g(n) + l)^2}{\\pi e} + \\frac{1}{l + 1} \\]\n\ngives the location of his opponent. Why does Arnar take the last step? Well, Arnar won’t tell you because he’s afraid you will be able to beat him in Association of Myths if he tells you all his secrets. He also requests that you give him the answer to at least two correct decimal places since his mouse can track changes as small as a hundredth of an in-game unit of length.\n\n-----Input-----\nThe first line of the input starts with three real numbers $a, b, c$, $-10^9 \\leq a \\leq b \\leq 10^9$ and $1 \\leq c \\leq 10^9$. These numbers will have at most 6 digits after the decimal point. The second line has four integers $t_1, t_2, t_3, t_4$, $1 \\leq t_ i \\leq 10^9$. The third line has five integers $n, k, r, s, l$, $1 \\leq n, k, r, s, l \\leq 10^3$.\n\n-----Output-----\nThe output should consist of one line containing the location of Arnar’s opponent as described above.\n\n-----Examples-----\nSample Input:\n-99.99 99.99 9999.99\n99 9 999 9999\n9 99 9 99 9\nSample Output:\n9.585073",
        "solutions": "",
        "difficulty": "competition",
        "input": "-99.99 99.99 9999.99\n99 9 999 9999\n9 99 9 99 9\n",
        "output": "9.585073\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/associationofmyths"
    },
    {
        "id": 319,
        "task_id": 3596,
        "test_case_id": 2,
        "question": "Arnar is playing his favourite video game Association of Myths. He’s playing a character by the name of Lumen. Lumen has a special ability where she can shoot a laser beam across the entire map killing all enemies it hits. The only drawback is that the beam is rather narrow and Arnar isn’t always sure where the enemies are located. He now needs your help to determine where the enemies are hiding so he can hit them (and hopefully win the game).\n\nArnar is very serious about this game and is therefore willing to meticulously research how the opponents move. He knows they only travel in a straight path called the ’medial path’. We can therefore describe their location as a function $f$ of a single variable. After much thought Arnar has concluded that $f$ can be described by\\[ c\\int _ a^b \\left(t_1\\Gamma (x) + \\sqrt [t_2]{\\log (\\operatorname {erf}(t_3 x))} - J_ k(x)^{t_4}\\right)dx \\]\n\nwhere $\\log $ is the natural logarithm,\\[ \\Gamma (z) = \\int _0^{\\infty } x^{z - 1} e^{-x} dx, \\]\\[ \\operatorname {erf}(x) = \\frac{2}{\\sqrt {\\pi }} \\int _0^x e^{-t^2} dt, \\]\n\nand\\[ J_ k(x) = \\frac{1}{\\pi } \\int _0^{\\pi } \\cos (k \\tau - x \\sin \\tau ) d \\tau . \\]\n\nArnar thinks that it maybe a bit tough for you to compute $f$ as previously described so he tells you it is enough to calculate the $r$-th degree Taylor polynomial around $0$, i.e.\\[ P(x) = \\sum _{i = 0}^r \\frac{f^{(i)}(0)}{i!}x^i. \\]\n\nArnar is a afraid he was too aggressive when approximating $f$ with $P$ so he would like to modify $P$ a little bit further. He knows that as the game progresses his opponent will have more in-game currency and will therefore buy better, more agile shoes. To account for this Arnar recursively defines a sequence of polynomials by\\[ P_0(x) = P(x), \\quad P_ n(x) = \\sum _{i = 0}^{r + n} P_{n - 1}(i) x^i. \\]\n\nArnar finally notes that during the endgame phase the opponent will probably have to go to the bathroom and to account for that he wants reduce the size of the final polynomial in his sequence. He therefore takes $P_ s$ and differentiates it $\\operatorname {deg}(P_ s) + 1$ times and calls the outcome $g$. Arnar is now satisfied that\\[ \\frac{(g(n) + l)^2}{\\pi e} + \\frac{1}{l + 1} \\]\n\ngives the location of his opponent. Why does Arnar take the last step? Well, Arnar won’t tell you because he’s afraid you will be able to beat him in Association of Myths if he tells you all his secrets. He also requests that you give him the answer to at least two correct decimal places since his mouse can track changes as small as a hundredth of an in-game unit of length.\n\n-----Input-----\nThe first line of the input starts with three real numbers $a, b, c$, $-10^9 \\leq a \\leq b \\leq 10^9$ and $1 \\leq c \\leq 10^9$. These numbers will have at most 6 digits after the decimal point. The second line has four integers $t_1, t_2, t_3, t_4$, $1 \\leq t_ i \\leq 10^9$. The third line has five integers $n, k, r, s, l$, $1 \\leq n, k, r, s, l \\leq 10^3$.\n\n-----Output-----\nThe output should consist of one line containing the location of Arnar’s opponent as described above.\n\n-----Examples-----\nSample Input:\n-99.99 99.99 9999.99\n99 9 999 9999\n9 99 9 99 9\nSample Output:\n9.585073",
        "solutions": "",
        "difficulty": "competition",
        "input": "-17.56 55.81 1000.7\n9 17 16 1000001\n17 2 14 3 9\n",
        "output": "9.585073\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/associationofmyths"
    },
    {
        "id": 320,
        "task_id": 3620,
        "test_case_id": 1,
        "question": "You are given a simple graph with $N$ nodes and $M$ edges. The graph has the special property that any connected component of size $s$ contains no more than $s + 2$ edges. You are also given two integers $k$ and $P$. Find the number of $k$-colourings of the graph, modulo $P$.\n\nRecall that a simple graph is an undirected graph with no self loops and no repeated edges. A $k$-colouring of a graph is a way to assign to each node of the graph exactly one of $k$ colours, such that if edge $(u, v)$ is present in the graph, then $u$ and $v$ receive different colors.\n\n-----Input-----\nThe first line of input consists of four integers, $N, M, k$, and $P$ ($1 \\leq N \\leq 50000$, $0 \\leq M \\leq 1.5 N$, $1 \\leq k \\leq 10^9$, $1 \\leq P \\leq 2 \\cdot 10^9$). The next $M$ lines of input each contains a pair of integers $A$ and $B$ ($1 \\leq A \\leq N$, $1 \\leq B \\leq N$), describing an edge in the graph connecting nodes $A$ and $B$.\n\n-----Output-----\nOutput the number of $k$-colourings of the given graph, modulo $P$.\n\n-----Examples-----\nSample Input:\n3 3 2 10000\n1 2\n2 3\n3 1\nSample Output:\n0",
        "solutions": "",
        "difficulty": "competition",
        "input": "3 3 2 10000\n1 2\n2 3\n3 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/kcolouring"
    },
    {
        "id": 321,
        "task_id": 3620,
        "test_case_id": 2,
        "question": "You are given a simple graph with $N$ nodes and $M$ edges. The graph has the special property that any connected component of size $s$ contains no more than $s + 2$ edges. You are also given two integers $k$ and $P$. Find the number of $k$-colourings of the graph, modulo $P$.\n\nRecall that a simple graph is an undirected graph with no self loops and no repeated edges. A $k$-colouring of a graph is a way to assign to each node of the graph exactly one of $k$ colours, such that if edge $(u, v)$ is present in the graph, then $u$ and $v$ receive different colors.\n\n-----Input-----\nThe first line of input consists of four integers, $N, M, k$, and $P$ ($1 \\leq N \\leq 50000$, $0 \\leq M \\leq 1.5 N$, $1 \\leq k \\leq 10^9$, $1 \\leq P \\leq 2 \\cdot 10^9$). The next $M$ lines of input each contains a pair of integers $A$ and $B$ ($1 \\leq A \\leq N$, $1 \\leq B \\leq N$), describing an edge in the graph connecting nodes $A$ and $B$.\n\n-----Output-----\nOutput the number of $k$-colourings of the given graph, modulo $P$.\n\n-----Examples-----\nSample Input:\n3 3 2 10000\n1 2\n2 3\n3 1\nSample Output:\n0",
        "solutions": "",
        "difficulty": "competition",
        "input": "3 3 4 13\n1 2\n2 3\n3 1\n",
        "output": "11\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/kcolouring"
    },
    {
        "id": 322,
        "task_id": 3692,
        "test_case_id": 1,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 1\n2 0 1\n4 0 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 323,
        "task_id": 3692,
        "test_case_id": 2,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 2\n3 0 2\n6 0 2\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 324,
        "task_id": 3692,
        "test_case_id": 3,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 2\n2 0 2\n1 1 2\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 325,
        "task_id": 3692,
        "test_case_id": 5,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "2\n-10 10 1\n10 -10 1\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 326,
        "task_id": 3692,
        "test_case_id": 6,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "2\n-6 6 9\n3 -6 6\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 327,
        "task_id": 3692,
        "test_case_id": 7,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "2\n-10 -10 10\n10 10 10\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 328,
        "task_id": 3692,
        "test_case_id": 8,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 1 5\n-7 7 10\n-3 -4 8\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 329,
        "task_id": 3692,
        "test_case_id": 9,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-2 8 10\n3 -2 5\n3 1 3\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 330,
        "task_id": 3692,
        "test_case_id": 10,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 2\n0 0 4\n3 0 2\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 331,
        "task_id": 3692,
        "test_case_id": 11,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n8 5 7\n7 3 7\n5 2 5\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 332,
        "task_id": 3692,
        "test_case_id": 12,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 5 7\n1 -2 7\n7 9 7\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 333,
        "task_id": 3692,
        "test_case_id": 13,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 -7 10\n-7 9 10\n-2 -1 4\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 334,
        "task_id": 3692,
        "test_case_id": 14,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-2 -3 5\n-6 1 7\n5 4 5\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 335,
        "task_id": 3692,
        "test_case_id": 15,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n3 -2 7\n-1 2 5\n-4 1 3\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 336,
        "task_id": 3692,
        "test_case_id": 16,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n4 5 10\n1 -1 5\n-1 -5 5\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 337,
        "task_id": 3692,
        "test_case_id": 17,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 0 5\n-2 1 5\n-5 4 7\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 338,
        "task_id": 3692,
        "test_case_id": 18,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 3 5\n1 -1 7\n2 5 10\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 339,
        "task_id": 3692,
        "test_case_id": 19,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 4 3\n5 6 4\n1 -5 9\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 340,
        "task_id": 3692,
        "test_case_id": 20,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 4 4\n2 4 2\n-1 0 6\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 341,
        "task_id": 3692,
        "test_case_id": 21,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-10 4 10\n10 4 10\n0 -7 10\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 342,
        "task_id": 3692,
        "test_case_id": 22,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 -5 3\n-3 -4 1\n-6 0 9\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 343,
        "task_id": 3692,
        "test_case_id": 23,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n4 0 1\n-1 1 9\n0 3 6\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 344,
        "task_id": 3692,
        "test_case_id": 24,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 -2 3\n-4 -6 3\n-6 -4 9\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 345,
        "task_id": 3692,
        "test_case_id": 25,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 6 4\n-1 4 7\n0 2 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 346,
        "task_id": 3692,
        "test_case_id": 26,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 -1 2\n-6 -3 10\n-1 3 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 347,
        "task_id": 3692,
        "test_case_id": 27,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-2 -5 4\n-5 -1 5\n-6 -2 9\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 348,
        "task_id": 3692,
        "test_case_id": 28,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n5 -2 3\n1 1 2\n4 -3 7\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 349,
        "task_id": 3692,
        "test_case_id": 29,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 -6 3\n-2 0 1\n1 -4 6\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 350,
        "task_id": 3692,
        "test_case_id": 30,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 -2 3\n-5 -4 4\n-6 -5 8\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 351,
        "task_id": 3692,
        "test_case_id": 31,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 3 4\n-2 0 8\n3 6 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 352,
        "task_id": 3692,
        "test_case_id": 32,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 -1 2\n-6 -5 10\n1 3 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 353,
        "task_id": 3692,
        "test_case_id": 33,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 2 1\n0 -6 9\n-5 -3 2\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 354,
        "task_id": 3692,
        "test_case_id": 34,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 -5 4\n6 5 2\n-6 -6 7\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 355,
        "task_id": 3692,
        "test_case_id": 35,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 -2 3\n-1 1 8\n-4 -3 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 356,
        "task_id": 3692,
        "test_case_id": 36,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 -1 8\n0 3 3\n2 2 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 357,
        "task_id": 3692,
        "test_case_id": 37,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n3 4 9\n2 -3 1\n-1 1 4\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 358,
        "task_id": 3692,
        "test_case_id": 38,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 -6 5\n-2 -2 10\n-3 4 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 359,
        "task_id": 3692,
        "test_case_id": 39,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 6 5\n1 -1 5\n-2 3 10\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 360,
        "task_id": 3692,
        "test_case_id": 40,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n3 -5 5\n-1 -2 10\n-5 1 5\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 361,
        "task_id": 3692,
        "test_case_id": 41,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 6\n-4 -3 1\n-3 4 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 362,
        "task_id": 3692,
        "test_case_id": 42,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 -2 10\n3 -1 3\n-1 1 5\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 363,
        "task_id": 3692,
        "test_case_id": 43,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 -1 10\n-5 2 5\n1 -6 5\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 364,
        "task_id": 3692,
        "test_case_id": 44,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 1 1\n-2 -6 7\n-6 -3 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 365,
        "task_id": 3692,
        "test_case_id": 45,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n3 -4 2\n-1 -1 3\n-5 2 8\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 366,
        "task_id": 3692,
        "test_case_id": 46,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n6 -1 1\n1 1 4\n-2 5 9\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 367,
        "task_id": 3692,
        "test_case_id": 47,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 -6 1\n-6 5 8\n-2 2 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 368,
        "task_id": 3692,
        "test_case_id": 48,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -6 8\n-4 -5 1\n-1 -4 6\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 369,
        "task_id": 3692,
        "test_case_id": 49,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 -5 7\n2 -3 6\n-2 0 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 370,
        "task_id": 3692,
        "test_case_id": 50,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 -5 1\n4 -3 3\n-6 -6 10\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 371,
        "task_id": 3692,
        "test_case_id": 51,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 -1 4\n-1 -5 1\n-5 0 9\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 372,
        "task_id": 3692,
        "test_case_id": 52,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -6 9\n4 -3 4\n-3 -1 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 373,
        "task_id": 3692,
        "test_case_id": 53,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 -2 7\n-6 -1 7\n-3 -5 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 374,
        "task_id": 3692,
        "test_case_id": 54,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 -2 8\n6 -5 3\n3 -1 8\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 375,
        "task_id": 3692,
        "test_case_id": 55,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 1 4\n-1 6 9\n-6 5 9\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 376,
        "task_id": 3692,
        "test_case_id": 56,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 -1 5\n-1 3 10\n4 5 5\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 377,
        "task_id": 3692,
        "test_case_id": 57,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-2 2 3\n0 -6 3\n-6 -1 8\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 378,
        "task_id": 3692,
        "test_case_id": 58,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 -3 9\n0 -2 7\n-6 -6 10\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 379,
        "task_id": 3692,
        "test_case_id": 59,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 -6 8\n-2 -1 7\n1 -5 2\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 380,
        "task_id": 3692,
        "test_case_id": 60,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 3 4\n1 4 4\n-6 -6 10\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 381,
        "task_id": 3692,
        "test_case_id": 61,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n6 2 6\n-6 5 7\n-2 -4 4\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 382,
        "task_id": 3692,
        "test_case_id": 62,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n5 2 4\n-3 6 4\n-6 -6 10\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 383,
        "task_id": 3692,
        "test_case_id": 63,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n5 -5 1\n-3 1 9\n2 -6 6\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 384,
        "task_id": 3692,
        "test_case_id": 64,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 6 4\n4 2 9\n-4 -6 9\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 385,
        "task_id": 3692,
        "test_case_id": 65,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -4 9\n0 4 1\n-1 3 1\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 386,
        "task_id": 3692,
        "test_case_id": 66,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 -6 4\n1 -3 1\n-2 1 4\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 387,
        "task_id": 3692,
        "test_case_id": 67,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 0 6\n-3 -6 6\n4 6 4\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 388,
        "task_id": 3692,
        "test_case_id": 68,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n6 -5 1\n3 1 9\n-6 -6 9\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 389,
        "task_id": 3692,
        "test_case_id": 69,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 -6 7\n-6 0 6\n-2 3 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 390,
        "task_id": 3692,
        "test_case_id": 70,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -6 9\n6 -5 3\n-5 -1 9\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 391,
        "task_id": 3692,
        "test_case_id": 71,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 -5 2\n-5 -6 3\n-2 -2 3\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 392,
        "task_id": 3692,
        "test_case_id": 72,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -6 9\n6 -4 1\n-3 -2 8\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 393,
        "task_id": 3692,
        "test_case_id": 73,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -2 1\n-3 -1 1\n-2 1 4\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 394,
        "task_id": 3692,
        "test_case_id": 74,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n5 -2 6\n-1 6 4\n2 2 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 395,
        "task_id": 3692,
        "test_case_id": 75,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 1 2\n-6 -1 6\n6 4 7\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 396,
        "task_id": 3692,
        "test_case_id": 76,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 4 4\n-6 -4 6\n-4 -2 4\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 397,
        "task_id": 3692,
        "test_case_id": 77,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n5 -6 6\n-3 0 4\n-4 6 9\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 398,
        "task_id": 3692,
        "test_case_id": 78,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 4 4\n3 -6 4\n-4 -4 6\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 399,
        "task_id": 3692,
        "test_case_id": 79,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n6 -3 6\n2 0 1\n-6 6 9\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 400,
        "task_id": 3692,
        "test_case_id": 80,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 6 9\n6 1 4\n2 0 1\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 401,
        "task_id": 3692,
        "test_case_id": 81,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 -5 2\n-6 3 2\n-3 -1 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 402,
        "task_id": 3692,
        "test_case_id": 82,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n5 -4 1\n3 -5 5\n-3 3 5\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 403,
        "task_id": 3692,
        "test_case_id": 83,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 3 1\n2 -6 7\n-3 6 6\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 404,
        "task_id": 3692,
        "test_case_id": 84,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 -4 2\n-6 -2 2\n0 0 3\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 405,
        "task_id": 3692,
        "test_case_id": 85,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -2 7\n5 0 2\n2 4 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 406,
        "task_id": 3692,
        "test_case_id": 86,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 6 4\n-2 3 1\n-1 -3 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 407,
        "task_id": 3692,
        "test_case_id": 87,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 -5 2\n-6 -6 9\n4 4 5\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 408,
        "task_id": 3692,
        "test_case_id": 88,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 3 6\n4 -3 2\n-2 -1 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 409,
        "task_id": 3692,
        "test_case_id": 89,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 5 6\n-3 -4 5\n-6 -6 6\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 410,
        "task_id": 3692,
        "test_case_id": 90,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-2 -5 3\n1 -1 2\n-3 4 6\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 411,
        "task_id": 3692,
        "test_case_id": 91,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -6 7\n1 4 2\n0 -5 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 412,
        "task_id": 3692,
        "test_case_id": 92,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 3 5\n5 -2 6\n-3 4 4\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 413,
        "task_id": 3692,
        "test_case_id": 93,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-2 0 2\n1 4 3\n-6 3 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 414,
        "task_id": 3692,
        "test_case_id": 94,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 3 4\n0 0 1\n-5 -4 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 415,
        "task_id": 3692,
        "test_case_id": 95,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 5 4\n-6 -6 7\n1 6 6\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 416,
        "task_id": 3692,
        "test_case_id": 96,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 -6 8\n5 6 8\n2 2 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 417,
        "task_id": 3692,
        "test_case_id": 97,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n6 1 2\n-6 -6 7\n5 -1 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 418,
        "task_id": 3692,
        "test_case_id": 98,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 6 4\n-3 -6 5\n4 2 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 419,
        "task_id": 3692,
        "test_case_id": 99,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-5 5 4\n2 3 3\n-6 -6 7\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 420,
        "task_id": 3692,
        "test_case_id": 100,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-6 5 2\n-6 -1 4\n2 5 6\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 421,
        "task_id": 3692,
        "test_case_id": 101,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 -2 5\n2 0 3\n2 -1 4\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 422,
        "task_id": 3692,
        "test_case_id": 102,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n4 -3 8\n3 -3 7\n-3 -3 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 423,
        "task_id": 3692,
        "test_case_id": 103,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 0 2\n4 0 4\n0 -4 4\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 424,
        "task_id": 3692,
        "test_case_id": 104,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-1 0 5\n5 0 5\n5 8 5\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 425,
        "task_id": 3692,
        "test_case_id": 105,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 0 1\n-1 0 1\n0 1 1\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 426,
        "task_id": 3692,
        "test_case_id": 106,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 0 2\n4 0 4\n0 -4 5\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 427,
        "task_id": 3692,
        "test_case_id": 107,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 0 2\n4 0 4\n0 -4 3\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 428,
        "task_id": 3692,
        "test_case_id": 108,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 0 2\n4 0 4\n0 -4 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 429,
        "task_id": 3692,
        "test_case_id": 109,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 0 2\n4 0 4\n0 -4 8\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 430,
        "task_id": 3692,
        "test_case_id": 110,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-9 0 9\n-9 10 10\n9 4 10\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 431,
        "task_id": 3692,
        "test_case_id": 111,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-9 10 10\n9 4 10\n0 -2 6\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 432,
        "task_id": 3692,
        "test_case_id": 112,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n9 5 10\n8 -2 9\n-9 -1 9\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 433,
        "task_id": 3692,
        "test_case_id": 113,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-4 -2 9\n8 4 9\n-10 10 10\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 434,
        "task_id": 3692,
        "test_case_id": 114,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n1 8 2\n3 8 1\n3 -2 9\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 435,
        "task_id": 3692,
        "test_case_id": 115,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 1\n0 3 2\n4 0 3\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 436,
        "task_id": 3692,
        "test_case_id": 116,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-3 0 5\n3 0 5\n0 0 4\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 437,
        "task_id": 3692,
        "test_case_id": 117,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n4 1 5\n-4 1 5\n0 0 4\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 438,
        "task_id": 3692,
        "test_case_id": 118,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 1\n0 1 1\n0 2 1\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 439,
        "task_id": 3692,
        "test_case_id": 119,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 5\n1 7 5\n7 7 5\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 440,
        "task_id": 3692,
        "test_case_id": 120,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "2\n0 0 2\n3 0 2\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 441,
        "task_id": 3692,
        "test_case_id": 121,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 2\n1 0 1\n-1 0 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 442,
        "task_id": 3692,
        "test_case_id": 122,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-2 0 2\n2 0 2\n0 0 4\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 443,
        "task_id": 3692,
        "test_case_id": 123,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n3 4 5\n-3 4 5\n0 -5 5\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 444,
        "task_id": 3692,
        "test_case_id": 124,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 1\n1 0 1\n2 0 1\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 445,
        "task_id": 3692,
        "test_case_id": 125,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n2 2 4\n8 2 4\n5 10 5\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 446,
        "task_id": 3692,
        "test_case_id": 126,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 5\n4 0 3\n8 0 5\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 447,
        "task_id": 3692,
        "test_case_id": 127,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 1\n2 0 3\n-2 0 3\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 448,
        "task_id": 3692,
        "test_case_id": 128,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 1\n2 0 1\n1 0 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 449,
        "task_id": 3692,
        "test_case_id": 129,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0 5\n8 0 5\n4 0 3\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 450,
        "task_id": 3692,
        "test_case_id": 130,
        "question": "Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement.\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n\n-----Input-----\n\nThe first line of input contains one integer n (1 ≤ n ≤ 3), denoting the number of circles.\n\nThe following n lines each contains three space-separated integers x, y and r ( - 10 ≤ x, y ≤ 10, 1 ≤ r ≤ 10), describing a circle whose center is (x, y) and the radius is r. No two circles have the same x, y and r at the same time.\n\n\n-----Output-----\n\nPrint a single integer — the number of regions on the plane.\n\n\n-----Examples-----\nInput\n3\n0 0 1\n2 0 1\n4 0 1\n\nOutput\n4\n\nInput\n3\n0 0 2\n3 0 2\n6 0 2\n\nOutput\n6\n\nInput\n3\n0 0 2\n2 0 2\n1 1 2\n\nOutput\n8\n\n\n\n-----Note-----\n\nFor the first example, $000$ \n\nFor the second example, [Image] \n\nFor the third example, $\\text{Q)}$",
        "solutions": "[\"from math import sqrt\\ndef pt(x):\\n    print(x)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\", \"from math import *\\neps = 1e-9\\nans = dict()\\nans[(0,0,0)] = 4\\nans[(0,0,1)] = 4\\nans[(0,1,0)] = 4\\nans[(1,0,0)] = 4\\nans[(0,1,1)] = 4\\nans[(1,0,1)] = 4\\nans[(1,1,0)] = 4\\nans[(1,1,1)] = 5\\nans[(0,0,2)] = 5\\nans[(0,2,0)] = 5\\nans[(2,0,0)] = 5\\nans[(0,1,2)] = 5\\nans[(0,2,1)] = 5\\nans[(1,0,2)] = 5\\nans[(1,2,0)] = 5\\nans[(2,0,1)] = 5\\nans[(2,1,0)] = 5\\nans[(1,1,2)] = 6\\nans[(1,2,1)] = 6\\nans[(2,1,1)] = 6\\nans[(0,2,2)] = 6\\nans[(2,0,2)] = 6\\nans[(2,2,0)] = 6\\nans[(1,2,2)] = 7\\nans[(2,1,2)] = 7\\nans[(2,2,1)] = 7\\nans[(2,2,2)] = 8\\n\\ndef dist(A, B):\\n    return ((A[0] - B[0]) ** 2 + (A[1] - B[1]) ** 2) ** 0.5\\n\\ndef equal(A, B):\\n    return dist(A, B) < eps\\n\\ndef belong(P, i):\\n    return abs(dist(P, (c[i][0], c[i][1])) - c[i][2]) < eps\\n\\ndef intersection(c1, c2):\\n    O1 = c1[0], c1[1]\\n    O2 = c2[0], c2[1]\\n    r1, r2 = c1[2], c2[2]\\n    OO = (O2[0]- O1[0], O2[1]- O1[1])\\n    d = dist(O1, O2)\\n    if d > r1 + r2 or d < abs(r1 - r2):\\n        return []\\n    alp = atan2(OO[1], OO[0])\\n    phi = acos((r1**2 + d**2 - r2**2) / (2 * r1 * d))\\n    P1 = (r1 * cos(alp + phi) + O1[0], r1 * sin(alp + phi) + O1[1])\\n    P2 = (r1 * cos(alp - phi) + O1[0], r1 * sin(alp - phi) + O1[1])\\n    if equal(P1, P2):\\n        return [P1]\\n    return [P1, P2]\\n\\ndef solve():\\n    if n == 1:\\n        return 2\\n    if n == 2:\\n        res = 3\\n        inter = intersection(c[0], c[1])\\n        if len(inter) == 2:\\n            res += 1\\n        return res\\n    cnt = 0\\n    inter = [0, 0, 0]\\n    p = []\\n    for i in range(3):\\n        for j in range(i + 1, 3):\\n            cur = intersection(c[i], c[j])\\n            for P in cur:\\n                p.append(P)\\n                inter[i + j - 1] += 1\\n    for P in p:\\n        flag = 1\\n        for i in range(3):\\n            if not belong(P, i):\\n                flag = 0\\n        if flag:\\n            cnt += 1\\n    res = ans[tuple(inter)] - cnt // 3\\n    return res\\n\\n\\nn = int(input())\\nc = [tuple(map(int, input().split())) for i in range(n)]\\nprint(solve())\\n\", \"from math import sqrt\\n\\nclass vector:\\n\\tdef __init__(self, _x = 0, _y = 0):\\n\\t\\tself.x = _x\\n\\t\\tself.y = _y\\n\\tdef len(self):\\n\\t\\treturn sqrt(self.x ** 2 + self.y ** 2)\\n\\tdef len_sq(self):\\n\\t\\treturn self.x ** 2 + self.y ** 2\\n\\tdef __mul__(self, other):\\n\\t\\tif (type(self) == type(other)):\\n\\t\\t\\treturn self.x * other.x + self.y * other.y\\n\\t\\treturn vector(self.x * other, self.y * other)\\n\\tdef __mod__(self, other):\\n\\t\\treturn self.x * other.y - self.y * other.x\\n\\tdef normed(self):\\n\\t\\tlength = self.len()\\n\\t\\treturn vector(self.x / length, self.y / length)\\n\\tdef normate(self):\\n\\t\\tself = self.normed()\\n\\tdef __str__(self):\\n\\t\\treturn \\\"(\\\" + str(self.x) + \\\", \\\" + str(self.y) + \\\")\\\"\\n\\tdef __add__(self, other):\\n\\t\\treturn vector(self.x + other.x, self.y + other.y);\\n\\tdef __sub__(self, other):\\n\\t\\treturn vector(self.x - other.x, self.y - other.y);\\n\\tdef __eq__(self, other):\\n\\t\\treturn self.x == other.x and self.y == other.y\\n\\tdef rot(self):\\n\\t\\treturn vector(self.y, -self.x)\\n\\nclass line:\\n\\tdef __init__(self, a = 0, b = 0, c = 0):\\n\\t\\tself.a = a\\n\\t\\tself.b = b\\n\\t\\tself.c = c\\n\\tdef intersect(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\tdx = self.c * other.b - self.b * other.c\\n\\t\\tdy = self.a * other.c - self.c * other.a\\n\\t\\treturn vector(dx / d, dy / d)\\n\\tdef fake(self, other):\\n\\t\\td = self.a * other.b - self.b * other.a\\n\\t\\treturn d\\n\\tdef __str__(self):\\n\\t\\treturn str(self.a) + \\\"*x + \\\" + str(self.b) + \\\"*y = \\\" + str(self.c) \\n\\ndef line_pt(A, B):\\n\\t\\td = (A - B).rot()\\n\\t\\treturn line(d.x, d.y, d * A)\\n\\nclass circle:\\n\\tdef __init__(self, O = vector(0, 0), r = 0):\\n\\t\\tself.O = O\\n\\t\\tself.r = r\\n\\tdef intersect(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn []\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn []\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\tM = rad_line.intersect(central)\\n\\t\\t# print(M)\\n\\t\\tif ((O1 - O2).len_sq() == r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn [M]\\n\\t\\td = (O2 - O1).normed().rot()\\n\\t\\tif (r1 ** 2 - (O1 - M).len_sq() < 0):\\n\\t\\t\\treturn []\\n\\t\\td = d * (sqrt(r1 ** 2 - (O1 - M).len_sq()))\\n\\t\\treturn [M + d, M - d]\\n\\tdef fake(self, other):\\n\\t\\tO1 = self.O\\n\\t\\tO2 = other.O\\n\\t\\tr1 = self.r\\n\\t\\tr2 = other.r\\n\\t\\tif (O1 == O2):\\n\\t\\t\\treturn 1\\n\\t\\tif ((O1 - O2).len_sq() > r1 ** 2 + r2 ** 2 + 2 * r1 * r2):\\n\\t\\t\\treturn 1\\n\\t\\trad_line = line(2 * (O2.x - O1.x), 2 * (O2.y - O1.y), r1 ** 2 - O1.len_sq() - r2 ** 2 + O2.len_sq())\\n\\t\\tcentral = line_pt(O1, O2)\\n\\t\\treturn rad_line.fake(central)\\n\\n\\n# a = vector(3, 4)\\n# b = vector(4, 4)\\n# print(circle(vector(1, 2), 3).intersect(circle(vector(2, 1), 6)))\\nn = int(input())\\narr = []\\nm = 1\\nfor i in range(n):\\n\\tx, y, r = map(int, input().split())\\n\\tarr.append(circle(vector(x, y), r))\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\tm *= arr[i].fake(arr[j])\\nfor i in range(n):\\n\\tarr[i].O = arr[i].O * m\\n\\tarr[i].r = arr[i].r * m\\n# print(m)\\ns = set()\\nV = 0\\nfor i in range(n):\\n\\tfor j in range(i + 1, n):\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, 6), round(e.y, 6)))\\nV += len(s)\\nE = 0\\n\\npar = [i for i in range(n)]\\n\\ndef get_par(v):\\n\\tif (par[v] != v):\\n\\t\\tpar[v] = get_par(par[v])\\n\\treturn par[v]\\ndef unite(v, u):\\n\\tpar[get_par(v)] = get_par(u)\\nfor i in range(n):\\n\\ts = set()\\n\\tfor j in range(n):\\t\\n\\t\\ttmp = arr[i].intersect(arr[j])\\n\\t\\tif (len(tmp)):\\n\\t\\t\\tunite(i, j)\\n\\t\\tfor e in tmp:\\n\\t\\t\\ts.add((round(e.x, \\t), round(e.y, \\t)))\\n\\tE += len(s)\\n# print(V, E)\\n# print(len({get_par(i) for i in range(n)}))\\nprint(E - V + 1 + len({get_par(i) for i in range(n)}))\", \"from math import sqrt\\npt = lambda *a, **k: print(*a, **k, flush=True)\\nrd = lambda: map(int, input().split())\\nn = int(input())\\ndef f(x1, y1, r1, x2, y2, r2):\\n    a = (r1 + r2) ** 2\\n    b = (r1 - r2) ** 2\\n    d = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    if d > a:\\n        return 1\\n    elif d == a:\\n        return 4\\n    elif d < b:\\n        return 3\\n    elif d == b:\\n        return 5\\n    else:\\n        return 2\\ndef g(x1, y1, r1, x2, y2, r2):\\n    ds = (x1 - x2) ** 2 + (y1 - y2) ** 2\\n    d = sqrt(ds)\\n    A = (r1 ** 2 - r2 ** 2 + ds) / (2 * d)\\n    h = sqrt(r1 ** 2 - A ** 2)\\n    x = x1 + A * (x2 - x1) / d  \\n    y = y1 + A * (y2 - y1) / d\\n    x3 = x - h * (y2 - y1) / d  \\n    y3 = y + h * (x2 - x1) / d\\n    x4 = x + h * (y2 - y1) / d  \\n    y4 = y - h * (x2 - x1) / d\\n    return x3, y3, x4, y4 \\nif n is 1:\\n    pt(2)\\nif n is 2:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    pt(4 if a is 2 else 3)\\nif n is 3:\\n    x1, y1, r1 = rd()\\n    x2, y2, r2 = rd()\\n    x3, y3, r3 = rd()\\n    a = f(x1, y1, r1, x2, y2, r2)\\n    b = f(x1, y1, r1, x3, y3, r3)\\n    c = f(x3, y3, r3, x2, y2, r2)\\n    t = [a, b, c]\\n    t.sort()\\n    a, b, c = t\\n    if a is 1 and b is 1 and c in [1, 3, 4, 5]:\\n        pt(4)\\n    if a is 1 and b is 1 and c is 2:\\n        pt(5)\\n    if a is 1 and b is 2 and c is 2:\\n        pt(6)\\n    if a is 1 and b is 2 and c in [3, 4, 5]:\\n        pt(5)\\n    if a is 1 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 2 and b is 2 and c is 2:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        r = 8\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        if abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            r -= 1\\n        pt(r)\\n    if a is 2 and b is 2 and c is 3:\\n        pt(6)\\n    if a is 2 and b is 2 and c in [4, 5]:\\n        x4, y4, x5, y5 = g(x1, y1, r1, x2, y2, r2)\\n        if abs((x4 - x3) ** 2 + (y4 - y3) ** 2 - r3 ** 2) < 1e-6 or abs((x5 - x3) ** 2 + (y5 - y3) ** 2 - r3 ** 2) < 1e-6:\\n            pt(6)\\n        else:\\n            pt(7)\\n    if a is 2 and b is 3:\\n        pt(5)\\n    if a is 2 and b in [4, 5]:\\n        pt(6)\\n    if a is 3 and b in [3, 4, 5]:\\n        pt(4)\\n    if a is 4 and b is 4 and c is 4:\\n        pt(5)\\n    if a is 4 and b is 4 and c is 5:\\n        pt(4)\\n    if a is 4 and b is 5 and c is 5:\\n        pt(5)\\n    if a is 5 and b is 5 and c is 5:\\n        pt(4)\\n\\n\\n\\n\\n# Made By Mostafa_Khaled\", \"from decimal import *\\n\\ngetcontext().prec = 40\\neps = Decimal('1e-10')\\n\\n\\nclass Circle:\\n\\tdef __init__(self, x, y, r):\\n\\t\\tself.x = x\\n\\t\\tself.y = y\\n\\t\\tself.r = r\\n\\n\\tdef contains(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd < (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef in_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r - c.r)**2 and self.r > c.r\\n\\n\\tdef ex_touches(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd == (self.r + c.r)**2\\n\\t\\n\\tdef intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn (self.r - c.r)**2 < dd < (self.r + c.r)**2\\n\\t\\n\\tdef not_intersects(self, c):\\n\\t\\tdd = (self.x - c.x)**2 + (self.y - c.y)**2  # dd = d*d\\n\\t\\treturn dd > (self.r + c.r)**2\\n\\t\\n\\tdef get_intersections(self, c):\\n\\t\\tx1, y1, r1, x2, y2, r2 = list(map(Decimal, [self.x, self.y, self.r, c.x, c.y, c.r]))\\n\\t\\t\\n\\t\\tRR = (x1-x2)**2 + (y1-y2)**2\\n\\t\\t\\n\\t\\trx1 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try1 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) + (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\t\\n\\t\\trx2 = (x1+x2)/2 + (r1**2-r2**2)/(2*RR)*(x2-x1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (y2-y1)\\n\\t\\try2 = (y1+y2)/2 + (r1**2-r2**2)/(2*RR)*(y2-y1) - (2*(r1**2+r2**2)/RR-(r1**2-r2**2)**2/(RR**2)-1).sqrt()/2 * (x1-x2)\\n\\t\\n\\t\\treturn {(rx1, ry1), (rx2, ry2)}\\n\\n\\tdef is_on(self, p):\\n\\t\\treturn abs((self.x - p[0])**2 + (self.y - p[1])**2 - self.r**2) < eps\\n\\t\\n\\tdef __repr__(self):\\n\\t\\treturn \\\"(%s, %s, %s)\\\" % (self.x, self.y, self.r)\\n\\n\\ndef count_regions(n, circles):\\n\\tif n == 1:\\n\\t\\treturn 2\\n\\n\\tif n == 2:\\n\\t\\treturn 3 + circles[0].intersects(circles[1])\\n\\n\\tif n == 3:\\n\\t\\tc0, c1, c2 = circles\\n\\t\\tif c0.not_intersects(c1):\\n\\t\\t\\tif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c2.not_intersects(c0):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.contains(c2) or c0.in_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\n\\t\\telif c0.contains(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.ex_touches(c2) or c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\telif c0.in_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.not_intersects(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 7/6, depends on intersections\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.ex_touches(c2) or c2.in_touches(c1))\\n\\n\\t\\telif c0.ex_touches(c1):\\n\\t\\t\\tif c0.in_touches(c2) or c0.contains(c2):\\n\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 5\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 4\\n\\t\\t\\t\\n\\t\\t\\telif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 4 + c1.intersects(c2)\\n\\t\\t\\t\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\t# intersects: 8/7/6?\\n\\t\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\t\\treturn 7 + all(not c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\telif c1.ex_touches(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5\\n\\n\\t\\telif c0.intersects(c1):\\n\\t\\t\\tif c0.not_intersects(c2):\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.contains(c2):\\n\\t\\t\\t\\t# [?] c1.intersects(c2) -> ?\\n\\t\\t\\t\\treturn 5 + c1.intersects(c2)\\n\\n\\t\\t\\telif c0.in_touches(c2) or c0.ex_touches(c2):\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tc0_x_c2 = c0.get_intersections(c2)\\n\\t\\t\\t\\t\\treturn 6 + all(not c1.is_on(p) for p in c0_x_c2)\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\treturn 5 + (c1.in_touches(c2) or c1.ex_touches(c2))\\n\\n\\t\\t\\telif c0.intersects(c2):\\n\\t\\t\\t\\tc0_x_c1 = c0.get_intersections(c1)\\n\\t\\t\\t\\tif c1.intersects(c2):\\n\\t\\t\\t\\t\\tif all(not c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 8\\n\\t\\t\\t\\t\\telif all(c2.is_on(p) for p in c0_x_c1):\\n\\t\\t\\t\\t\\t\\treturn 6\\n\\t\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\t\\treturn 7\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telif c1.in_touches(c2) or c1.ex_touches(c2) or c2.in_touches(c1):\\n\\t\\t\\t\\t\\treturn 7 - any(c2.is_on(p) for p in c0_x_c1)\\n\\t\\t\\t\\t\\n\\t\\t\\t\\telse:  # if c1.contains(c2) or c2.contains(c1) or c1.not_intersects(c2):\\n\\t\\t\\t\\t\\treturn 6\\n\\n\\t\\treturn 4\\n\\n\\treturn 0\\n\\n\\ndef main():\\n\\tn = int(input())\\n\\tcircles = [tuple(map(int, input().split())) for c in range(n)]\\n\\tcircles.sort(key=lambda c: (-c[2], c[0], c[1]))\\n\\tcircles = [Circle(*u) for u in circles]\\n\\t# print(n, circles)\\n\\tprint(count_regions(n, circles))\\n\\n\\ndef __starting_point():\\n\\tmain()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n-10 0 2\n-8 2 2\n-4 -3 5\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/933/C"
    },
    {
        "id": 451,
        "task_id": 3858,
        "test_case_id": 1,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "4\n0 0\n0 1\n1 0\n1 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 452,
        "task_id": 3858,
        "test_case_id": 2,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "5\n0 0\n0 1\n0 2\n0 3\n1 1\n",
        "output": "11\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 453,
        "task_id": 3858,
        "test_case_id": 4,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "2\n11 22\n31 45\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 454,
        "task_id": 3858,
        "test_case_id": 5,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "3\n0 0\n9998 9999\n9997 9998\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 455,
        "task_id": 3858,
        "test_case_id": 6,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "125\n1364 2136\n4530 5609\n8555 7238\n7151 4606\n6464 8941\n9723 9780\n2138 3308\n3013 3268\n372 7398\n7906 6497\n8085 2898\n852 8202\n7683 1159\n4426 2151\n3228 7259\n4624 4992\n3047 8020\n5973 1520\n8096 8598\n8901 1437\n5875 5712\n2761 4610\n3764 4068\n8056 8770\n0 980\n7599 5592\n2956 4617\n982 7168\n2081 7506\n5698 4797\n4410 9630\n5992 8674\n5555 7530\n8275 6351\n3971 7666\n1969 6692\n4647 4732\n6426 1730\n1097 6826\n2556 3490\n3841 9172\n2360 2787\n3656 3563\n2705 9433\n2983 2679\n9495 9171\n5641 7285\n9094 8708\n6736 7439\n2865 1935\n6486 4239\n9412 3100\n4937 1834\n8944 1268\n4989 111\n9556 7366\n1621 8676\n1458 3599\n2071 9576\n5187 8977\n8665 5641\n3071 2684\n279 4821\n4054 8994\n5279 7412\n8214 5502\n8035 7310\n9091 2020\n7412 1330\n8301 3337\n2687 6900\n3827 1039\n8530 3861\n4497 1728\n4808 7849\n8543 8999\n5156 5496\n1000 7111\n5170 5394\n6771 644\n4416 945\n2135 227\n8537 5525\n9007 4934\n9958 7909\n5447 1099\n4441 1513\n2611 4978\n3848 2601\n1623 1316\n9885 6460\n529 8860\n7356 4389\n3519 7455\n7491 5680\n6238 3819\n9235 5319\n3813 9390\n6319 1437\n4770 5104\n9114 207\n8125 7004\n7935 7836\n2690 1163\n2818 6985\n7796 2314\n671 3304\n1312 4954\n3141 8129\n5035 1480\n2654 1172\n7578 2933\n5991 4846\n2817 104\n2794 297\n3975 4631\n3590 7886\n9421 8883\n9315 7984\n4956 6140\n9021 6492\n8422 8135\n4068 2707\n5375 9468\n358 6479\n",
        "output": "37437132\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 456,
        "task_id": 3858,
        "test_case_id": 7,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "8\n5747 4703\n6868 9505\n4873 6907\n1866 3861\n3527 8028\n5870 2600\n397 5742\n519 3575\n",
        "output": "219\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 457,
        "task_id": 3858,
        "test_case_id": 8,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "158\n11 7\n8 6\n11 1\n7 3\n3 3\n5 13\n9 11\n5 14\n2 8\n9 14\n9 7\n13 14\n2 6\n4 12\n12 7\n11 0\n9 8\n1 13\n4 4\n12 3\n1 1\n1 6\n1 3\n8 14\n2 13\n6 7\n9 6\n13 3\n10 3\n6 2\n8 12\n11 10\n7 10\n0 7\n8 9\n10 4\n12 9\n7 9\n0 9\n13 4\n0 0\n13 7\n11 6\n11 2\n9 13\n3 7\n0 4\n6 1\n10 14\n0 6\n7 8\n8 11\n2 2\n7 14\n13 13\n13 12\n14 10\n0 13\n14 0\n2 5\n6 9\n14 12\n2 1\n10 5\n14 4\n12 13\n3 13\n3 2\n13 0\n4 11\n8 1\n4 8\n14 5\n6 13\n0 8\n13 11\n3 12\n1 12\n2 9\n14 3\n6 3\n3 9\n13 8\n4 13\n4 9\n6 14\n2 3\n11 13\n12 5\n4 2\n12 4\n1 2\n0 11\n1 5\n10 1\n10 8\n14 11\n12 8\n1 8\n6 12\n11 5\n12 10\n13 10\n2 10\n6 10\n3 1\n0 1\n14 1\n6 11\n6 6\n1 7\n3 4\n11 8\n14 9\n11 11\n3 11\n12 1\n5 0\n9 12\n7 4\n9 5\n5 7\n1 10\n9 2\n5 5\n5 3\n4 5\n9 0\n12 2\n12 12\n10 9\n8 2\n13 2\n3 6\n12 11\n9 4\n2 11\n14 8\n5 1\n12 0\n8 13\n1 9\n12 14\n4 14\n7 12\n10 13\n7 0\n4 3\n8 4\n1 11\n4 1\n9 1\n8 3\n1 4\n0 12\n2 0\n1 0\n6 8\n",
        "output": "835295061\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 458,
        "task_id": 3858,
        "test_case_id": 9,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "135\n3 17\n7 5\n3 11\n8 9\n6 9\n6 8\n2 16\n2 0\n7 2\n7 4\n2 10\n8 13\n6 13\n4 1\n7 14\n8 14\n3 6\n6 2\n5 12\n7 12\n4 17\n4 16\n4 6\n4 10\n1 15\n4 13\n1 7\n2 11\n6 7\n3 9\n4 2\n4 14\n7 10\n8 17\n8 18\n0 13\n7 11\n9 17\n4 19\n5 7\n5 6\n1 0\n9 4\n4 3\n3 14\n9 3\n1 4\n7 7\n1 5\n9 11\n6 10\n8 0\n0 4\n9 9\n1 18\n9 1\n1 1\n0 3\n9 16\n2 17\n0 18\n8 5\n4 9\n5 0\n5 8\n7 1\n9 2\n2 9\n5 19\n2 12\n3 5\n3 8\n2 2\n5 10\n9 15\n9 6\n0 6\n2 13\n0 0\n5 15\n7 15\n9 19\n1 8\n3 0\n6 3\n8 1\n1 12\n8 19\n7 9\n1 2\n8 11\n9 18\n9 13\n9 10\n6 4\n2 5\n9 0\n5 18\n5 5\n0 15\n9 14\n2 1\n6 16\n0 17\n4 7\n3 13\n5 17\n9 8\n3 7\n0 5\n0 1\n7 18\n0 14\n4 4\n4 5\n9 5\n4 18\n6 11\n4 11\n0 12\n1 14\n8 10\n8 2\n8 3\n1 19\n1 9\n1 17\n1 13\n2 4\n1 11\n2 15\n0 7\n3 2\n3 10\n6 12\n",
        "output": "409953897\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 459,
        "task_id": 3858,
        "test_case_id": 10,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "145\n7 16\n15 17\n8 1\n10 8\n15 15\n5 15\n0 18\n2 5\n11 18\n18 18\n11 10\n10 0\n3 8\n4 5\n12 0\n13 13\n12 10\n4 10\n14 1\n5 16\n6 15\n18 1\n5 18\n15 3\n15 8\n0 17\n9 19\n9 2\n4 8\n1 2\n13 0\n13 2\n8 10\n15 2\n13 16\n1 11\n3 5\n14 14\n3 7\n5 3\n9 7\n12 1\n6 1\n19 8\n9 11\n16 15\n16 8\n1 16\n11 8\n18 10\n13 9\n0 16\n15 12\n16 14\n17 13\n3 2\n6 18\n5 9\n4 19\n11 0\n10 19\n19 6\n7 17\n0 0\n9 12\n11 7\n8 5\n6 7\n4 7\n14 12\n17 8\n7 8\n2 10\n10 12\n6 0\n3 16\n8 2\n5 11\n19 19\n16 16\n14 11\n1 1\n8 0\n15 7\n0 8\n18 12\n18 4\n12 11\n2 16\n1 12\n2 0\n9 10\n4 0\n18 11\n3 9\n3 11\n6 16\n17 4\n3 4\n8 6\n6 3\n6 4\n8 3\n12 3\n11 16\n4 2\n12 7\n2 19\n8 8\n0 11\n5 19\n19 16\n3 13\n15 1\n14 2\n7 9\n2 1\n17 6\n3 17\n17 14\n11 19\n2 18\n12 17\n16 17\n5 8\n2 6\n3 14\n17 15\n10 2\n17 9\n8 16\n13 8\n14 16\n3 10\n15 14\n2 17\n0 15\n4 9\n10 18\n15 4\n19 1\n16 6\n10 13\n6 13\n0 14\n",
        "output": "989773047\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 460,
        "task_id": 3858,
        "test_case_id": 11,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "82\n12 23\n2 13\n3 26\n27 23\n29 17\n6 3\n29 10\n25 14\n28 11\n12 6\n6 6\n28 29\n24 26\n20 1\n1 3\n13 23\n16 5\n11 0\n8 25\n16 2\n29 26\n2 12\n7 20\n25 6\n6 20\n12 27\n18 3\n12 18\n28 26\n18 2\n1 16\n22 8\n0 17\n2 1\n9 16\n21 20\n24 17\n15 5\n7 15\n26 23\n1 25\n15 17\n4 28\n13 21\n1 9\n18 28\n29 23\n20 19\n26 4\n23 11\n3 19\n20 27\n18 4\n29 15\n24 24\n0 24\n7 19\n21 19\n10 22\n13 27\n3 28\n16 23\n26 17\n11 18\n28 20\n16 13\n22 19\n15 21\n14 20\n23 29\n3 29\n24 4\n26 0\n25 17\n27 19\n19 9\n3 7\n21 29\n26 16\n7 14\n15 18\n17 11\n",
        "output": "529806400\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 461,
        "task_id": 3858,
        "test_case_id": 12,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "148\n4 930\n5 863\n3 949\n0 650\n7 778\n5 797\n0 371\n5 952\n9 57\n3 937\n8 8\n9 517\n4 114\n4 489\n0 535\n8 114\n5 906\n9 108\n3 842\n3 497\n4 427\n9 7\n0 831\n9 224\n9 912\n7 356\n4 180\n3 938\n6 303\n2 338\n0 963\n5 429\n2 65\n4 139\n2 577\n8 243\n4 553\n8 48\n6 381\n7 467\n9 861\n2 27\n7 521\n9 27\n8 850\n0 996\n7 615\n0 725\n7 764\n0 68\n8 488\n0 493\n6 536\n4 125\n7 971\n7 876\n1 94\n4 400\n6 129\n1 186\n8 959\n4 983\n7 348\n0 383\n8 886\n1 205\n7 773\n1 913\n9 866\n7 722\n3 955\n8 495\n4 291\n3 301\n4 739\n3 254\n7 662\n6 919\n7 211\n9 47\n7 557\n7 747\n7 130\n7 863\n9 481\n1 474\n5 343\n7 853\n5 122\n0 394\n4 9\n3 614\n5 10\n4 223\n4 420\n6 202\n6 880\n4 88\n5 772\n2 3\n9 494\n2 947\n7 182\n8 416\n5 647\n9 769\n6 811\n7 994\n0 434\n6 867\n3 440\n8 904\n9 838\n0 773\n1 795\n3 768\n6 333\n9 210\n7 322\n0 147\n1 376\n4 418\n8 617\n5 803\n9 602\n3 910\n5 499\n5 74\n0 97\n8 215\n2 680\n2 299\n1 500\n8 610\n1 562\n8 842\n3 533\n7 883\n5 130\n4 133\n1 171\n6 984\n8 564\n2 171\n3 445\n8 396\n1 195\n5 357\n",
        "output": "926071364\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 462,
        "task_id": 3858,
        "test_case_id": 13,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "160\n993 1\n325 7\n965 2\n185 5\n42 2\n541 2\n358 0\n406 9\n225 3\n545 0\n491 3\n729 7\n555 2\n241 3\n34 3\n88 4\n960 8\n331 0\n170 7\n302 8\n791 3\n671 0\n27 6\n902 8\n572 4\n852 7\n860 7\n176 6\n949 1\n128 8\n879 9\n131 9\n687 5\n962 8\n692 3\n628 5\n85 5\n519 2\n452 6\n141 2\n422 3\n155 2\n653 2\n431 2\n546 8\n49 8\n410 7\n154 6\n159 6\n115 7\n249 6\n281 5\n243 2\n487 6\n139 1\n683 0\n464 8\n832 0\n203 7\n886 4\n24 4\n694 8\n204 4\n551 5\n595 2\n626 6\n19 3\n972 7\n974 9\n652 0\n767 0\n314 6\n943 0\n800 6\n206 5\n197 7\n861 3\n639 4\n696 4\n149 5\n718 0\n276 6\n193 5\n969 3\n45 4\n342 6\n822 2\n988 6\n415 8\n901 1\n210 2\n319 5\n23 3\n345 0\n401 9\n148 9\n710 0\n589 3\n733 7\n884 0\n682 1\n204 2\n823 3\n718 2\n723 1\n948 5\n329 9\n280 2\n693 9\n25 3\n750 1\n338 8\n614 0\n54 7\n852 4\n245 3\n421 0\n805 2\n589 8\n809 2\n450 7\n525 5\n570 7\n648 7\n670 7\n841 8\n227 0\n770 9\n67 9\n638 0\n959 0\n948 6\n890 6\n928 4\n626 4\n914 3\n240 7\n751 2\n334 4\n14 2\n7 0\n591 6\n801 3\n44 2\n247 8\n742 7\n373 4\n404 3\n371 8\n456 9\n894 7\n128 5\n699 7\n927 5\n114 6\n131 5\n247 4\n763 9\n473 6\n540 5\n",
        "output": "342590318\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 463,
        "task_id": 3858,
        "test_case_id": 14,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "130\n657 0\n7528 4\n6543 3\n1512 4\n5702 4\n5810 2\n9055 0\n9445 0\n3204 4\n2603 3\n19 2\n912 2\n5146 2\n581 2\n6036 2\n2835 1\n7952 4\n7579 2\n5494 0\n6614 3\n2646 2\n8448 2\n4250 1\n9695 0\n601 1\n4670 3\n2829 4\n7017 2\n3007 1\n7048 1\n1273 4\n5066 1\n829 1\n7755 1\n9829 3\n699 3\n2707 3\n4547 3\n9171 1\n4558 1\n451 2\n3122 2\n5988 2\n8409 4\n5134 0\n3337 0\n9762 3\n8487 2\n5158 1\n6337 3\n2779 0\n953 1\n5367 1\n7875 0\n5278 3\n9252 2\n1906 3\n2855 0\n1756 4\n8363 3\n5681 4\n8740 1\n2548 4\n7551 4\n3184 0\n7807 1\n3962 1\n5118 3\n5678 3\n5451 1\n810 1\n3952 1\n7002 1\n690 2\n7325 3\n8463 3\n8541 4\n7906 4\n3843 0\n9138 4\n3231 4\n3506 1\n7826 2\n1076 1\n9198 4\n9679 2\n6060 2\n3769 3\n1153 2\n4907 1\n490 0\n8636 2\n8557 3\n4169 0\n7425 4\n1000 2\n4829 1\n8373 1\n5677 1\n4395 2\n9870 0\n8690 4\n6196 2\n8435 2\n1342 4\n8030 1\n4608 4\n6376 0\n237 4\n5741 1\n8360 2\n8726 3\n5480 4\n9820 3\n8095 4\n3700 0\n3338 3\n9718 1\n1183 1\n2661 0\n3124 1\n927 0\n1811 2\n2531 4\n3818 4\n7311 4\n8168 4\n5582 1\n2090 2\n723 4\n",
        "output": "86218700\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 464,
        "task_id": 3858,
        "test_case_id": 15,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "21\n4 4895\n3 1687\n4 180\n0 281\n2 6355\n3 7778\n4 3913\n2 7173\n3 9846\n4 3361\n0 6969\n2 1966\n4 1237\n3 7758\n3 7126\n3 5765\n4 2453\n4 2775\n3 1840\n1 8718\n0 40\n",
        "output": "2096720\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 465,
        "task_id": 3858,
        "test_case_id": 16,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "162\n6067 0\n9538 0\n2700 0\n2929 0\n3693 0\n4802 0\n188 0\n8512 0\n424 0\n7761 0\n5119 0\n8329 0\n1879 0\n2719 0\n3415 0\n8224 0\n3689 0\n9010 0\n2765 0\n9973 0\n1169 0\n2251 0\n892 0\n1406 0\n6459 0\n7958 0\n6775 0\n7415 0\n1181 0\n55 0\n2834 0\n666 0\n3991 0\n1985 0\n9264 0\n7483 0\n6201 0\n6409 0\n6610 0\n1776 0\n3997 0\n9051 0\n2780 0\n171 0\n9421 0\n7123 0\n5189 0\n1347 0\n2959 0\n4408 0\n4818 0\n7981 0\n9964 0\n8409 0\n8192 0\n1571 0\n8953 0\n3129 0\n3265 0\n5211 0\n6138 0\n1269 0\n8332 0\n9201 0\n8304 0\n7919 0\n1272 0\n926 0\n3726 0\n6910 0\n8200 0\n5758 0\n1197 0\n3462 0\n6154 0\n1826 0\n2894 0\n9055 0\n255 0\n9246 0\n1908 0\n9712 0\n5585 0\n1813 0\n3606 0\n9776 0\n4583 0\n6630 0\n9967 0\n8970 0\n3690 0\n1032 0\n1675 0\n1110 0\n5623 0\n272 0\n1384 0\n1008 0\n7799 0\n2470 0\n6717 0\n9620 0\n2627 0\n9861 0\n6632 0\n7622 0\n5541 0\n7 0\n7295 0\n4752 0\n4781 0\n8116 0\n1036 0\n6309 0\n7261 0\n4564 0\n804 0\n877 0\n8124 0\n2685 0\n1200 0\n9169 0\n7294 0\n417 0\n8336 0\n1790 0\n8539 0\n9786 0\n9382 0\n6134 0\n3101 0\n8327 0\n2401 0\n8504 0\n3139 0\n5848 0\n2112 0\n701 0\n7744 0\n2046 0\n914 0\n477 0\n4262 0\n4527 0\n1858 0\n1267 0\n4788 0\n7259 0\n8789 0\n9938 0\n2371 0\n6194 0\n7594 0\n3473 0\n1622 0\n2603 0\n3428 0\n1606 0\n8683 0\n1444 0\n6670 0\n350 0\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 466,
        "task_id": 3858,
        "test_case_id": 17,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "107\n0 1472\n0 8250\n0 2504\n0 6651\n0 6696\n0 8316\n0 1379\n0 1958\n0 2317\n0 1432\n0 1515\n0 8595\n0 7215\n0 3925\n0 2564\n0 3066\n0 8738\n0 2668\n0 5918\n0 9541\n0 6068\n0 7818\n0 1436\n0 3317\n0 8597\n0 509\n0 7603\n0 3067\n0 9582\n0 1663\n0 1081\n0 320\n0 950\n0 3880\n0 4695\n0 2061\n0 5178\n0 4644\n0 9322\n0 338\n0 7857\n0 2647\n0 3223\n0 2804\n0 1434\n0 9622\n0 6132\n0 8026\n0 2590\n0 6113\n0 3426\n0 275\n0 982\n0 9433\n0 727\n0 5065\n0 9918\n0 7398\n0 2337\n0 1361\n0 4881\n0 9051\n0 9179\n0 9203\n0 7880\n0 5888\n0 2849\n0 9439\n0 6945\n0 729\n0 2048\n0 7586\n0 5243\n0 4199\n0 628\n0 4248\n0 6090\n0 1990\n0 1306\n0 2038\n0 6980\n0 1226\n0 1500\n0 556\n0 2732\n0 4005\n0 3963\n0 1946\n0 9955\n0 472\n0 2874\n0 9839\n0 3490\n0 5149\n0 918\n0 9144\n0 513\n0 1728\n0 3792\n0 5608\n0 856\n0 2448\n0 3034\n0 7480\n0 8970\n0 874\n0 3997\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 467,
        "task_id": 3858,
        "test_case_id": 18,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n6961 9533\n1847 2184\n9391 1072\n8146 7944\n3403 1703\n2280 4570\n8872 8035\n9417 8677\n8552 1719\n6602 3237\n6223 2519\n654 7141\n8494 2638\n7204 8528\n1352 3164\n1638 4388\n1102 7239\n3795 5213\n6836 786\n9662 2985\n6241 9368\n1325 8963\n1237 5004\n8000 5104\n9833 4105\n8543 3581\n9753 6278\n7003 8220\n9412 9666\n3212 4503\n4223 8775\n1098 4945\n8829 7823\n8103 3960\n6632 876\n6716 1114\n2133 7555\n2651 3042\n7027 3091\n5032 1117\n7000 5645\n9411 6945\n6464 3506\n3033 39\n8277 3731\n8592 4447\n9833 3439\n118 6314\n7718 4686\n8237 2805\n34 1721\n2548 3168\n6338 9994\n6123 808\n8687 1813\n6822 994\n231 8537\n5106 526\n9925 6198\n4860 6977\n7041 2027\n6107 3539\n1283 9212\n1739 328\n5704 5778\n8573 8264\n3914 9078\n3331 5537\n4148 2968\n4799 8662\n6944 7358\n4798 8939\n7779 6390\n8994 8909\n3418 2226\n8887 4015\n1160 4480\n7383 2846\n6161 8475\n5223 8030\n869 3315\n2351 8805\n1932 4674\n6569 2224\n2494 2320\n75 777\n3748 9561\n7070 7926\n2550 5020\n5512 1836\n8370 9272\n7304 4219\n4334 7856\n6631 4129\n8657 7662\n7097 3238\n6105 1730\n714 1984\n8861 8112\n9133 3809\n8028 9213\n2763 7401\n1254 5363\n4679 6518\n1964 6520\n3579 7145\n5608 8240\n7315 3800\n5347 5042\n1932 863\n9766 2340\n9314 5466\n1993 3695\n8716 3096\n8205 9135\n7079 5755\n4801 1233\n9388 8387\n263 9395\n129 951\n8688 9247\n9947 2963\n2872 8678\n7698 2296\n729 7562\n8953 8477\n1368 9869\n8183 5996\n7378 5883\n2699 4468\n5268 3412\n9679 9013\n9119 568\n6197 3309\n3452 2846\n4884 4591\n8736 8550\n4672 9856\n2462 9764\n3458 8913\n9948 6393\n8401 5933\n9005 6620\n5555 792\n1279 7412\n5019 5339\n143 9553\n1941 275\n2709 440\n9150 9920\n1925 889\n2269 6409\n409 2076\n8792 7880\n1868 6440\n896 4303\n7689 8580\n3547 5453\n9013 1443\n4232 8082\n670 1405\n8061 9996\n1083 6641\n3453 9438\n8009 2419\n495 3764\n5700 5918\n5903 531\n2502 6726\n5974 6018\n3517 6942\n3837 1479\n4594 7244\n1695 5569\n5329 5141\n1152 3953\n5164 8656\n1831 2573\n9056 1059\n1009 1759\n7453 3545\n3628 9711\n959 8512\n184 8688\n6275 3680\n2599 9281\n4113 4054\n4877 7020\n1990 8352\n9108 3672\n4874 2952\n7684 5333\n2155 3270\n6072 6174\n2079 4442\n9653 5999\n753 7604\n3255 4274\n9312 5726\n8700 2299\n",
        "output": "982904631\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 468,
        "task_id": 3858,
        "test_case_id": 19,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n8020 8648\n1485 5804\n4219 3090\n4654 4159\n8033 8567\n8223 9458\n1963 7916\n7492 6451\n254 9361\n1212 5158\n9786 3104\n8105 7262\n3825 4811\n2489 472\n7353 8583\n3143 5503\n2546 7122\n5653 4976\n4084 8170\n9211 9891\n9053 2480\n2543 3626\n6411 8124\n123 9443\n2473 7482\n4144 2040\n4966 7434\n1136 772\n8820 2632\n2819 3111\n3453 5746\n7041 4928\n4908 9027\n8330 6781\n9983 6550\n8617 7489\n386 6154\n4338 315\n4480 922\n1356 1246\n9864 8440\n1836 241\n9865 4003\n4741 9508\n8021 223\n8584 609\n4307 8106\n4860 4654\n482 6030\n4092 7789\n6811 2631\n1079 5596\n9413 5949\n9425 4637\n5038 356\n2752 1297\n5974 4831\n193 2027\n7865 7343\n6538 8405\n1266 7843\n9916 3120\n8963 7472\n6406 1692\n9664 115\n6123 6842\n2984 3\n3766 6913\n6909 3043\n8109 315\n5861 7851\n9568 6233\n6216 9297\n2731 6637\n5000 7974\n1736 6380\n5369 1469\n576 2473\n979 4080\n3461 8248\n457 7919\n8014 2550\n350 9040\n1647 8359\n3109 2464\n5141 9920\n3519 491\n5683 798\n5282 3462\n5242 5553\n5362 8016\n7222 4959\n6005 2371\n2958 8641\n5174 1072\n875 5303\n4824 7984\n3734 790\n6348 682\n2703 6454\n1674 9159\n4372 4931\n4043 3770\n6463 7718\n6303 1586\n32 655\n3026 9883\n9598 852\n6907 8207\n2993 1956\n2947 8477\n3136 3697\n68 1868\n677 647\n1802 4690\n1273 227\n1698 5674\n3324 2014\n2413 6578\n8567 8362\n5094 170\n1881 1461\n6965 5839\n4743 3174\n851 5906\n4201 4835\n3230 1422\n3905 1120\n6985 6776\n5390 5726\n5203 3203\n8188 6473\n4601 1460\n395 7880\n1103 7645\n5263 9057\n2858 4993\n4419 7710\n6489 1740\n4740 4142\n7684 1580\n2380 2329\n4361 3048\n8995 5052\n9012 4151\n8659 4467\n3523 1348\n3116 7417\n8924 2718\n6060 459\n2374 6846\n7219 4242\n7740 2089\n309 4755\n6215 2734\n9273 4589\n6896 6865\n9327 3487\n6421 7196\n8341 2298\n5348 8541\n8902 2450\n1722 2448\n2391 330\n4507 3003\n2817 7393\n5542 6553\n4141 9982\n372 1858\n9819 7523\n8853 6798\n8238 2586\n557 7414\n3011 3282\n6938 9386\n3104 1748\n1528 2932\n2553 761\n8116 594\n1391 7099\n5527 6558\n5815 2071\n3534 7925\n3310 787\n8791 9227\n6534 6861\n8460 4165\n6744 733\n688 8411\n2240 9750\n1852 8054\n9491 6435\n2932 4110\n7304 3359\n2484 9260\n624 2597\n5553 4111\n8761 6013\n8979 7839\n5428 5275\n",
        "output": "982904631\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 469,
        "task_id": 3858,
        "test_case_id": 20,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n5 13\n0 10\n3 10\n4 3\n14 4\n6 13\n10 11\n9 10\n12 3\n2 9\n12 6\n2 4\n3 6\n10 7\n7 4\n1 7\n8 5\n14 8\n6 4\n13 1\n5 10\n0 14\n11 0\n12 7\n0 4\n3 7\n13 8\n6 3\n10 5\n14 9\n6 5\n13 9\n13 11\n13 10\n6 7\n10 10\n8 1\n9 6\n8 7\n11 7\n2 3\n4 14\n9 0\n11 11\n1 13\n12 0\n12 5\n6 11\n2 13\n0 3\n5 1\n13 14\n10 13\n8 2\n7 10\n10 14\n10 0\n9 14\n9 1\n13 5\n10 12\n3 14\n3 5\n2 6\n7 11\n7 7\n4 7\n1 9\n4 5\n12 14\n9 3\n13 6\n10 9\n2 1\n12 11\n11 1\n1 6\n4 12\n11 10\n3 4\n1 4\n1 14\n5 14\n0 1\n13 0\n2 5\n3 11\n8 11\n11 14\n4 6\n2 8\n10 3\n8 0\n4 10\n1 8\n9 7\n12 9\n14 6\n10 6\n5 11\n8 8\n5 6\n5 8\n12 2\n5 5\n11 6\n5 0\n1 1\n14 12\n4 8\n4 13\n10 1\n10 8\n4 4\n13 4\n8 9\n11 2\n9 2\n3 3\n14 3\n7 3\n9 11\n0 7\n6 10\n9 8\n1 5\n11 3\n12 1\n8 14\n2 2\n9 5\n7 1\n12 13\n6 12\n7 12\n12 10\n13 3\n11 12\n3 9\n11 8\n1 0\n13 13\n4 2\n14 7\n3 0\n12 4\n12 12\n10 2\n2 12\n13 2\n0 8\n2 14\n3 8\n14 1\n3 13\n5 9\n4 11\n7 13\n0 11\n11 4\n4 1\n3 1\n13 12\n9 12\n2 11\n8 6\n14 0\n8 4\n9 4\n5 3\n2 10\n7 0\n0 13\n1 2\n14 13\n0 6\n8 12\n11 5\n0 12\n7 2\n13 7\n7 9\n11 9\n5 4\n6 9\n7 6\n3 2\n5 2\n1 3\n14 11\n6 0\n12 8\n1 11\n9 13\n9 9\n10 4\n14 14\n2 7\n1 10\n5 7\n",
        "output": "982368519\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 470,
        "task_id": 3858,
        "test_case_id": 21,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n1 8\n2 11\n4 1\n1 18\n0 11\n5 12\n6 0\n2 12\n6 15\n9 15\n5 10\n6 8\n3 13\n5 17\n3 9\n2 1\n6 18\n0 1\n7 15\n3 1\n2 0\n8 11\n6 9\n6 11\n8 2\n7 1\n6 6\n2 15\n0 2\n7 10\n7 4\n8 14\n9 6\n1 2\n5 13\n5 18\n1 14\n3 6\n6 7\n6 14\n0 7\n3 10\n7 14\n7 6\n9 0\n0 14\n3 7\n1 19\n9 19\n3 5\n8 7\n2 4\n7 5\n9 8\n3 11\n0 3\n7 3\n9 13\n7 16\n1 9\n2 17\n6 12\n3 12\n0 12\n7 7\n3 2\n2 5\n8 12\n4 11\n9 12\n2 13\n2 10\n9 4\n2 6\n1 15\n7 2\n8 6\n1 5\n0 19\n9 18\n7 8\n3 16\n2 14\n1 16\n9 10\n5 9\n7 18\n8 8\n2 18\n2 8\n8 5\n6 10\n5 14\n0 15\n7 12\n8 3\n4 12\n1 0\n8 18\n4 4\n8 4\n1 1\n5 11\n0 0\n7 0\n2 3\n9 9\n3 14\n4 7\n4 18\n9 7\n9 11\n1 7\n5 8\n9 16\n7 19\n8 0\n3 8\n9 14\n4 13\n6 3\n8 10\n4 14\n5 16\n7 17\n5 0\n2 2\n5 1\n9 1\n1 13\n2 19\n0 10\n6 2\n1 10\n4 10\n3 3\n1 6\n3 4\n9 2\n0 8\n0 9\n9 5\n8 9\n4 6\n9 17\n8 15\n0 13\n5 15\n0 16\n2 7\n1 4\n5 5\n9 3\n4 17\n0 18\n4 2\n8 19\n4 5\n1 11\n6 17\n8 13\n1 3\n4 15\n6 1\n6 4\n5 6\n5 4\n3 15\n6 19\n3 0\n4 16\n2 9\n5 19\n4 3\n4 19\n7 11\n0 6\n2 16\n0 4\n5 2\n8 16\n6 5\n7 13\n4 0\n6 13\n1 12\n8 17\n4 9\n5 3\n0 17\n1 17\n3 19\n7 9\n0 5\n4 8\n3 18\n5 7\n3 17\n6 16\n8 1\n",
        "output": "972358979\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 471,
        "task_id": 3858,
        "test_case_id": 22,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n3 16\n9 0\n9 7\n8 10\n10 9\n0 16\n13 2\n17 15\n18 19\n19 10\n7 11\n16 14\n9 16\n1 3\n2 12\n13 7\n6 15\n13 19\n11 5\n3 17\n3 6\n17 18\n5 6\n14 2\n10 19\n13 1\n4 0\n4 18\n14 16\n16 0\n3 3\n17 11\n9 5\n2 10\n17 1\n7 2\n15 19\n11 10\n1 1\n17 5\n10 4\n15 8\n12 13\n5 7\n7 8\n17 13\n12 1\n2 19\n2 7\n12 2\n17 7\n4 7\n11 19\n9 14\n16 4\n12 10\n3 14\n10 3\n15 7\n3 8\n11 7\n6 13\n8 6\n11 9\n4 4\n11 16\n10 7\n14 15\n14 6\n10 11\n2 6\n9 18\n4 5\n0 3\n0 7\n7 3\n9 19\n18 9\n5 9\n17 12\n0 13\n12 5\n13 13\n17 14\n14 18\n18 16\n5 11\n12 14\n15 11\n0 2\n1 13\n6 9\n13 16\n0 6\n3 0\n10 6\n19 5\n12 3\n2 14\n6 8\n6 7\n18 5\n6 14\n10 16\n11 4\n16 9\n19 13\n9 12\n18 2\n18 12\n8 5\n7 15\n17 2\n1 12\n17 16\n11 17\n15 16\n7 1\n8 7\n7 19\n12 8\n16 6\n13 4\n6 5\n1 19\n5 19\n5 10\n19 12\n15 5\n6 4\n12 9\n18 13\n11 11\n9 13\n10 15\n8 3\n15 4\n14 19\n15 0\n1 15\n7 17\n8 11\n16 17\n17 3\n7 12\n14 7\n12 18\n10 10\n8 19\n0 5\n19 2\n13 10\n3 5\n8 0\n18 4\n10 0\n4 12\n15 15\n15 13\n13 6\n12 4\n18 6\n0 9\n18 14\n10 18\n2 9\n6 3\n6 16\n4 2\n5 1\n4 10\n1 18\n14 9\n9 3\n10 13\n10 14\n4 3\n0 19\n15 14\n13 0\n19 7\n7 4\n15 3\n2 1\n0 17\n1 17\n0 1\n12 16\n15 10\n11 14\n1 4\n7 9\n12 12\n18 3\n18 8\n10 2\n10 1\n0 11\n14 11\n5 16\n",
        "output": "982714030\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 472,
        "task_id": 3858,
        "test_case_id": 23,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n10 28\n16 4\n17 18\n11 8\n14 8\n8 26\n3 29\n25 5\n9 6\n13 22\n0 24\n21 6\n25 10\n28 5\n11 16\n19 13\n22 1\n1 10\n9 17\n25 0\n23 29\n28 15\n7 27\n0 7\n22 24\n18 6\n7 25\n27 16\n3 0\n1 13\n16 7\n12 14\n9 2\n18 7\n11 13\n4 8\n21 3\n10 9\n28 6\n22 23\n13 13\n24 8\n10 24\n1 8\n3 12\n26 15\n0 0\n1 1\n29 13\n10 25\n17 4\n10 12\n17 28\n21 12\n17 13\n16 19\n8 15\n4 19\n3 2\n17 29\n22 2\n10 3\n16 28\n3 20\n12 25\n9 0\n4 25\n8 27\n19 2\n1 9\n2 8\n4 10\n13 27\n28 16\n8 29\n26 12\n5 1\n24 9\n3 26\n11 12\n14 5\n6 5\n21 15\n3 27\n18 27\n4 1\n8 12\n20 2\n24 4\n20 12\n29 10\n27 7\n3 6\n23 17\n21 26\n13 28\n2 16\n18 13\n9 18\n13 21\n13 18\n7 7\n16 13\n18 4\n5 2\n5 25\n10 5\n13 20\n23 28\n11 7\n24 21\n28 18\n24 28\n7 6\n20 24\n22 15\n15 1\n7 10\n13 10\n11 28\n7 0\n0 1\n18 24\n16 29\n5 17\n11 5\n23 3\n0 21\n5 20\n22 11\n8 7\n5 8\n27 1\n3 23\n17 12\n8 25\n4 16\n9 28\n15 27\n28 26\n13 6\n14 11\n20 21\n6 6\n15 7\n8 28\n29 6\n9 8\n25 15\n16 21\n4 13\n2 14\n4 3\n6 28\n5 13\n13 17\n9 12\n20 1\n26 19\n24 6\n28 29\n9 16\n2 25\n2 4\n6 29\n28 3\n7 15\n25 20\n27 13\n1 25\n0 18\n3 13\n12 9\n3 24\n2 2\n2 18\n6 25\n18 21\n0 23\n29 15\n1 12\n4 14\n14 29\n28 28\n25 17\n26 1\n20 0\n21 2\n8 13\n14 24\n17 3\n15 11\n1 16\n17 26\n24 29\n16 14\n16 27\n14 19\n19 15\n7 21\n",
        "output": "982874858\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 473,
        "task_id": 3858,
        "test_case_id": 24,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n2 57\n0 96\n7 282\n0 90\n1 757\n2 975\n0 820\n7 280\n7 745\n1 901\n0 246\n1 403\n5 291\n5 501\n1 467\n5 239\n2 625\n8 472\n3 668\n3 286\n9 209\n3 154\n6 38\n5 661\n7 265\n8 568\n1 821\n7 561\n0 751\n5 56\n1 618\n9 335\n0 707\n4 733\n8 209\n9 692\n6 93\n9 813\n8 688\n5 628\n4 492\n6 74\n0 670\n3 915\n0 542\n0 490\n5 404\n3 353\n2 793\n3 28\n8 394\n4 525\n3 931\n3 505\n9 402\n0 128\n8 627\n4 674\n0 860\n9 721\n1 187\n3 151\n4 199\n6 815\n5 420\n8 716\n6 134\n0 539\n4 95\n6 322\n5 105\n1 693\n6 923\n3 264\n7 105\n8 142\n2 693\n4 806\n7 722\n2 201\n1 940\n6 171\n6 571\n0 709\n3 950\n4 965\n6 992\n2 446\n8 276\n2 754\n6 178\n0 506\n2 17\n2 356\n1 428\n3 67\n4 121\n4 967\n6 123\n5 360\n5 256\n6 477\n2 222\n3 805\n3 520\n0 367\n2 725\n1 824\n7 540\n4 1\n6 692\n2 466\n5 542\n0 942\n5 110\n6 869\n2 388\n4 173\n6 491\n0 526\n8 707\n9 601\n8 536\n6 628\n3 23\n5 434\n7 736\n2 364\n1 284\n8 180\n2 741\n9 840\n4 941\n2 515\n6 403\n8 563\n4 580\n9 346\n5 411\n5 531\n8 355\n5 227\n8 265\n4 267\n4 831\n8 371\n0 10\n6 374\n5 998\n1 670\n6 47\n7 80\n2 457\n9 295\n5 846\n9 438\n5 408\n3 81\n3 416\n9 819\n8 349\n5 757\n5 63\n0 573\n5 993\n5 152\n0 664\n0 995\n2 737\n2 537\n0 736\n9 238\n7 598\n3 331\n6 11\n7 758\n5 250\n8 250\n0 61\n0 806\n2 940\n5 188\n8 403\n4 675\n7 74\n7 884\n4 37\n2 553\n2 900\n9 144\n4 29\n1 799\n1 266\n2 264\n3 275\n1 90\n6 326\n9 215\n6 354\n5 740\n",
        "output": "805058255\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 474,
        "task_id": 3858,
        "test_case_id": 25,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n695 3\n894 6\n88 5\n335 6\n379 1\n267 9\n427 5\n747 9\n914 7\n799 5\n286 5\n798 3\n103 1\n870 9\n519 7\n975 6\n299 7\n638 7\n771 6\n65 7\n989 7\n557 3\n176 3\n498 2\n678 0\n268 7\n747 7\n341 9\n346 9\n602 4\n155 4\n228 5\n644 7\n152 7\n745 7\n408 4\n904 6\n935 7\n452 9\n524 0\n731 6\n543 7\n431 3\n189 8\n568 8\n554 8\n947 9\n33 3\n60 4\n857 7\n261 8\n860 3\n39 9\n490 6\n353 9\n295 2\n996 5\n819 0\n843 7\n738 3\n209 1\n848 1\n290 8\n318 4\n851 7\n311 3\n796 4\n309 2\n810 9\n655 8\n450 2\n678 6\n202 9\n315 3\n964 5\n506 4\n372 9\n70 0\n621 3\n987 1\n427 3\n868 5\n127 8\n737 0\n909 1\n646 8\n996 9\n12 9\n522 9\n635 2\n846 1\n703 9\n800 5\n638 3\n783 0\n205 5\n426 6\n778 6\n371 0\n431 5\n192 5\n803 2\n846 3\n354 9\n25 7\n657 4\n812 3\n45 8\n161 1\n269 3\n919 3\n618 7\n503 4\n563 8\n798 1\n53 2\n938 8\n749 4\n354 3\n494 0\n557 0\n358 8\n14 8\n859 4\n386 9\n668 2\n316 9\n326 2\n748 7\n456 5\n563 6\n765 4\n495 4\n839 2\n137 9\n412 2\n612 5\n427 2\n951 4\n7 2\n78 7\n748 1\n542 4\n105 3\n206 8\n510 8\n93 7\n50 8\n841 5\n904 3\n808 3\n53 0\n115 7\n80 3\n660 1\n527 2\n938 1\n504 5\n992 6\n422 5\n999 9\n463 3\n374 6\n503 6\n968 8\n942 9\n824 9\n299 5\n646 2\n412 9\n607 7\n387 1\n783 6\n607 4\n234 9\n509 1\n184 2\n649 6\n723 8\n438 6\n401 7\n284 9\n943 3\n841 7\n92 9\n427 7\n702 1\n364 7\n816 6\n766 6\n553 7\n977 3\n194 6\n864 2\n77 2\n687 4\n399 7\n927 9\n652 1\n471 6\n",
        "output": "140964608\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 475,
        "task_id": 3858,
        "test_case_id": 26,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n6366 3\n1993 0\n62 4\n9870 0\n2915 4\n6306 3\n9366 1\n156 1\n7103 1\n6049 0\n4758 4\n5184 4\n8914 0\n6299 1\n7805 2\n6376 1\n8046 2\n8040 3\n1677 3\n8411 0\n6654 1\n1710 0\n7197 2\n542 3\n1456 4\n4452 3\n4715 1\n3991 4\n7077 3\n885 3\n1443 1\n3853 0\n3835 2\n5640 1\n7522 1\n4012 1\n8655 1\n4189 2\n6252 1\n1913 1\n2699 2\n1597 1\n5731 1\n1193 0\n4531 3\n4032 2\n5502 3\n2098 1\n3461 0\n674 1\n3818 3\n578 3\n9164 4\n3122 1\n2729 4\n1507 1\n2098 0\n7111 4\n5319 3\n1802 3\n5088 0\n1664 3\n7981 4\n6838 2\n5279 1\n3592 3\n814 4\n5133 4\n9611 3\n3912 4\n7276 3\n6493 2\n9495 3\n3826 4\n4285 2\n792 4\n6728 3\n6755 4\n9472 1\n8667 4\n3800 0\n1027 1\n8599 1\n3037 3\n7458 2\n198 2\n7321 3\n8226 4\n22 3\n4678 2\n9772 4\n5582 1\n1516 3\n9669 0\n4626 0\n2639 1\n4521 0\n7103 2\n9361 0\n4168 4\n1753 3\n3975 3\n1495 3\n4280 1\n8328 2\n6816 0\n8669 1\n1453 0\n5393 0\n3647 4\n1261 2\n5691 1\n7030 0\n1903 4\n6050 3\n3628 2\n3290 2\n9708 1\n2855 2\n6680 2\n1439 0\n6574 3\n8487 4\n8631 1\n4888 0\n6378 2\n2159 2\n7148 4\n3389 3\n9813 2\n4951 0\n7160 3\n239 4\n4522 0\n3433 0\n6491 1\n2371 4\n6840 2\n9987 0\n7017 3\n641 2\n4680 4\n7439 0\n2923 2\n9959 2\n2542 4\n8028 2\n64 1\n8056 1\n2250 4\n4025 3\n1841 1\n2373 0\n3335 3\n7805 1\n1441 3\n1719 4\n8781 3\n2279 0\n4902 2\n8067 1\n6857 2\n4877 3\n2288 1\n4571 4\n5838 0\n6319 2\n6233 4\n1742 4\n6950 3\n9239 0\n3743 0\n3521 4\n1560 2\n7120 3\n1842 1\n490 1\n2509 2\n4891 0\n4102 4\n4568 3\n3493 4\n8526 3\n495 0\n6568 1\n4505 2\n6456 1\n9873 0\n8998 3\n1653 4\n8146 1\n5901 0\n7476 3\n7675 0\n7822 4\n3893 1\n4668 2\n5881 0\n6746 3\n2125 0\n",
        "output": "227970128\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 476,
        "task_id": 3858,
        "test_case_id": 27,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n1 4063\n2 6541\n4 773\n3 1561\n4 8820\n4 7914\n3 461\n4 9241\n2 9316\n3 9144\n1 9978\n4 7653\n0 1957\n3 7761\n4 7469\n2 3266\n1 2791\n0 3554\n3 5397\n2 8273\n2 8797\n2 3229\n1 3281\n4 6968\n2 6777\n0 238\n4 8659\n0 2587\n2 6715\n1 7213\n2 6595\n3 2522\n3 4024\n2 9561\n0 434\n1 8702\n3 3548\n2 7434\n1 6943\n2 1688\n2 8453\n0 2064\n4 4561\n1 4542\n3 3565\n1 979\n2 3249\n3 5237\n3 4021\n3 1960\n0 4453\n0 6870\n1 344\n2 5373\n3 7008\n4 1010\n3 4022\n2 473\n4 7271\n0 8354\n3 9607\n2 6335\n2 6207\n1 3568\n3 5884\n3 9629\n0 8275\n2 6791\n1 8146\n4 5257\n3 7912\n1 2690\n2 4296\n1 7999\n3 8232\n2 6973\n2 8097\n2 8582\n2 9111\n2 3972\n0 5776\n4 3211\n0 7724\n2 9110\n3 2939\n4 9355\n1 1612\n4 1768\n0 4472\n3 437\n0 336\n2 9512\n4 9164\n4 9930\n2 7908\n4 4614\n2 8336\n0 1255\n4 1980\n1 193\n3 7428\n3 8319\n0 3329\n2 9604\n3 2810\n2 5860\n2 643\n3 3885\n3 8895\n2 3839\n1 8414\n4 3860\n3 3937\n3 6704\n0 2517\n1 4363\n4 1953\n0 8360\n1 3579\n0 4382\n1 9577\n1 6643\n0 6776\n0 6699\n2 4429\n2 5532\n1 339\n1 741\n3 619\n3 7246\n3 6396\n1 5750\n4 3850\n1 7721\n1 9906\n0 8903\n1 9373\n0 3467\n0 5760\n1 521\n4 3719\n2 2644\n2 9537\n0 5568\n4 3736\n2 6778\n0 3460\n0 435\n3 6248\n1 3427\n0 9911\n4 8016\n3 3703\n1 9767\n4 3882\n3 2257\n1 1948\n1 323\n1 2739\n1 3183\n1 1813\n0 8279\n3 4914\n4 2297\n4 7605\n2 2175\n4 1298\n0 6302\n2 4613\n1 4131\n1 2343\n4 5626\n2 1074\n3 7174\n0 7271\n3 7141\n3 7317\n0 1390\n4 5700\n0 2372\n0 4599\n2 9926\n1 3095\n0 8564\n4 8785\n4 652\n0 451\n0 5450\n1 713\n3 5808\n4 8120\n2 2149\n2 756\n2 9746\n0 9616\n4 9131\n2 320\n2 8167\n3 2886\n1 1003\n",
        "output": "303575002\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 477,
        "task_id": 3858,
        "test_case_id": 28,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n3976 0\n5853 0\n7950 0\n2031 0\n9080 0\n7576 0\n1884 0\n800 0\n6329 0\n1361 0\n4001 0\n9314 0\n8634 0\n8737 0\n3491 0\n35 0\n9465 0\n5435 0\n4786 0\n4762 0\n5107 0\n8813 0\n7597 0\n5552 0\n5180 0\n7912 0\n8322 0\n8744 0\n8135 0\n9565 0\n2866 0\n5040 0\n9356 0\n6045 0\n2345 0\n5412 0\n3600 0\n7341 0\n9730 0\n7997 0\n4289 0\n3945 0\n6962 0\n1359 0\n1964 0\n9729 0\n2103 0\n2121 0\n4903 0\n3925 0\n6854 0\n344 0\n3983 0\n272 0\n4286 0\n90 0\n9849 0\n6865 0\n6694 0\n1054 0\n9788 0\n4346 0\n5585 0\n1531 0\n617 0\n2013 0\n3763 0\n1917 0\n3724 0\n9287 0\n4411 0\n8650 0\n4447 0\n592 0\n4280 0\n7988 0\n4946 0\n6714 0\n7456 0\n8937 0\n5732 0\n2653 0\n4754 0\n2236 0\n6737 0\n8565 0\n7370 0\n3392 0\n3059 0\n6731 0\n1908 0\n515 0\n3665 0\n455 0\n9608 0\n3317 0\n9081 0\n6926 0\n1084 0\n3471 0\n3280 0\n732 0\n9772 0\n7538 0\n7595 0\n3678 0\n9525 0\n4888 0\n8054 0\n8000 0\n4143 0\n7591 0\n5153 0\n5576 0\n8700 0\n2654 0\n4448 0\n5138 0\n5961 0\n2646 0\n8538 0\n1671 0\n8382 0\n7817 0\n8576 0\n9190 0\n4266 0\n9195 0\n8259 0\n7446 0\n2873 0\n7589 0\n5340 0\n1349 0\n2443 0\n7441 0\n1267 0\n1508 0\n3171 0\n2161 0\n7729 0\n677 0\n4542 0\n6677 0\n2316 0\n8994 0\n8008 0\n1710 0\n752 0\n8566 0\n7009 0\n1168 0\n5852 0\n2716 0\n3130 0\n7363 0\n374 0\n234 0\n2093 0\n9487 0\n1225 0\n8130 0\n9009 0\n5484 0\n3093 0\n7458 0\n7596 0\n1212 0\n7504 0\n4414 0\n8849 0\n299 0\n4522 0\n1811 0\n225 0\n1602 0\n4619 0\n9820 0\n8958 0\n6816 0\n8604 0\n8159 0\n5410 0\n467 0\n5241 0\n586 0\n9034 0\n9686 0\n7650 0\n3776 0\n1830 0\n1520 0\n2555 0\n9856 0\n6222 0\n4594 0\n7694 0\n9155 0\n9741 0\n1880 0\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 478,
        "task_id": 3858,
        "test_case_id": 29,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n0 3658\n0 9296\n0 4021\n0 5700\n0 5001\n0 8057\n0 1595\n0 8662\n0 4461\n0 8091\n0 1268\n0 6492\n0 2822\n0 4606\n0 259\n0 5219\n0 9184\n0 2537\n0 2240\n0 637\n0 5310\n0 2363\n0 6949\n0 6004\n0 3190\n0 2944\n0 3529\n0 4138\n0 9952\n0 492\n0 7289\n0 5269\n0 1638\n0 174\n0 5379\n0 271\n0 2187\n0 7065\n0 450\n0 762\n0 1273\n0 2335\n0 2490\n0 2635\n0 2470\n0 6813\n0 3394\n0 838\n0 865\n0 1239\n0 5056\n0 3754\n0 9191\n0 3494\n0 3688\n0 3886\n0 9034\n0 3771\n0 9108\n0 2356\n0 4863\n0 8624\n0 7162\n0 2266\n0 1605\n0 9454\n0 7653\n0 6523\n0 9756\n0 3352\n0 6289\n0 4267\n0 3565\n0 9113\n0 7521\n0 2340\n0 224\n0 5955\n0 3511\n0 5116\n0 5504\n0 5398\n0 4633\n0 6687\n0 3018\n0 1055\n0 8128\n0 9589\n0 6909\n0 1394\n0 4814\n0 1707\n0 6502\n0 3289\n0 2792\n0 8891\n0 5268\n0 7028\n0 6967\n0 3422\n0 5831\n0 3283\n0 171\n0 786\n0 7403\n0 609\n0 487\n0 4619\n0 7069\n0 3589\n0 6797\n0 9622\n0 364\n0 8253\n0 820\n0 2789\n0 3941\n0 7091\n0 7944\n0 8553\n0 8266\n0 4670\n0 8237\n0 7632\n0 3530\n0 592\n0 3837\n0 8131\n0 9208\n0 7733\n0 7432\n0 6378\n0 1034\n0 8552\n0 3723\n0 5207\n0 9651\n0 2957\n0 4916\n0 8002\n0 976\n0 9521\n0 9301\n0 3988\n0 4098\n0 1573\n0 6167\n0 4694\n0 5413\n0 3745\n0 4700\n0 7400\n0 4468\n0 4589\n0 5513\n0 9435\n0 1862\n0 8012\n0 6336\n0 6371\n0 3546\n0 3915\n0 2897\n0 7816\n0 803\n0 3425\n0 1624\n0 2749\n0 9642\n0 6237\n0 3342\n0 7615\n0 721\n0 2277\n0 7215\n0 6513\n0 5464\n0 575\n0 2781\n0 7806\n0 2230\n0 9568\n0 6090\n0 3695\n0 3137\n0 50\n0 452\n0 6957\n0 2727\n0 9619\n0 5394\n0 8917\n0 9750\n0 8545\n0 6320\n0 5698\n0 8082\n0 9700\n0 5871\n0 8532\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 479,
        "task_id": 3858,
        "test_case_id": 30,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n4 3690\n7939 8\n9566 9566\n8892 8892\n4 788\n4 3498\n4815 4815\n7455 8\n8465 8704\n5278 5278\n9273 7436\n4 7675\n3799 8\n8557 8557\n2732 8\n4179 8\n1184 6828\n3511 8\n4 5138\n6549 6549\n7271 8\n6527 8\n4 5796\n5886 5886\n4494 4494\n4 409\n4 5531\n9228 9767\n6574 4090\n4775 9696\n2254 4781\n4 3572\n4 9660\n4 4051\n9754 9996\n4 8272\n1981 8\n4 6155\n8223 8\n1751 8\n4 9269\n9028 9028\n6692 3041\n4 4176\n4945 8\n2962 8\n7962 7962\n2240 8\n3530 8\n9750 8\n3391 5\n4112 8017\n4 410\n6254 6254\n662 662\n4624 4624\n2707 2707\n5076 5076\n4 803\n4679 4679\n328 3164\n4 9379\n4 3076\n6992 8\n1085 1085\n4 1908\n7135 2915\n1417 1417\n4 3562\n6053 6053\n3145 8\n5484 8\n4 6060\n64 9265\n4 1109\n7746 7746\n5271 8\n4942 4942\n5452 8\n2620 7485\n6833 6833\n1090 2641\n1899 4046\n7441 8\n8802 8\n6332 6332\n3610 3610\n4 2865\n3956 3956\n4 6394\n4619 4619\n7027 8\n4 1145\n5801 8881\n856 856\n9254 8\n3074 8\n9588 9588\n4 6456\n5969 5969\n644 8\n1645 1645\n4 1393\n5286 1946\n4275 8\n9120 8634\n4 4921\n4 4365\n7086 9010\n526 9568\n5466 5466\n6838 6838\n2817 2074\n1861 8\n7362 7362\n7400 7400\n3903 8\n6638 8\n7916 7916\n4665 4541\n1900 8\n268 8017\n2687 2687\n4 4383\n5798 4682\n1921 1316\n578 8\n4 1041\n4 8421\n4 3895\n6909 6909\n1245 8\n5086 1796\n5814 9194\n4 2812\n4 4712\n2569 2569\n4003 4003\n382 4969\n1561 2216\n368 368\n78 78\n4 9752\n990 8\n9034 8\n5685 8\n4 457\n471 1373\n8708 8\n9091 9091\n4 3620\n7150 7150\n4 1219\n3972 8461\n6748 8\n3075 8\n4 2723\n4541 8\n4 5925\n2076 1358\n4209 4209\n4 9144\n5275 8\n1080 1080\n4 4424\n8393 4199\n4 5088\n5813 5813\n4 6990\n2698 3406\n6156 8\n4247 8\n4961 4961\n4592 8005\n7382 7662\n1703 2997\n2328 2328\n1802 1802\n4 2497\n4 1943\n4484 4484\n4 5320\n4 6648\n1133 8\n4 5956\n357 357\n2434 9301\n5626 417\n40 8\n2498 2393\n3337 3337\n673 673\n975 8\n224 8843\n4 7249\n1836 1836\n4 8957\n4 3671\n6812 8\n4 314\n",
        "output": "794059416\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 480,
        "task_id": 3858,
        "test_case_id": 31,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n4545 4224\n4243 4244\n2 7270\n2 4467\n4 1622\n950 4015\n6 333\n7299 261\n6 8179\n2201 6\n2872 2872\n1729 9\n6017 10\n3024 9\n5 6548\n5 6588\n5068 4563\n3 9331\n3 2012\n5 9849\n2 4916\n1352 6\n199 506\n4041 4045\n9237 6\n2 660\n3603 3600\n780 5838\n6661 6660\n5 2626\n6 3352\n6 1333\n2518 8\n3081 3959\n3475 3477\n4297 4296\n9164 1373\n6 1704\n6449 9\n6 4225\n8677 5481\n8419 8418\n8589 6\n4 7257\n5678 3821\n6671 6671\n6850 7\n7916 6587\n3243 3751\n7681 7681\n6779 6776\n3659 3661\n4 6847\n3 2353\n5 5536\n3667 10\n4 7823\n5314 6\n7050 8\n6 3621\n6393 7\n5533 1054\n1610 8\n8694 3838\n2858 5065\n5419 9\n1851 9\n1497 848\n9643 9643\n6 4377\n2734 5165\n3 9890\n3671 1472\n4663 4660\n1729 1729\n8313 5415\n2 5197\n3613 9\n7579 10\n2 7675\n1125 1126\n4 3428\n4926 7\n6712 7\n2 4551\n5843 5841\n5 7740\n9987 7\n5165 3757\n5 2912\n7502 7501\n6097 6098\n534 6383\n9617 4519\n3174 6\n1618 6497\n3074 5991\n8031 7296\n9083 8762\n9836 6\n6 7217\n2 4246\n849 10\n3409 6\n5361 6\n7137 8\n907 8\n3 8568\n7638 10\n930 9983\n6255 364\n2684 2684\n5302 7460\n5226 3731\n1119 9\n9978 2209\n7978 7360\n8191 7335\n872 9\n9199 9199\n5144 8415\n4568 4570\n469 470\n8575 8135\n7548 8\n7697 7699\n5 5610\n5620 7\n2152 7\n1122 7\n8081 6\n9275 9966\n49 6\n1363 1367\n3821 3820\n1962 6\n2 2211\n5 9604\n4 7992\n2213 7447\n5370 9\n6 8651\n964 6\n490 9\n3 8008\n9293 8\n5166 3091\n1343 1340\n8559 8558\n7799 10\n6 8122\n380 378\n9926 2021\n2070 1417\n9857 9853\n643 645\n3730 427\n5 9747\n7911 3720\n7589 9\n9870 9872\n3691 3690\n7309 7189\n351 352\n6303 7\n4 6534\n560 8\n7531 8\n6 693\n1674 1673\n4 3110\n625 8\n8220 8221\n3 1746\n5509 7\n1828 2186\n9947 9\n3363 2992\n7919 9185\n4 4209\n2450 7\n2463 2465\n6742 369\n130 9\n6 3298\n2 5855\n2176 2174\n1778 4135\n6705 6703\n4241 7727\n3 2489\n9395 9399\n2 6940\n2 8747\n5011 5013\n4431 4432\n3 5190\n6932 10\n5499 4895\n4787 4789\n",
        "output": "982870470\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 481,
        "task_id": 3858,
        "test_case_id": 32,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n2993 8\n2 5254\n4576 2677\n75 8859\n6292 6\n2 5731\n8047 8050\n977 8871\n4760 4760\n4379 7948\n8910 8909\n7567 2685\n5082 5081\n187 7767\n3 8853\n3020 1497\n767 769\n6489 8\n3 5996\n2 4954\n6 2802\n3 5392\n9947 9945\n8174 2330\n2 8795\n6525 6525\n9097 8142\n780 8\n6414 10\n7978 7977\n6506 791\n7168 7169\n5 9818\n5400 7261\n4495 4496\n2599 8386\n6727 7329\n9702 5442\n6255 1610\n9080 9529\n9885 3213\n3 46\n3538 4832\n3 6830\n7355 1987\n9823 1725\n550 9647\n939 6\n3337 3335\n5364 5388\n4 8857\n8702 5157\n8876 7\n3280 10\n1585 1581\n6638 4993\n2738 2740\n9345 10\n6 9986\n2444 2443\n6992 2724\n8587 10\n6 9792\n9574 7\n9792 9793\n4952 4948\n8919 8918\n916 7465\n5 7793\n4395 6853\n5592 2630\n2 5260\n9208 9207\n2 4349\n2341 2342\n9254 6344\n1406 1403\n4607 7\n7291 3978\n4281 10\n8551 4723\n5215 5217\n4 2782\n3 1820\n2 7713\n4887 6044\n7862 10\n6649 6645\n1346 9\n5 5595\n6 254\n2 2170\n6561 8302\n6404 6406\n5 9219\n6 1233\n4140 9\n1657 3465\n9894 7342\n5 8772\n1587 2475\n5872 10\n7480 7478\n3 8818\n3468 3465\n3854 3858\n9892 10\n3942 10\n4 6217\n5 2224\n6664 6666\n5539 7\n6187 6187\n7398 286\n6847 8\n5538 5537\n21 24\n6 8959\n5 8942\n3454 8\n1909 10\n5 8809\n2147 2148\n6949 6949\n1208 8\n2661 7456\n8722 6\n3345 3346\n3969 3973\n8334 8336\n3 245\n240 1008\n4 762\n5 5885\n4 8053\n1207 9\n4 4442\n4 1432\n6071 6074\n1805 9\n1616 1617\n2911 7\n2 811\n3820 3822\n8345 8344\n2119 2121\n3999 560\n6638 6636\n5390 9061\n7598 8\n6817 6481\n5469 9\n2 4646\n4746 4453\n9924 9924\n7972 10\n2 1564\n7821 6113\n7249 7\n4 8362\n2 187\n9167 7\n2156 2156\n7611 10\n5605 1788\n1837 1861\n4468 7\n6267 155\n2 7636\n6 7223\n8970 8\n5026 10\n3913 4042\n2279 7\n1593 6\n6844 5290\n4980 4255\n9882 9881\n5282 6\n8754 8\n4 6498\n9889 6476\n4 9822\n4329 2674\n5 9277\n5 8346\n3 8437\n9516 10\n3929 3929\n967 6\n7923 7926\n9203 9204\n3 9476\n3 5684\n4 3215\n913 10\n7672 10\n1321 643\n6 1162\n3 1732\n",
        "output": "982820926\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 482,
        "task_id": 3858,
        "test_case_id": 33,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n2 3427\n2391 10\n7511 8\n3338 3339\n9093 7\n6868 9\n9803 2101\n7346 9\n3 6139\n1095 1664\n2124 10\n794 794\n6782 9\n7693 7694\n4344 6\n8952 8952\n6322 6\n2 4405\n4558 7\n6024 1395\n6857 9\n923 924\n9432 2640\n8351 7\n2206 6\n111 6\n5003 5867\n4908 4908\n3 6739\n8120 8119\n9573 7\n3009 3006\n2 8273\n2780 2782\n473 9157\n2 7186\n3117 4826\n6 9647\n530 9\n1149 2231\n4219 7\n2317 9\n4 5762\n8841 8\n7695 1188\n3 9800\n5914 6127\n5873 10\n123 123\n3430 7\n2916 6445\n6 6209\n476 6\n1657 8976\n1163 1161\n6 3341\n9910 8\n4176 4175\n9110 9110\n5540 9169\n2386 6\n6238 6932\n9834 10\n1206 9\n8813 4228\n9055 9054\n6 5034\n1695 6\n99 10\n8696 5509\n9440 9440\n6018 8\n2 9150\n7593 4647\n4810 4809\n6351 1135\n3834 8868\n8978 6\n3091 3090\n5 4939\n2 2611\n5 8814\n8372 8372\n2 488\n7577 4192\n4298 7\n2176 2174\n3 9605\n5 5238\n98 4179\n6 5942\n6 3673\n4 8588\n4 8457\n4 6230\n492 8006\n2 4752\n5 9577\n3104 3108\n9202 8\n3 3322\n2 136\n3 736\n9477 7639\n5 154\n6 9963\n1584 9673\n7377 1096\n4021 158\n5057 6\n1493 6\n6 7322\n3 5034\n2350 2353\n6 8383\n4 5034\n5733 6\n6753 6755\n1862 4764\n9298 7\n5047 10\n3359 3360\n1271 6\n3871 3867\n3 3075\n313 8\n7854 10\n1607 7\n9428 10\n4 4035\n6361 6361\n7210 2271\n366 746\n4955 4953\n1371 1371\n1997 9908\n4 4777\n5319 5317\n3 8742\n9846 9847\n6058 491\n8633 10\n5863 5863\n7583 10\n4 3888\n5813 8186\n5142 7\n2 562\n1464 1468\n3761 3758\n7254 6\n5 3475\n6848 10\n4031 4029\n6331 6332\n312 312\n2962 7\n9855 8\n9894 9893\n8095 9467\n5303 5306\n2 3282\n5025 3139\n8047 7\n6339 4723\n9219 10\n3 9884\n4285 4288\n3 4755\n2227 2228\n4 8463\n952 8\n622 6977\n4557 10\n9799 7985\n899 9\n5 4288\n3 9145\n1645 7\n9418 9414\n2 6080\n1878 1877\n8161 8189\n9375 9378\n7481 10\n7063 10\n9496 9496\n473 6\n9511 1444\n5 355\n3260 3262\n4262 9\n5114 1042\n1069 1069\n4 6856\n5904 6\n1740 10\n3 7800\n2075 7015\n4 7708\n",
        "output": "982774695\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 483,
        "task_id": 3858,
        "test_case_id": 34,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n5 1625\n7694 8\n1778 1780\n8535 8535\n1071 9\n5489 4617\n4821 4821\n4541 8\n9427 9428\n2731 6\n8320 6806\n5598 8\n2425 2425\n2 6758\n4020 4018\n359 4008\n5927 8013\n7557 2658\n2 6960\n7019 4059\n9129 9130\n8545 8\n4 4229\n5768 1092\n7607 7608\n8412 8413\n1339 1341\n191 1305\n6553 1406\n6 6678\n5 5501\n4445 4445\n8329 3094\n6029 7\n3 6995\n9490 7\n3 5602\n3690 6436\n9825 7\n8254 8255\n599 8\n6122 9\n8128 7\n9588 10\n3002 9514\n7458 7460\n7082 7082\n1927 7685\n4 1083\n3 346\n2297 1767\n3810 8621\n9072 9068\n712 713\n5966 7\n9454 10\n5292 5291\n6214 9\n7529 3258\n1067 3946\n4124 4122\n6327 6327\n5461 10\n6 6866\n2156 8468\n3586 8\n3650 8495\n5903 6\n6819 5222\n9764 1788\n3563 1764\n3617 3781\n1243 10\n3 296\n6101 9194\n6 2200\n4 2951\n7425 7424\n2083 4457\n4 1125\n8583 8580\n4831 4830\n3 3636\n7036 8905\n5744 7\n3471 6\n1522 7\n4 2993\n8987 8988\n4295 4038\n9488 9485\n4 5738\n5051 5050\n5 3418\n1479 4757\n790 793\n107 109\n8651 1758\n4056 4054\n5 2172\n520 523\n6215 6219\n3 7878\n6206 6206\n8393 8659\n4830 4833\n670 1806\n2721 2720\n4 6501\n4574 4570\n980 4103\n3306 3306\n4 1462\n5989 5990\n3 6576\n5117 10\n8241 8\n9538 3425\n3 5467\n3558 3561\n2 1360\n4373 4377\n3 1450\n3 9310\n4570 4568\n7854 9\n1361 9\n1804 9750\n4759 8\n5559 1196\n287 8876\n995 995\n5 5711\n328 7\n6 4563\n1294 8435\n1533 6631\n9405 8\n5890 2680\n6966 5713\n4 4782\n2 2118\n6864 9572\n254 7\n6113 6112\n8729 7\n2444 2445\n5 1075\n6115 6114\n4820 4823\n6654 6655\n5 5362\n2 4324\n708 708\n2866 2866\n6427 2158\n2871 63\n3125 10\n5137 10\n2 7618\n722 1361\n4 3514\n5402 293\n5089 3931\n2 5153\n2 5586\n512 508\n3987 4007\n2276 1255\n9361 4369\n410 5723\n70 71\n2097 2096\n6181 9857\n8696 8698\n7746 7746\n7866 8104\n9808 9809\n6 4836\n8177 303\n6964 8047\n2120 2118\n9448 3388\n5500 5499\n546 6\n4661 4659\n6474 8389\n9936 9936\n9379 9375\n7187 7189\n3269 7\n5007 5006\n3827 9\n4008 6\n6 2462\n4 8019\n6146 9\n2486 9558\n6 7845\n9804 274\n",
        "output": "982885412\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 484,
        "task_id": 3858,
        "test_case_id": 35,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n130 6901\n73 5329\n176 979\n55 3025\n34 1156\n38 1444\n25 625\n187 4972\n24 576\n152 3106\n161 5923\n83 6889\n138 9045\n157 4651\n105 1026\n134 7957\n87 7569\n190 6103\n39 1521\n160 5602\n82 6724\n46 2116\n120 4401\n14 196\n64 4096\n35 1225\n124 5377\n144 738\n117 3690\n91 8281\n106 1237\n192 6867\n6 36\n136 8497\n153 3411\n145 1027\n65 4225\n172 9586\n17 289\n199 9604\n146 1318\n101 202\n107 1450\n60 3600\n163 6571\n32 1024\n53 2809\n59 3481\n184 3859\n100 1\n47 2209\n28 784\n114 2997\n104 817\n102 405\n33 1089\n69 4761\n93 8649\n58 3364\n78 6084\n54 2916\n174 279\n8 64\n80 6400\n171 9243\n88 7744\n197 8812\n116 3457\n30 900\n185 4228\n179 2044\n2 4\n11 121\n112 2545\n1 1\n167 7891\n3 9\n57 3249\n90 8100\n7 49\n183 3492\n26 676\n56 3136\n151 2803\n15 225\n75 5625\n111 2322\n162 6246\n118 3925\n68 4624\n98 9604\n63 3969\n67 4489\n181 2764\n45 2025\n196 8419\n188 5347\n125 5626\n142 166\n29 841\n84 7056\n126 5877\n12 144\n77 5929\n37 1369\n95 9025\n191 6484\n189 5724\n79 6241\n155 4027\n103 610\n99 9801\n51 2601\n113 2770\n122 4885\n94 8836\n119 4162\n148 1906\n5 25\n41 1681\n21 441\n149 2203\n43 1849\n115 3226\n182 3127\n110 2101\n123 5130\n13 169\n137 8770\n193 7252\n173 9931\n109 1882\n19 361\n139 9322\n156 4338\n165 7227\n48 2304\n23 529\n0 0\n4 16\n72 5184\n198 9207\n71 5041\n147 1611\n86 7396\n40 1600\n194 7639\n127 6130\n154 3718\n22 484\n61 3721\n42 1764\n66 4356\n159 5283\n49 2401\n180 2403\n158 4966\n92 8464\n16 256\n81 6561\n150 2502\n121 4642\n62 3844\n76 5776\n52 2704\n169 8563\n133 7690\n89 7921\n20 400\n170 8902\n97 9409\n36 1296\n50 2500\n164 6898\n175 628\n141 9882\n131 7162\n168 8226\n85 7225\n74 5476\n166 7558\n143 451\n27 729\n135 8226\n10 100\n108 1665\n186 4599\n132 7425\n195 8028\n9 81\n70 4900\n18 324\n44 1936\n177 1332\n140 9601\n31 961\n129 6642\n128 6385\n178 1687\n96 9216\n",
        "output": "982904590\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 485,
        "task_id": 3858,
        "test_case_id": 36,
        "question": "You are given N points (x_i,y_i) located on a two-dimensional plane.\nConsider a subset S of the N points that forms a convex polygon.\nHere, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.\nFor example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.\nFor a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.\nCompute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo 998244353.\n\n-----Constraints-----\n - 1≤N≤200\n - 0≤x_i,y_i<10^4 (1≤i≤N)\n - If i≠j, x_i≠x_j or y_i≠y_j.\n - x_i and y_i are integers.\n\n-----Input-----\nThe input is given from Standard Input in the following format:\nN\nx_1 y_1\nx_2 y_2\n:\nx_N y_N\n\n-----Output-----\nPrint the sum of all the scores modulo 998244353.\n\n-----Sample Input-----\n4\n0 0\n0 1\n1 0\n1 1\n\n-----Sample Output-----\n5\n\nWe have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\nfrom fractions import gcd\\nfrom collections import Counter\\n\\n\\\"\\\"\\\"\\n\\u9069\\u5f53\\u306b\\u90e8\\u5206\\u96c6\\u5408X\\u3092\\u3068\\u308a\\u3001\\u51f8\\u5305 S \\u3068\\u3057\\u3066\\u3001S\\u306b1\\u70b9\\u8a08\\u4e0a\\u3059\\u308c\\u3070\\u3088\\u3044\\n\\u3053\\u308c\\u3060\\u30682^N\\u70b9\\u5f97\\u3089\\u308c\\u308b\\n\\u305f\\u3060\\u3057\\u3001\\u51f8\\u5305\\u306e\\u9762\\u7a4d\\u304c0\\u3068\\u306a\\u308b\\u5834\\u5408\\u304c\\u4f8b\\u5916\\n\\u7a7a\\u96c6\\u5408\\u30011\\u70b9\\u306e\\u5834\\u5408\\u3068\\u3001\\u7dda\\u5206\\u306e\\u5834\\u5408\\u3092\\u9664\\u5916\\u3059\\u308b\\n\\n\\\"\\\"\\\"\\n\\nMOD = 998244353\\nN = int(input())\\nXY = [[int(x) for x in input().split()] for _ in range(N)]\\n\\nanswer = pow(2,N,MOD)\\nanswer -= N + 1# \\u7a7a\\u30011\\u70b9\\nfor i,(x,y) in enumerate(XY):\\n    # i \\u3092\\u9078\\u3073\\u3001i+1\\u756a\\u76ee\\u4ee5\\u4e0a\\u306e\\u3046\\u3061\\u3044\\u304f\\u3064\\u304b\\u3092\\u9078\\u3093\\u3067\\u7dda\\u5206\\u3068\\u3059\\u308b\\n    pts = []\\n    for x1, y1 in XY[i+1:]:\\n        dx, dy = x1-x, y1-y\\n        g = gcd(dx, dy)\\n        dx //= g\\n        dy //= g\\n        # \\u6a19\\u6e96\\u5316\\n        if dx < 0:\\n            dx, dy = -dx, -dy\\n        elif dx == 0:\\n            dy = 1\\n        pts.append((dx,dy))\\n    c = Counter(pts)\\n    for v in c.values():\\n        answer -= pow(2,v,MOD) - 1\\n\\nanswer %= MOD\\nprint(answer)\", \"import math\\nmod = 998244353\\nn = int(input())\\np = [list(map(int, input().split())) for i in range(n)]\\npow2 = [1]\\nfor i in range(n):\\n    pow2.append(pow2[-1] * 2 % mod)\\nused = [[False] * n for i in range(n)]\\nret = (pow2[n] - 1 - n - n * (n - 1) / 2) % mod\\nfor i in range(n):\\n    for j in range(i):\\n        if used[i][j]:\\n            continue\\n        inline = [i, j]\\n        for k in range(n):\\n            if k == i or k == j:\\n                continue\\n            if (p[i][1] - p[k][1]) * (p[j][0] - p[k][0]) == (p[j][1] - p[k][1]) * (p[i][0] - p[k][0]):\\n                inline.append(k)\\n        for k in range(len(inline)):\\n            for l in range(len(inline)):\\n                used[inline[k]][inline[l]] = True\\n        v = len(inline)\\n        ret = (ret + mod - pow2[v] + 1 + v + v * (v - 1) // 2) % mod\\nprint((int(ret)))\\n\", \"#!/usr/bin/env python3\\n\\n\\ndef main():\\n\\n    N = int(input())\\n    points = []\\n    for i in range(N):\\n        points.append(list(map(int, input().split())))\\n\\n    x = [p[0] for p in points]\\n    y = [p[1] for p in points]\\n\\n    M = 998244353\\n\\n    c = 1 + N + N * (N - 1) // 2\\n    for i in range(N):\\n        xi = x[i]\\n        yi = y[i]\\n        dx = [xj - xi for xj in x]\\n        dy = [yj - yi for yj in y]\\n\\n        for j in range(i + 1, N):\\n            xj = dx[j]\\n            yj = dy[j]\\n            cc = 1\\n            for k in range(j + 1, N):\\n                if xj * dy[k] - dx[k] * yj == 0:\\n                    cc *= 2\\n                    cc %= M\\n            c += cc - 1\\n\\n    r = 1\\n    for i in range(N):\\n        r *= 2\\n        r %= M\\n\\n    r -= c\\n    r %= M\\n\\n    print(r)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"from collections import Counter\\nfrom fractions import gcd\\n\\nn = int(input())\\nxys = [tuple(map(int, input().split())) for _ in range(n)]\\nMOD = 998244353\\n\\nexcludes = 0\\nfor i, (x1, y1) in enumerate(xys):\\n    slopes = []\\n    for x2, y2 in xys[i + 1:]:\\n        dx, dy = x2 - x1, y2 - y1\\n        if dx == 0:\\n            slopes.append(1j)\\n        elif dy == 0:\\n            slopes.append(1)\\n        else:\\n            m = gcd(dx, dy)\\n            slopes.append(dx // m + dy // m * 1j)\\n    for c in list(Counter(slopes).values()):\\n        if c > 1:\\n            excludes += 2 ** c - c - 1\\n    excludes %= MOD\\n\\nprint(((pow(2, n, MOD) - excludes - (n * (n - 1) // 2) - n - 1) % MOD))\\n\", \"def gcd(a,b):\\n    if b == 0:\\n        return a\\n    return gcd(b,a%b)\\ndef inpl(): return [int(i) for i in input().split()]\\nfrom collections import defaultdict\\nH = defaultdict(lambda: 0)\\nN = int(input())\\ndot = [()]*N\\nmod = 998244353\\nfor i in range(N):\\n    x, y = inpl()\\n    dot[i] = (x, y)\\nfor i in range(N):\\n    for j in range(i+1,N):\\n        x1, y1 = dot[i]\\n        x2, y2 = dot[j]\\n        A = y1 - y2\\n        B = x2 - x1\\n        C = y1*x2 - x1*y2\\n        if A < 0:\\n            A *= -1\\n            B *= -1\\n            C *= -1\\n        gcdabc = gcd(gcd(A,B), C)\\n        A //= gcdabc\\n        B //= gcdabc\\n        C //= gcdabc\\n        H[(A, B, C)] += 1\\nans = (2**N - N - 1 )%mod\\nfor i in H.values():\\n    i = int((1+(1+8*i)**(1/2))/2)\\n    ans -= (2**i - i - 1)%mod\\nprint(ans%mod)\", \"from collections import Counter\\n\\ndef __starting_point():\\n    N = int(input())\\n    xy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\n    modulo_num = 998244353\\n\\n    duplicate_list = [0] * (N + 1)\\n    for i in range(N):\\n        xi, yi = xy_list[i]\\n        gradient_list = []\\n        for j in range(N):\\n            xj, yj = xy_list[j]\\n            if xi != xj:\\n                gradient_list.append((yj - yi) / (xj - xi))\\n            elif yi != yj:\\n                gradient_list.append(100000)\\n\\n        counter = Counter(gradient_list)\\n        for k in list(counter.values()):\\n            duplicate_list[k + 1] += 1\\n\\n    ans = pow(2, N, modulo_num)\\n    ans -= 1\\n    ans -= N\\n    for i in range(2, N + 1):\\n        cnt = duplicate_list[i] // i\\n        ans -= cnt * (pow(2, i, modulo_num) - i - 1)\\n    ans = ans % modulo_num\\n    print(ans)\\n\\n__starting_point()\", \"# seishin.py\\nN = int(input())\\nP = [list(map(int, input().split())) for i in range(N)]\\n\\nMOD = 998244353\\n\\nans = pow(2, N, MOD) - 1 - N\\nu = set()\\nfor i in range(N):\\n    xi, yi = P[i]\\n    for j in range(i+1, N):\\n        xj, yj = P[j]\\n        if (i, j) in u:\\n            continue\\n        cnt = 0\\n        Q = {i, j}\\n        for k in range(N):\\n            xk, yk = P[k]\\n            if (xj - xi)*(yk - yi) == (xk - xi)*(yj - yi):\\n                cnt += 1\\n                Q.add(k)\\n        for p in Q:\\n            for q in Q:\\n                u.add((p, q))\\n        ans -= pow(2, cnt, MOD) - cnt - 1\\nprint(ans % MOD)\", \"from fractions import Fraction\\nfrom collections import defaultdict\\nfrom itertools import combinations\\nclass Combination:\\n    \\\"\\\"\\\"\\n    O(n)\\u306e\\u524d\\u8a08\\u7b97\\u30921\\u56de\\u884c\\u3046\\u3053\\u3068\\u3067\\uff0cO(1)\\u3067nCr mod m\\u3092\\u6c42\\u3081\\u3089\\u308c\\u308b\\n    n_max = 10**6\\u306e\\u3068\\u304d\\u524d\\u51e6\\u7406\\u306f\\u7d04950ms (PyPy\\u306a\\u3089\\u7d04340ms, 10**7\\u3067\\u7d041800ms)\\n    \\u4f7f\\u7528\\u4f8b\\uff1a\\n    comb = Combination(1000000)\\n    print(comb(5, 3))  # 10\\n    \\\"\\\"\\\"\\n    def __init__(self, n_max, mod=10**9+7):\\n        self.mod = mod\\n        self.modinv = self.make_modinv_list(n_max)\\n        self.fac, self.facinv = self.make_factorial_list(n_max)\\n\\n    def __call__(self, n, r):\\n        return self.fac[n] * self.facinv[r] % self.mod * self.facinv[n-r] % self.mod\\n\\n    def make_factorial_list(self, n):\\n        # \\u968e\\u4e57\\u306e\\u30ea\\u30b9\\u30c8\\u3068\\u968e\\u4e57\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        # self.make_modinv_list()\\u304c\\u5148\\u306b\\u5b9f\\u884c\\u3055\\u308c\\u3066\\u3044\\u308b\\u5fc5\\u8981\\u304c\\u3042\\u308b\\n        fac = [1]\\n        facinv = [1]\\n        for i in range(1, n+1):\\n            fac.append(fac[i-1] * i % self.mod)\\n            facinv.append(facinv[i-1] * self.modinv[i] % self.mod)\\n        return fac, facinv\\n\\n    def make_modinv_list(self, n):\\n        # 0\\u304b\\u3089n\\u307e\\u3067\\u306emod\\u9006\\u5143\\u306e\\u30ea\\u30b9\\u30c8\\u3092\\u8fd4\\u3059 O(n)\\n        modinv = [0] * (n+1)\\n        modinv[1] = 1\\n        for i in range(2, n+1):\\n            modinv[i] = self.mod - self.mod//i * modinv[self.mod%i] % self.mod\\n        return modinv\\n\\nN = int(input())\\nXY = [list(map(int, input().split())) for i in range(N)]\\n\\nD = defaultdict(int)\\nfor (x1, y1), (x2, y2) in combinations(XY, 2):\\n    if x1 != x2:\\n        a = Fraction(y2-y1, x2-x1)\\n        b = Fraction(y1) - a * Fraction(x1)\\n        D[(a, b)] += 1\\n    else:\\n        D[(None, x1)] += 1\\n\\nmod = 998244353\\ncomb = Combination(N+1, mod)\\nA = [0] * (N+1)\\ncoA = [0] * (N+1)\\nans = 0\\nfor i in range(3, N+1):\\n    ans += comb(N, i)\\n    ans %= mod\\nfor i in range(3, N+1):\\n    for j in range(i-2):\\n        A[i] += (i-2-j)*(1<<j)\\n    coA[i] = coA[i-1]+A[i]\\nfor (a, b), n in D.items():\\n    if n==1:\\n        continue\\n    n = int((n<<1)**0.5)+1\\n    ans -= coA[n]\\n    ans %= mod\\nprint(ans)\", \"from collections import *\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)\\nint1 = lambda x: int(x) - 1\\np2D = lambda x: print(*x, sep=\\\"\\\\n\\\")\\ndef MI(): return map(int, sys.stdin.readline().split())\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\n\\ndef gcd(a, b):\\n    if b == 0: return a\\n    return gcd(b, a % b)\\n\\ndef red(a, b, c):\\n    if a == 0 and b < 0: b, c = -b, -c\\n    if a < 0: a, b, c = -a, -b, -c\\n    g = gcd(a, gcd(abs(b), abs(c)))\\n    return a // g, b // g, c // g\\n\\ndef main():\\n    md = 998244353\\n    n = int(input())\\n    xy = LLI(n)\\n    cnt_online = {}\\n    # \\u5404\\uff12\\u70b9\\u3092\\u7d50\\u3076\\u76f4\\u7dda\\u3092ax+by+c=0\\u306e(a,b,c)\\u3067\\u8868\\u3057\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u306b\\u3044\\u304f\\u3064\\u306e\\u70b9\\u304c\\u3042\\u308b\\u304b\\u30ab\\u30a6\\u30f3\\u30c8\\u3059\\u308b\\n    for i in range(n):\\n        x0, y0 = xy[i]\\n        counted = set()\\n        for j in range(i):\\n            x1, y1 = xy[j]\\n            a = y0 - y1\\n            b = x1 - x0\\n            c = -a * x0 - b * y0\\n            a, b, c = red(a, b, c)\\n            if (a, b, c) in counted: continue\\n            counted.add((a, b, c))\\n            cnt_online.setdefault((a, b, c), 1)\\n            cnt_online[(a, b, c)] += 1\\n    # print(cnt_online)\\n    # \\u5404\\u76f4\\u7dda\\u4e0a\\u3067\\u30012\\u70b9\\u4ee5\\u4e0a\\u3067\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u70b9\\u306e\\u9078\\u3073\\u65b9\\u3092\\u6570\\u3048\\u308b\\n    sum_online = 0\\n    for plot_n in cnt_online.values():\\n        sum_online += pow(2, plot_n, md) - 1 - plot_n\\n        sum_online %= md\\n    # \\u3059\\u3079\\u3066\\u306e2\\u70b9\\u4ee5\\u4e0a\\u306e\\u9078\\u3073\\u65b9\\u304b\\u3089\\u3001\\u591a\\u89d2\\u5f62\\u304c\\u3067\\u304d\\u306a\\u3044\\u3082\\u306e\\u3092\\u5f15\\u304f\\n    ans = pow(2, n, md) - 1 - n - sum_online\\n    print(ans % md)\\n\\nmain()\\n\", \"# E\\nfrom collections import Counter\\n\\nN = int(input())\\nxy_list = [list(map(int, input().split())) for _ in range(N)]\\n\\nM = 998244353\\n\\nres = pow(2, N, M)\\n# 0 points\\nres -= 1\\n# 1 points\\nres -= N\\n\\n# all lines\\nline_cnt = [0]*(N+1)\\n\\nfor i in range(N):\\n    xi, yi = xy_list[i]\\n    angle_list = []\\n    for j in range(N):\\n        xj, yj = xy_list[j]\\n        if xj != xi:\\n            angle_list.append((yj-yi)/(xj-xi))\\n        elif yj != yi:\\n            angle_list.append(10000.0)\\n        \\n    # count points in same line\\n    cnt_i = Counter(angle_list)\\n    for k in cnt_i.values():\\n        line_cnt[k+1] += 1\\n\\nfor i in range(2, N+1):\\n    cnt = line_cnt[i] // i\\n    res -= cnt*(pow(2, i, M)-i-1)\\n\\nres = res % M\\nprint(res)\", \"import sys\\nfrom collections import defaultdict\\n\\n# sys.stdin = open('e1.in')\\n\\n\\ndef read_int_list():\\n    return list(map(int, input().split()))\\n\\n\\ndef read_int():\\n    return int(input())\\n\\n\\ndef read_str_list():\\n    return input().split()\\n\\n\\ndef read_str():\\n    return input()\\n\\n\\ndef gcd(a, b):\\n    if b == 0:\\n        return a\\n    return gcd(b, a % b)\\n\\n\\ndef solve():\\n    def collinear(x0, y0, x1, y1):\\n        return x0 * y1 == x1 * y0\\n\\n    def aligned(i, j, k):\\n        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])\\n\\n    n = read_int()\\n    mod = 998244353\\n    res = pow(2, n, mod) - n - 1\\n    x, y = list(zip(*[read_int_list() for i in range(n)]))\\n    lines = defaultdict(set)\\n    for i in range(n):\\n        for j in range(i + 1, n):\\n            a = y[i] - y[j]\\n            b = x[j] - x[i]\\n            g = gcd(a, b)\\n            a //= g\\n            b //= g\\n            if a < 0 or (a == 0 and b < 0):\\n                a, b = -a, -b\\n            c = -(a * x[i] + b * y[i])\\n            lines[(a, b, c)].add(i)\\n            lines[(a, b, c)].add(j)\\n    for k, v in list(lines.items()):\\n        m = len(v)\\n        res -= pow(2, m, mod) - m - 1\\n    res %= mod\\n    return res\\n\\n\\ndef main():\\n    res = solve()\\n    print(res)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "competition",
        "input": "200\n56 106\n104 202\n68 130\n185 361\n92 178\n49 92\n23 43\n43 80\n168 327\n133 259\n122 238\n89 172\n16 29\n147 285\n9 15\n22 41\n51 96\n157 305\n158 307\n83 160\n150 291\n137 267\n60 114\n97 188\n96 186\n74 142\n129 251\n105 204\n176 343\n75 144\n106 206\n29 54\n202 395\n190 371\n71 136\n35 66\n170 331\n93 180\n100 194\n197 385\n134 261\n67 128\n163 317\n130 253\n153 297\n146 283\n28 52\n66 126\n24 45\n204 399\n33 62\n177 345\n98 190\n124 242\n14 25\n196 383\n30 56\n55 104\n110 214\n47 88\n112 218\n8 13\n95 184\n79 152\n62 118\n160 311\n77 148\n27 50\n125 244\n88 170\n128 249\n41 76\n156 303\n201 393\n10 17\n161 313\n166 323\n144 279\n18 33\n73 140\n123 240\n152 295\n116 226\n15 27\n139 271\n171 333\n65 124\n154 299\n99 192\n173 337\n188 367\n107 208\n78 150\n141 273\n32 60\n80 154\n17 31\n193 377\n72 138\n192 375\n40 74\n172 335\n64 122\n7 11\n61 116\n138 269\n175 341\n169 329\n199 389\n54 102\n3 3\n101 196\n70 134\n52 98\n120 234\n21 39\n19 35\n149 289\n36 68\n31 58\n178 347\n58 110\n85 164\n126 245\n111 216\n108 210\n119 232\n195 381\n136 265\n34 64\n20 37\n63 120\n84 162\n48 90\n189 369\n45 84\n165 321\n86 166\n145 281\n184 359\n6 9\n5 7\n183 357\n53 100\n82 158\n42 78\n114 222\n191 373\n25 46\n187 365\n38 72\n194 379\n50 94\n76 146\n4 5\n181 353\n90 174\n174 339\n87 168\n155 301\n26 48\n167 325\n117 228\n142 275\n11 19\n148 287\n151 293\n143 277\n91 176\n179 349\n81 156\n180 351\n37 70\n135 263\n200 391\n182 355\n69 132\n162 315\n132 257\n115 224\n59 112\n103 200\n102 198\n109 212\n13 23\n127 247\n94 182\n113 220\n12 21\n46 86\n118 230\n164 319\n198 387\n159 309\n44 82\n57 108\n203 397\n186 363\n121 236\n131 255\n",
        "output": "553862327\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/arc082/tasks/arc082_c"
    },
    {
        "id": 486,
        "task_id": 4080,
        "test_case_id": 1,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n",
        "output": "6\n2\n1 4 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 487,
        "task_id": 4080,
        "test_case_id": 2,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n",
        "output": "7\n2\n2 3 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 488,
        "task_id": 4080,
        "test_case_id": 3,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "1 0\n1000000\n",
        "output": "0\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 489,
        "task_id": 4080,
        "test_case_id": 4,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "1 0\n-1\n",
        "output": "0\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 490,
        "task_id": 4080,
        "test_case_id": 5,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "1 1\n-10\n1 1\n",
        "output": "0\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 491,
        "task_id": 4080,
        "test_case_id": 6,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "1 2\n-7\n1 1\n1 1\n",
        "output": "0\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 492,
        "task_id": 4080,
        "test_case_id": 7,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "1 5\n9\n1 1\n1 1\n1 1\n1 1\n1 1\n",
        "output": "0\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 493,
        "task_id": 4080,
        "test_case_id": 8,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "1 50\n5\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n",
        "output": "0\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 494,
        "task_id": 4080,
        "test_case_id": 9,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "1 100\n-4\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n",
        "output": "0\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 495,
        "task_id": 4080,
        "test_case_id": 10,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "2 0\n9 0\n",
        "output": "9\n0\n\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 496,
        "task_id": 4080,
        "test_case_id": 11,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "2 1\n5 8\n1 1\n",
        "output": "4\n1\n1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 497,
        "task_id": 4080,
        "test_case_id": 12,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "2 2\n-4 7\n2 2\n2 2\n",
        "output": "11\n0\n\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 498,
        "task_id": 4080,
        "test_case_id": 13,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "2 5\n-1 -7\n1 2\n2 2\n2 2\n2 2\n1 1\n",
        "output": "9\n3\n2 4 3 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 499,
        "task_id": 4080,
        "test_case_id": 14,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "2 50\n-8 -6\n2 2\n2 2\n1 2\n1 1\n2 2\n1 2\n2 2\n2 2\n1 1\n1 1\n1 1\n2 2\n2 2\n2 2\n1 1\n1 1\n1 1\n2 2\n2 2\n1 1\n2 2\n2 2\n2 2\n1 1\n1 2\n2 2\n1 1\n2 2\n1 1\n2 2\n1 2\n1 1\n2 2\n1 2\n2 2\n1 1\n1 2\n1 1\n2 2\n2 2\n2 2\n1 2\n2 2\n2 2\n2 2\n1 1\n1 1\n1 2\n2 2\n1 2\n",
        "output": "23\n25\n2 13 1 19 45 43 44 8 7 23 30 12 21 39 5 14 22 18 49 41 35 40 26 28 33 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 500,
        "task_id": 4080,
        "test_case_id": 15,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "5 0\n0 -8 -5 -6 10\n",
        "output": "18\n0\n\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 501,
        "task_id": 4080,
        "test_case_id": 16,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "5 1\n8 -5 -4 4 -8\n1 3\n",
        "output": "16\n0\n\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 502,
        "task_id": 4080,
        "test_case_id": 17,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "5 2\n-1 -5 0 5 -1\n3 4\n2 5\n",
        "output": "10\n0\n\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 503,
        "task_id": 4080,
        "test_case_id": 18,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "5 5\n2 6 -2 1 -6\n3 5\n4 5\n4 4\n2 3\n5 5\n",
        "output": "15\n4\n5 1 3 2 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 504,
        "task_id": 4080,
        "test_case_id": 19,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "5 50\n-5 4 -4 -1 5\n4 5\n4 5\n1 1\n1 5\n1 1\n1 1\n3 5\n4 5\n5 5\n1 1\n3 3\n1 4\n1 3\n4 5\n4 5\n5 5\n1 3\n2 3\n3 4\n5 5\n1 2\n4 4\n2 2\n1 3\n5 5\n2 5\n2 2\n3 3\n1 1\n4 4\n2 3\n3 4\n5 5\n5 5\n5 5\n4 4\n5 5\n1 4\n3 3\n3 3\n4 4\n1 1\n4 5\n2 5\n3 3\n1 1\n5 5\n3 4\n2 4\n5 5\n",
        "output": "25\n30\n36 13 21 46 38 32 30 28 41 40 42 6 29 22 27 19 31 24 48 5 12 39 10 45 49 18 17 3 11 23 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 505,
        "task_id": 4080,
        "test_case_id": 20,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "50 0\n-7 8 -10 -6 10 -2 -2 -4 0 0 1 9 -8 -7 -6 3 2 -3 10 -5 -9 7 -4 -9 1 -3 3 -3 -1 7 -4 3 -8 6 -3 -9 9 -10 -6 -10 5 3 -3 -3 3 -5 -3 -9 -7 2\n",
        "output": "20\n0\n\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 506,
        "task_id": 4080,
        "test_case_id": 21,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "50 1\n-8 8 -6 -4 -1 8 6 7 -7 3 -2 -3 6 10 2 8 -2 -4 2 9 0 7 0 9 5 6 -2 -1 -7 -9 -6 1 9 -1 -6 -5 -4 2 -2 4 8 7 -9 7 6 3 -8 -3 4 0\n27 41\n",
        "output": "20\n1\n1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 507,
        "task_id": 4080,
        "test_case_id": 22,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "50 2\n5 -5 8 5 6 6 10 -4 8 -10 9 -10 -5 -7 -7 -3 5 -9 7 9 0 -2 8 10 1 -9 -7 -4 8 9 -7 -5 2 -9 -1 -1 8 -6 -5 10 -5 -2 10 5 0 10 -9 -9 3 -2\n2 21\n28 38\n",
        "output": "21\n2\n2 1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 508,
        "task_id": 4080,
        "test_case_id": 23,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "50 5\n0 2 1 1 -2 7 -3 3 8 -1 6 8 -8 -3 0 1 6 1 -5 -1 -4 -3 -4 -4 -6 6 -9 5 0 -7 -8 3 8 -4 3 6 -10 -3 6 1 4 1 2 -7 5 -5 5 -2 4 -5\n30 34\n8 25\n48 50\n19 48\n48 49\n",
        "output": "19\n4\n4 5 3 1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 509,
        "task_id": 4080,
        "test_case_id": 24,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "50 50\n1 -9 8 -9 5 -4 -4 3 -2 -1 3 -8 -6 -6 2 -5 -1 -2 -5 10 2 0 10 10 5 -3 7 -1 -9 9 -10 4 7 0 -6 0 -1 4 8 0 -3 -8 10 10 6 6 -3 -5 -3 -8\n28 41\n28 49\n49 49\n6 38\n22 30\n50 50\n41 45\n30 32\n17 44\n24 25\n27 39\n17 39\n50 50\n10 14\n11 12\n18 28\n11 20\n36 42\n45 49\n29 36\n32 44\n19 41\n6 36\n39 40\n32 44\n41 47\n6 47\n44 48\n31 48\n45 48\n43 48\n1 14\n46 50\n50 50\n38 40\n18 29\n19 29\n20 50\n3 21\n47 50\n6 24\n38 49\n14 48\n28 39\n36 50\n10 32\n5 12\n14 18\n13 23\n13 24\n",
        "output": "34\n48\n11 48 21 10 13 40 12 38 41 9 20 6 18 4 42 47 43 8 15 37 19 7 5 22 44 31 30 29 36 49 25 26 34 3 23 50 27 2 28 46 45 17 1 33 24 35 14 16 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 510,
        "task_id": 4080,
        "test_case_id": 25,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "100 0\n6 0 3 6 1 -7 9 -2 -8 -4 -6 -2 7 -3 0 4 -8 10 -2 5 -2 3 9 -7 -2 0 -4 -7 3 7 7 -2 -5 2 9 7 -9 1 2 -8 9 0 8 2 -4 3 -10 0 3 8 3 -6 -8 2 0 10 0 -3 1 2 7 -6 -9 -4 7 7 -2 -4 2 -4 -4 -6 2 0 10 -6 -1 -6 -10 -2 3 8 -10 -10 10 0 1 8 -8 5 9 -6 -2 0 5 -6 -6 6 -10 -6\n",
        "output": "20\n0\n\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 511,
        "task_id": 4080,
        "test_case_id": 26,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "100 1\n-3 4 8 -1 0 7 9 8 10 3 8 7 -8 -7 -9 -7 -8 5 7 5 -10 -5 9 -2 3 5 -10 -10 -2 7 10 9 8 8 2 3 -1 -8 -6 2 -5 4 6 -1 -5 7 10 -2 -3 2 -5 1 2 8 -7 -3 6 -6 -2 -3 -4 -1 -8 4 -10 6 -4 8 4 -9 -2 7 -4 -6 7 -1 1 -7 -4 2 -4 6 -6 -8 -9 -8 -1 -10 -1 -4 -4 -5 -9 -1 7 2 -7 -7 -2 -1\n83 92\n",
        "output": "21\n1\n1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 512,
        "task_id": 4080,
        "test_case_id": 27,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "100 2\n-7 3 0 -4 -10 1 -4 -2 3 7 2 -1 2 -3 -1 -6 -5 4 -2 2 -10 7 -8 -1 -5 10 2 9 9 3 8 -1 5 -3 -10 7 -10 5 -9 -9 -1 -5 1 -3 10 5 5 -8 -8 4 -5 0 -5 10 -9 1 0 8 -4 0 6 0 -10 8 7 -8 8 7 6 3 -9 -10 1 -5 9 -4 -4 0 -10 -8 -6 -4 3 -3 1 0 0 5 5 4 -8 1 5 2 1 1 10 -3 -7 0\n69 88\n96 96\n",
        "output": "21\n2\n2 1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 513,
        "task_id": 4080,
        "test_case_id": 28,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "100 5\n-4 10 -7 9 7 2 8 0 4 -4 -5 1 -5 1 6 6 0 -3 8 -4 -10 10 9 6 9 9 0 1 9 5 7 7 7 5 -10 -7 -7 0 6 -9 -9 -2 -8 6 -2 7 -2 -10 -3 -3 -9 2 -9 -1 8 -9 -3 3 -9 -8 3 -9 -4 0 0 1 -8 -5 8 -4 -5 -4 6 8 3 9 6 5 5 -10 7 -1 0 3 6 6 10 -8 -9 2 3 -8 -3 -1 5 7 3 1 -8 -6\n27 59\n64 94\n79 87\n9 57\n37 91\n",
        "output": "23\n5\n3 4 5 2 1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 514,
        "task_id": 4080,
        "test_case_id": 29,
        "question": "The only difference between easy and hard versions is a number of elements in the array.\n\nYou are given an array $a$ consisting of $n$ integers. The value of the $i$-th element of the array is $a_i$.\n\nYou are also given a set of $m$ segments. The $j$-th segment is $[l_j; r_j]$, where $1 \\le l_j \\le r_j \\le n$.\n\nYou can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $a = [0, 0, 0, 0, 0]$ and the given segments are $[1; 3]$ and $[2; 4]$ then you can choose both of them and the array will become $b = [-1, -2, -2, -1, 0]$.\n\nYou have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $a$ and obtain the array $b$ then the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ will be maximum possible.\n\nNote that you can choose the empty set.\n\nIf there are multiple answers, you can print any.\n\nIf you are Python programmer, consider using PyPy instead of Python when you submit your code.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n \\le 10^5, 0 \\le m \\le 300$) — the length of the array $a$ and the number of segments, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($-10^6 \\le a_i \\le 10^6$), where $a_i$ is the value of the $i$-th element of the array $a$.\n\nThe next $m$ lines are contain two integers each. The $j$-th of them contains two integers $l_j$ and $r_j$ ($1 \\le l_j \\le r_j \\le n$), where $l_j$ and $r_j$ are the ends of the $j$-th segment.\n\n\n-----Output-----\n\nIn the first line of the output print one integer $d$ — the maximum possible value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ if $b$ is the array obtained by applying some subset of the given segments to the array $a$.\n\nIn the second line of the output print one integer $q$ ($0 \\le q \\le m$) — the number of segments you apply.\n\nIn the third line print $q$ distinct integers $c_1, c_2, \\dots, c_q$ in any order ($1 \\le c_k \\le m$) — indices of segments you apply to the array $a$ in such a way that the value $\\max\\limits_{i=1}^{n}b_i - \\min\\limits_{i=1}^{n}b_i$ of the obtained array $b$ is maximum possible.\n\nIf there are multiple answers, you can print any.\n\n\n-----Examples-----\nInput\n5 4\n2 -2 3 1 2\n1 3\n4 5\n2 5\n1 3\n\nOutput\n6\n2\n4 1 \n\nInput\n5 4\n2 -2 3 1 4\n3 5\n3 4\n2 4\n2 5\n\nOutput\n7\n2\n3 2 \n\nInput\n1 0\n1000000\n\nOutput\n0\n0\n\n\n\n\n-----Note-----\n\nIn the first example the obtained array $b$ will be $[0, -4, 1, 1, 2]$ so the answer is $6$.\n\nIn the second example the obtained array $b$ will be $[2, -3, 1, -1, 4]$ so the answer is $7$.\n\nIn the third example you cannot do anything so the answer is $0$.",
        "solutions": "[\"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [0]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.min(0,n)\\n    Ai = -Aneg.min(i,i+1)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [j for j in range(m) if inter_copy[j][0]<=besta_ind<inter_copy[j][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls.\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\ninter2 = []\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    inter2.append((r,l))\\n\\ninter_copy = inter[:]\\n\\ninter.sort()\\ninter2.sort()\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nj1 = 0\\nj2 = 0\\nfor i in range(n):\\n    # Only segments containing i should be active\\n    # Activate\\n    while j1<m and inter[j1][0]<=i:\\n        l,r = inter[j1]\\n        Aneg.add(l,r,1)\\n        j1 += 1\\n    # Deactivate\\n    while j2<m and inter2[j2][0]<=i:\\n        r,l = inter2[j2]\\n        Aneg.add(l,r,-1)\\n        j2 += 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i]-(j1-j2)\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter_copy[i][0]<=besta_ind<inter_copy[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nbucketsl = [[] for _ in range(n)]\\nbucketsr = [[] for _ in range(n)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    bucketsl[l].append(r)\\n    if r<n:\\n        bucketsr[r].append(l)\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    # Activate\\n    for r in bucketsl[i]:\\n        Aneg.add(i,r,1)\\n        active_intervals += 1\\n    # Deactivate\\n    for l in bucketsr[i]:\\n        Aneg.add(l,i,-1)\\n        active_intervals -= 1\\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\n\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\n\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\n\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(list(range(m))):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\\n\", \"import sys\\nimport copy\\ninput = sys.stdin.readline\\n\\nn,m=map(int,input().split())\\nA=list(map(int,input().split()))\\n\\nLR=[list(map(int,input().split())) for i in range(m)]\\n\\n\\nMLIST=[]\\nfor l,r in LR:\\n    MLIST.append(l)\\n    MLIST.append(r)\\n\\nMLIST=list(set(MLIST))\\nMLIST.sort()\\nMLIST.append(10**9)\\n\\n\\nMDICT=dict()\\nMLEN=len(MLIST)\\nfor i in range(MLEN):\\n    MDICT[MLIST[i]]=i\\n\\n\\nMINUSLIST=[[0]*(MLEN*2-1) for i in range(MLEN)]\\n\\nfor l,r in LR:\\n    for j in range(MDICT[l],MDICT[r]+1):\\n        for k in range(MDICT[l]*2+1,MDICT[r]*2+2):\\n            MINUSLIST[j][k]+=1\\n            \\n#print(MLIST)\\n#print(MINUSLIST)\\n\\n\\nA_m=[[float(\\\"inf\\\"),-float(\\\"inf\\\")] for i in range(MLEN*2-1)]\\n\\nif MLIST[0]!=1:\\n    A_m[0]=[min(A[:MLIST[0]-1]),max(A[:MLIST[0]-1])]\\n#if MLIST[MLEN-2]!=n:\\n#    A_m[MLEN*2-2]=[min(A[MLIST[MLEN-2]:]),max(A[MLIST[MLEN-2]:])]\\n\\nfor i in range(MLEN-1):\\n    x=MLIST[i]-1\\n    y=MLIST[i+1]-1\\n\\n    A_m[2*i+1]=[A[x],A[x]]\\n    if x+1!=y and x!=n-1:\\n        A_m[2*i+2]=[min(A[x+1:y]),max(A[x+1:y])]\\n    \\ndef MINUS(A,B):\\n    MIN=float(\\\"inf\\\")\\n    MAX=-float(\\\"inf\\\")\\n\\n    for i in range(MLEN*2-1):\\n        if MIN>A[i][0]-B[i]:\\n            MIN=A[i][0]-B[i]\\n        if MAX<A[i][1]-B[i]:\\n            MAX=A[i][1]-B[i]\\n\\n    return MAX-MIN\\n\\nANSLIST=[0]*MLEN\\n\\nfor i in range(MLEN):\\n    ANSLIST[i]=MINUS(A_m,MINUSLIST[i])\\n\\nprint(max(ANSLIST))\\ntarget=MLIST[ANSLIST.index(max(ANSLIST))]\\n\\nAN=[]\\nfor i in range(m):\\n    z,w=LR[i]\\n    if z<=target<=w:\\n        AN.append(i+1)\\n\\nprint(len(AN))\\nfor a in AN:\\n    print(a,end=\\\" \\\")\\n\\n\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"import sys\\n\\n# a very nice implementation of a minimum segment tree with \\n# some inspiration taken from https://codeforces.com/blog/entry/18051\\n# this implementation should be able to be modified to do pretty\\n# much anything one would want to do with segment trees apart from\\n# persistance.\\n# note that especially in python this implementation is much much better\\n# than most other approches because how slow python can be with function\\n# calls. /pajenegod\\n\\n# currently it allows for two operations, both running in o(log n),\\n# 'add(l,r,value)' adds value to [l,r)\\n# 'find_min(l,r)' finds the index with the smallest value\\n\\nbig = 10**9\\n\\nclass super_seg:\\n    def __init__(self,data):\\n        n = len(data)\\n        m = 1\\n        while m<n: m *= 2\\n        \\n        self.n = n\\n        self.m = m\\n        self.data = [big]*(2*m)\\n        for i in range(n):\\n            self.data[i+m] = data[i]\\n        for i in reversed(range(m)):\\n            self.data[i] = min(self.data[2*i], self.data[2*i+1])\\n        self.query = [0]*(2*m)\\n    \\n    # push the query on seg_ind to its children\\n    def push(self,seg_ind):\\n        # let the children know of the queries\\n        q = self.query[seg_ind]\\n\\n        self.query[2*seg_ind]   += q\\n        self.query[2*seg_ind+1] += q\\n        \\n        self.data[2*seg_ind]   += q\\n        self.data[2*seg_ind+1] += q\\n\\n        # remove queries from seg_ind\\n        self.data[seg_ind] = min(self.data[2*seg_ind],self.data[2*seg_ind+1])\\n        self.query[seg_ind] = 0\\n\\n    # updates the node seg_ind to know of all queries\\n    # applied to it via its ancestors\\n    def update(self,seg_ind):\\n        # find all indecies to be updated\\n        seg_ind //= 2\\n        inds = []\\n        while seg_ind>0:\\n            inds.append(seg_ind)\\n            seg_ind//=2\\n       \\n        # push the queries down the segment tree\\n        for ind in reversed(inds):\\n            self.push(ind)\\n\\n    # make the changes to seg_ind be known to its ancestors\\n    def build(self,seg_ind):\\n        seg_ind//=2\\n        while seg_ind>0:\\n            self.data[seg_ind] = min(self.data[2*seg_ind], self.data[2*seg_ind+1]) + self.query[seg_ind]\\n            seg_ind //= 2\\n\\n    # lazily add value to [l,r)\\n    def add(self,l,r,value):\\n        l += self.m\\n        r += self.m\\n        \\n        l0 = l\\n        r0 = r\\n\\n        while l<r:\\n            if l%2==1:\\n                self.query[l]+= value\\n                self.data[l] += value\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                self.query[r]+= value\\n                self.data[r] += value\\n            l//=2\\n            r//=2\\n\\n        # tell all nodes above of the updated\\n        # area of the updates\\n        self.build(l0)\\n        self.build(r0-1)\\n    \\n    # min of data[l,r)\\n    def min(self,l,r):\\n        l += self.m\\n        r += self.m\\n\\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n        \\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        return min(self.data[ind] for ind in segs)\\n\\n\\n    # find index of smallest value in data[l,r)\\n    def find_min(self,l,r):\\n        l += self.m\\n        r += self.m\\n        \\n        # apply all the lazily stored queries\\n        self.update(l)\\n        self.update(r-1)\\n\\n        segs = []\\n        while l<r:\\n            if l%2==1:\\n                segs.append(l)\\n                l+=1\\n            if r%2==1:\\n                r-=1\\n                segs.append(r)\\n            l//=2\\n            r//=2\\n\\n        ind = min(segs, key=lambda i:self.data[i])\\n        mini = self.data[ind]\\n       \\n        # dig down in search of mini\\n        while ind<self.m:\\n            self.push(ind)\\n\\n            if self.data[2*ind]==mini:\\n                ind *= 2\\n            else:\\n                ind = 2*ind+1\\n        \\n        return ind-self.m,mini\\n\\n\\nn,m = [int(x) for x in input().split()]\\nA = [int(x) for x in input().split()]\\n\\ninter = []\\nupdate = [[] for _ in range(n+1)]\\nfor _ in range(m):\\n    l,r = [int(x) for x in input().split()]\\n    l -= 1\\n    inter.append((l,r))\\n    update[l].append((l,r))\\n    update[r].append((l,r))\\n\\nAneg = super_seg([-a for a in A])\\nbesta = -1\\nbesta_ind = -1\\n\\nactive_intervals = 0\\nfor i in range(n):\\n    for l,r in update[i]:\\n        Aneg.add(l,r,1 if l==i else -1)\\n        active_intervals += 1 if l==i else -1 \\n    Amax = -Aneg.data[1]\\n    Ai = A[i] - active_intervals\\n    if Amax-Ai>besta:\\n        besta = Amax-Ai\\n        besta_ind = i\\n\\nints = [i for i in range(m) if inter[i][0]<=besta_ind<inter[i][1]]\\nprint(besta)\\nprint(len(ints))\\nprint(*[x+1 for x in ints])\", \"def main():\\n    n, m = list(map(int, input().split()))\\n    aa = list(map(int, input().split()))\\n    res = max(aa) - min(aa)\\n    ll, rr = [n + 1], [n + 1]\\n    segments = res_segments = [False] * (m + 1)\\n    bounds = {0, n, n + 1}\\n    for _ in range(m):\\n        l, r = list(map(int, input().split()))\\n        l -= 1\\n        ll.append(l)\\n        rr.append(r)\\n        bounds.add(l)\\n        bounds.add(r)\\n    xlat = sorted(bounds)\\n    mi, ma = [], []\\n    for l, r in zip(xlat, xlat[1:-1]):\\n        t = aa[l:r]\\n        mi.append(min(t))\\n        ma.append(max(t))\\n    bounds = {x: i for i, x in enumerate(xlat)}\\n    for xx in (ll, rr):\\n        for i, x in enumerate(xx):\\n            xx[i] = bounds[x]\\n    il, ir = (sorted(list(range(m + 1)), key=xx.__getitem__, reverse=True) for xx in (ll, rr))\\n    for i in range(len(xlat) - 1):\\n        while True:\\n            k = il[-1]\\n            lo = ll[k]\\n            if lo > i:\\n                break\\n            segments[k] = True\\n            for j in range(lo, rr[k]):\\n                mi[j] -= 1\\n                ma[j] -= 1\\n            del il[-1]\\n\\n        x = max(ma) - min(mi)\\n        if res < x:\\n            res = x\\n            res_segments = segments[:]\\n        while True:\\n            k = ir[-1]\\n            hi = rr[k]\\n            if hi > i:\\n                break\\n            segments[k] = False\\n            for j in range(ll[k], hi):\\n                mi[j] += 1\\n                ma[j] += 1\\n            del ir[-1]\\n\\n    print(res)\\n    segments = [i for i, f in enumerate(res_segments) if f]\\n    print(len(segments))\\n    print(*segments)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = list(map(int, input().split()))\\nA = list(map(int, input().split()))\\nLf = [[] for _ in range(n)]\\nRb = [[] for _ in range(n)]\\nLR = []\\nfor i in range(m):\\n    l, r = list(map(int, input().split()))\\n    l, r = l-1, r-1\\n    Lf[r].append(l)\\n    Rb[l].append(r)\\n    LR.append((l, r))\\n\\nminus = [0]*n\\nINF = 10**18\\nans = [-INF]*n\\nmn = A[0]\\n\\nfor i in range(n):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for l in Lf[i]:\\n        for j in range(l, i+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\n\\nminus = [0]*n\\nmn = A[n-1]\\nfor i in reversed(list(range(n))):\\n    ans[i] = max(ans[i], A[i]-mn)\\n    for r in Rb[i]:\\n        for j in range(i, r+1):\\n            minus[j] -= 1\\n            mn = min(mn, A[j]+minus[j])\\n    mn = min(mn, A[i]+minus[i])\\nans_ = max(ans)\\nres = []\\nfor i in range(n):\\n    if ans[i] == ans_:\\n        for j in range(m):\\n            l, r = LR[j]\\n            if not (l <= i and i <= r):\\n                res.append(j+1)\\n        break\\nprint(ans_)\\nprint(len(res))\\nprint(*res)\\n\"]",
        "difficulty": "introductory",
        "input": "69 1\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n1 3\n",
        "output": "1\n1\n1 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1108/E2"
    },
    {
        "id": 515,
        "task_id": 4813,
        "test_case_id": 1,
        "question": "Hangman is a (somewhat macabre) word-guessing game that can be played by two people. Player $1$ thinks of a word consisting of $n$ letters, and draws a row of $n$ dashes on a piece of paper. These dashes correspond to the letters of the word, in the same order. Player $2$ then attempts to discover the word by making a sequence of letter guesses. For each letter guessed by Player $2$:\n - If the letter occurs one or more times in the word, Player $1$ writes the letter above each corresponding dash.\n - If the letter does not occur in the word, Player $1$ adds one component to a drawing of a stick-figure man hanging on a gallows. The drawing (initially empty) has $10$ possible components: base, vertical beam, horizontal beam, rope, head, torso, right leg, left leg, right arm, left arm.\n\nIf Player $2$ guesses all the letters in the word before the drawing of the hanging man is complete, then Player $2$ wins (and Player $1$ loses). Otherwise, Player $2$ loses (and Player $1$ wins).\n\nNed loves playing hangman, especially as Player $2$, but he finds that he is not a very good letter guesser. To improve his chances, he decides to adopt a new strategy. For each word selected by Player $1$, Ned chooses a random permutation of the letters of the alphabet, and then simply guesses letters in that order until he either wins or loses. Given the word and Ned’s permutation of the alphabet, determine the outcome of the game.\n\n-----Input-----\nThe input consists of two lines representing a single game of Hangman. The first line contains the word to be guessed, a non-empty string of uppercase English alphabet letters (A–Z) of maximum length $16$. The second line contains a permutation of the $26$ letters of the English alphabet, also uppercase.\n\n-----Output-----\nIf Ned wins the game by guessing letters in the order given by the permutation (proceeding from left to right), output “WIN”. Otherwise, output “LOSE”.\n\n-----Examples-----\nSample Input 1:\nHANGMAN\nABCDEFGHIJKLMNOPQRSTUVWXYZ\nSample Output 1:\nWIN\n\nSample Input 2:\nBANANA\nABCDEFGHIJKLMNOPQRSTUVWXYZ\nSample Output 2:\nLOSE",
        "solutions": "",
        "difficulty": "introductory",
        "input": "HANGMAN\nABCDEFGHIJKLMNOPQRSTUVWXYZ\n",
        "output": "WIN\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/hangman"
    },
    {
        "id": 516,
        "task_id": 4813,
        "test_case_id": 2,
        "question": "Hangman is a (somewhat macabre) word-guessing game that can be played by two people. Player $1$ thinks of a word consisting of $n$ letters, and draws a row of $n$ dashes on a piece of paper. These dashes correspond to the letters of the word, in the same order. Player $2$ then attempts to discover the word by making a sequence of letter guesses. For each letter guessed by Player $2$:\n - If the letter occurs one or more times in the word, Player $1$ writes the letter above each corresponding dash.\n - If the letter does not occur in the word, Player $1$ adds one component to a drawing of a stick-figure man hanging on a gallows. The drawing (initially empty) has $10$ possible components: base, vertical beam, horizontal beam, rope, head, torso, right leg, left leg, right arm, left arm.\n\nIf Player $2$ guesses all the letters in the word before the drawing of the hanging man is complete, then Player $2$ wins (and Player $1$ loses). Otherwise, Player $2$ loses (and Player $1$ wins).\n\nNed loves playing hangman, especially as Player $2$, but he finds that he is not a very good letter guesser. To improve his chances, he decides to adopt a new strategy. For each word selected by Player $1$, Ned chooses a random permutation of the letters of the alphabet, and then simply guesses letters in that order until he either wins or loses. Given the word and Ned’s permutation of the alphabet, determine the outcome of the game.\n\n-----Input-----\nThe input consists of two lines representing a single game of Hangman. The first line contains the word to be guessed, a non-empty string of uppercase English alphabet letters (A–Z) of maximum length $16$. The second line contains a permutation of the $26$ letters of the English alphabet, also uppercase.\n\n-----Output-----\nIf Ned wins the game by guessing letters in the order given by the permutation (proceeding from left to right), output “WIN”. Otherwise, output “LOSE”.\n\n-----Examples-----\nSample Input 1:\nHANGMAN\nABCDEFGHIJKLMNOPQRSTUVWXYZ\nSample Output 1:\nWIN\n\nSample Input 2:\nBANANA\nABCDEFGHIJKLMNOPQRSTUVWXYZ\nSample Output 2:\nLOSE",
        "solutions": "",
        "difficulty": "introductory",
        "input": "BANANA\nABCDEFGHIJKLMNOPQRSTUVWXYZ\n",
        "output": "LOSE\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/hangman"
    },
    {
        "id": 517,
        "task_id": 4813,
        "test_case_id": 3,
        "question": "Hangman is a (somewhat macabre) word-guessing game that can be played by two people. Player $1$ thinks of a word consisting of $n$ letters, and draws a row of $n$ dashes on a piece of paper. These dashes correspond to the letters of the word, in the same order. Player $2$ then attempts to discover the word by making a sequence of letter guesses. For each letter guessed by Player $2$:\n - If the letter occurs one or more times in the word, Player $1$ writes the letter above each corresponding dash.\n - If the letter does not occur in the word, Player $1$ adds one component to a drawing of a stick-figure man hanging on a gallows. The drawing (initially empty) has $10$ possible components: base, vertical beam, horizontal beam, rope, head, torso, right leg, left leg, right arm, left arm.\n\nIf Player $2$ guesses all the letters in the word before the drawing of the hanging man is complete, then Player $2$ wins (and Player $1$ loses). Otherwise, Player $2$ loses (and Player $1$ wins).\n\nNed loves playing hangman, especially as Player $2$, but he finds that he is not a very good letter guesser. To improve his chances, he decides to adopt a new strategy. For each word selected by Player $1$, Ned chooses a random permutation of the letters of the alphabet, and then simply guesses letters in that order until he either wins or loses. Given the word and Ned’s permutation of the alphabet, determine the outcome of the game.\n\n-----Input-----\nThe input consists of two lines representing a single game of Hangman. The first line contains the word to be guessed, a non-empty string of uppercase English alphabet letters (A–Z) of maximum length $16$. The second line contains a permutation of the $26$ letters of the English alphabet, also uppercase.\n\n-----Output-----\nIf Ned wins the game by guessing letters in the order given by the permutation (proceeding from left to right), output “WIN”. Otherwise, output “LOSE”.\n\n-----Examples-----\nSample Input 1:\nHANGMAN\nABCDEFGHIJKLMNOPQRSTUVWXYZ\nSample Output 1:\nWIN\n\nSample Input 2:\nBANANA\nABCDEFGHIJKLMNOPQRSTUVWXYZ\nSample Output 2:\nLOSE",
        "solutions": "",
        "difficulty": "introductory",
        "input": "RAINBOWS\nUSIANBVLOJRKWXZCTQGHPFMYDE\n",
        "output": "WIN\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/hangman"
    },
    {
        "id": 518,
        "task_id": 4866,
        "test_case_id": 1,
        "question": "HiQ recently got an assignment from a client to create a clone of the immensely popular website https://IsItHalloween.com. The website is a very simple one. People will visit the site occasionally to see if it is Halloween. Whenever it is, the website should print out yup, otherwise it should print out nope on the screen.\n\nSince HiQ is such a popular firm, they don’t have time to complete this assignment right now. Their frontend engineers have already programmed the frontend of the website that prints out yup or nope, but not the backend microservice that determines whether it is indeed Halloween or not. Do you have time to help them?\n\nThe behaviour of the server should be as follows: it gets as input the current date in the format FEB 9, where FEB is the month given in three letters (JAN, FEB, MAR, APR, MAY, JUN, JUL, AUG, SEP, OCT, NOV, DEC) and 9 is the day of the month starting at 1. It should then determine if this date represents October 31 or December 25 (since $31_8 = 25_{10}$).\n\n-----Input-----\nThe input consists of a single line containing a date of the format FEB 9, with the month and date separated by a single space.\n\n-----Output-----\nIf the date is October 31 or December 25, output yup. Otherwise, output nope.\n\n-----Examples-----\nSample Input:\nOCT 31\nSample Output:\nyup",
        "solutions": "",
        "difficulty": "introductory",
        "input": "OCT 31\n",
        "output": "yup\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/isithalloween"
    },
    {
        "id": 519,
        "task_id": 4866,
        "test_case_id": 2,
        "question": "HiQ recently got an assignment from a client to create a clone of the immensely popular website https://IsItHalloween.com. The website is a very simple one. People will visit the site occasionally to see if it is Halloween. Whenever it is, the website should print out yup, otherwise it should print out nope on the screen.\n\nSince HiQ is such a popular firm, they don’t have time to complete this assignment right now. Their frontend engineers have already programmed the frontend of the website that prints out yup or nope, but not the backend microservice that determines whether it is indeed Halloween or not. Do you have time to help them?\n\nThe behaviour of the server should be as follows: it gets as input the current date in the format FEB 9, where FEB is the month given in three letters (JAN, FEB, MAR, APR, MAY, JUN, JUL, AUG, SEP, OCT, NOV, DEC) and 9 is the day of the month starting at 1. It should then determine if this date represents October 31 or December 25 (since $31_8 = 25_{10}$).\n\n-----Input-----\nThe input consists of a single line containing a date of the format FEB 9, with the month and date separated by a single space.\n\n-----Output-----\nIf the date is October 31 or December 25, output yup. Otherwise, output nope.\n\n-----Examples-----\nSample Input:\nOCT 31\nSample Output:\nyup",
        "solutions": "",
        "difficulty": "introductory",
        "input": "JUN 24\n",
        "output": "nope\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/isithalloween"
    },
    {
        "id": 520,
        "task_id": 4941,
        "test_case_id": 1,
        "question": "A confused Dutchman trying to speak English could say “I am in the war”, even though there is no hostile activity going on. The confusion1 here is that the English sentence “I am confused” is translated in Dutch as “Ik ben in de war”, which is phonetically (“sounding”) quite close to the first sentence. Such confusion leads to much enjoyment, but can complicate matters a bit. \n\nGiven a sentence in Dutch and a dictionary containing both correct translations as well as phonetic (incorrect) translations of individual words, find the translation of the sentence and indicate whether it is correct, or in case there is more than one find the total number of correct and incorrect translations. A sentence is correctly translated when each word of the sentence is correctly translated.\n\n-----Input-----\nThe input consists of:\n - One line with an integer $n$ ($1 \\leq n \\leq 20$), the number of words in the Dutch sentence.\n - One line with $n$ words, the Dutch sentence $s$.\n - One line with an integer $m$ ($1 \\leq m \\leq 10^5$), the number of words in the dictionary.\n - $m$ lines, each with three strings $d$, $e$ and $c$, a Dutch word, the English translation, and “correct” if this is the correct translation or “incorrect” otherwise.\n\nA word consists of between $1$ and $20$ lowercase letters. Each word in $s$ appears at least once as a Dutch word in the dictionary, no word appears more than $8$ times as a Dutch word in the dictionary, and each combination of a Dutch and English word appears at most once.\n\n-----Output-----\nIn case there is only a single translation of $s$, output one line with the translation followed by one line with “correct” or “incorrect”. In case there is more than one translation, output one line with the number of possible correct translations followed by “correct”, and one line with the number of possible incorrect translations followed by “incorrect”.\n\n-----Examples-----\nSample Input:\n7\nals mollen mollen mollen mollen mollen mollen\n4\nals when correct\nmollen moles correct\nmollen destroy correct\nmollen mills incorrect\nSample Output:\n64 correct\n665 incorrect",
        "solutions": "",
        "difficulty": "introductory",
        "input": "7\nals mollen mollen mollen mollen mollen mollen\n4\nals when correct\nmollen moles correct\nmollen destroy correct\nmollen mills incorrect\n",
        "output": "64 correct\n665 incorrect\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/dunglish"
    },
    {
        "id": 521,
        "task_id": 4941,
        "test_case_id": 2,
        "question": "A confused Dutchman trying to speak English could say “I am in the war”, even though there is no hostile activity going on. The confusion1 here is that the English sentence “I am confused” is translated in Dutch as “Ik ben in de war”, which is phonetically (“sounding”) quite close to the first sentence. Such confusion leads to much enjoyment, but can complicate matters a bit. \n\nGiven a sentence in Dutch and a dictionary containing both correct translations as well as phonetic (incorrect) translations of individual words, find the translation of the sentence and indicate whether it is correct, or in case there is more than one find the total number of correct and incorrect translations. A sentence is correctly translated when each word of the sentence is correctly translated.\n\n-----Input-----\nThe input consists of:\n - One line with an integer $n$ ($1 \\leq n \\leq 20$), the number of words in the Dutch sentence.\n - One line with $n$ words, the Dutch sentence $s$.\n - One line with an integer $m$ ($1 \\leq m \\leq 10^5$), the number of words in the dictionary.\n - $m$ lines, each with three strings $d$, $e$ and $c$, a Dutch word, the English translation, and “correct” if this is the correct translation or “incorrect” otherwise.\n\nA word consists of between $1$ and $20$ lowercase letters. Each word in $s$ appears at least once as a Dutch word in the dictionary, no word appears more than $8$ times as a Dutch word in the dictionary, and each combination of a Dutch and English word appears at most once.\n\n-----Output-----\nIn case there is only a single translation of $s$, output one line with the translation followed by one line with “correct” or “incorrect”. In case there is more than one translation, output one line with the number of possible correct translations followed by “correct”, and one line with the number of possible incorrect translations followed by “incorrect”.\n\n-----Examples-----\nSample Input:\n7\nals mollen mollen mollen mollen mollen mollen\n4\nals when correct\nmollen moles correct\nmollen destroy correct\nmollen mills incorrect\nSample Output:\n64 correct\n665 incorrect",
        "solutions": "",
        "difficulty": "introductory",
        "input": "5\nde zuigers zijn buiten werking\n6\nzijn are correct\nbanaan banana correct\nde the correct\nzuigers suckers incorrect\nbuiten out correct\nwerking working incorrect\n",
        "output": "the suckers are out working\nincorrect\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/dunglish"
    },
    {
        "id": 522,
        "task_id": 4662,
        "test_case_id": 1,
        "question": "=====Problem Statement=====\nYou are given a valid XML document, and you have to print the maximum level of nesting in it. Take the depth of the root as 0.\n\n=====Input Format=====\nThe first line contains N, the number of lines in the XML document.\nThe next N lines follow containing the XML document.\n\n=====Output Format=====\nOutput a single line, the integer value of the maximum level of nesting in the XML document.",
        "solutions": "[\"# Enter your code here. Read input from STDIN. Print output to STDOUT\\nxml_str=\\\"\\\"\\nn=int(input())\\nfor i in range(0,n):\\n    tmp_str=input()\\n    xml_str=xml_str+tmp_str\\n    \\nimport xml.etree.ElementTree as etree\\ntree = etree.ElementTree(etree.fromstring(xml_str))\\nroot=tree.getroot()\\nar=[]\\ndef cnt_node(node):\\n    return max( [0] + [cnt_node(child)+1 for child in node])\\ncnt=cnt_node(root)\\nprint(cnt)\\n\", \"maxdepth = 0\\ndef depth(elem, level):\\n    #print(elem)\\n    if level == -1:\\n        level = 0\\n    nonlocal maxdepth\\n    if level > maxdepth:\\n        maxdepth = level\\n    for el in elem:\\n        depth(el, level+1)\\n\\n\"]",
        "difficulty": "introductory",
        "input": "6\n<feed xml:lang='en'>\n  <title>HackerRank</title>\n  <subtitle lang='en'>Programming challenges</subtitle>\n  <link rel='alternate' type='text/html' href='http://hackerrank.com/'/>\n  <updated>2013-12-25T12:00:00</updated>\n</feed>",
        "output": "1",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "\nimport xml.etree.ElementTree as etree\n\nmaxdepth = 0\ndef depth(elem, level):\n    global maxdepth\n    # your code goes here\n\nif __name__ == '__main__':\n    n = int(input())\n    xml = \"\"\n    for i in range(n):\n        xml =  xml + input() + \"\\n\"\n    tree = etree.ElementTree(etree.fromstring(xml))\n    depth(tree.getroot(), -1)\n    print(maxdepth)",
        "url": "https://www.hackerrank.com/challenges/xml2-find-the-maximum-depth/problem"
    },
    {
        "id": 523,
        "task_id": 4662,
        "test_case_id": 2,
        "question": "=====Problem Statement=====\nYou are given a valid XML document, and you have to print the maximum level of nesting in it. Take the depth of the root as 0.\n\n=====Input Format=====\nThe first line contains N, the number of lines in the XML document.\nThe next N lines follow containing the XML document.\n\n=====Output Format=====\nOutput a single line, the integer value of the maximum level of nesting in the XML document.",
        "solutions": "[\"# Enter your code here. Read input from STDIN. Print output to STDOUT\\nxml_str=\\\"\\\"\\nn=int(input())\\nfor i in range(0,n):\\n    tmp_str=input()\\n    xml_str=xml_str+tmp_str\\n    \\nimport xml.etree.ElementTree as etree\\ntree = etree.ElementTree(etree.fromstring(xml_str))\\nroot=tree.getroot()\\nar=[]\\ndef cnt_node(node):\\n    return max( [0] + [cnt_node(child)+1 for child in node])\\ncnt=cnt_node(root)\\nprint(cnt)\\n\", \"maxdepth = 0\\ndef depth(elem, level):\\n    #print(elem)\\n    if level == -1:\\n        level = 0\\n    nonlocal maxdepth\\n    if level > maxdepth:\\n        maxdepth = level\\n    for el in elem:\\n        depth(el, level+1)\\n\\n\"]",
        "difficulty": "introductory",
        "input": "11\n<feed xml:lang='en'>\n  <title>HackerRank</title>\n  <subtitle lang='en'>Programming challenges</subtitle>\n  <link rel='alternate' type='text/html' href='http://hackerrank.com/'/>\n  <updated>2013-12-25T12:00:00</updated>\n  <entry>\n  \t<author gender='male'>Harsh</author>\n    <question type='hard'>XML 1</question>\n    <description type='text'>This is related to XML parsing</description>\n  </entry>\n</feed>",
        "output": "2",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "\nimport xml.etree.ElementTree as etree\n\nmaxdepth = 0\ndef depth(elem, level):\n    global maxdepth\n    # your code goes here\n\nif __name__ == '__main__':\n    n = int(input())\n    xml = \"\"\n    for i in range(n):\n        xml =  xml + input() + \"\\n\"\n    tree = etree.ElementTree(etree.fromstring(xml))\n    depth(tree.getroot(), -1)\n    print(maxdepth)",
        "url": "https://www.hackerrank.com/challenges/xml2-find-the-maximum-depth/problem"
    },
    {
        "id": 524,
        "task_id": 581,
        "test_case_id": 1,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "3\n1 2\n1 3\n",
        "output": "3\n2 3 3\n2 1 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 525,
        "task_id": 581,
        "test_case_id": 2,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "5\n1 2\n1 3\n2 4\n2 5\n",
        "output": "9\n3 5 5\n4 3 3\n4 1 1\n4 2 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 526,
        "task_id": 581,
        "test_case_id": 4,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "4\n1 3\n1 4\n1 2\n",
        "output": "5\n3 4 4\n2 3 3\n2 1 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 527,
        "task_id": 581,
        "test_case_id": 5,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "4\n2 1\n1 3\n3 4\n",
        "output": "6\n4 2 2\n4 1 1\n4 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 528,
        "task_id": 581,
        "test_case_id": 6,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "4\n4 3\n3 2\n2 1\n",
        "output": "6\n4 1 1\n4 2 2\n4 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 529,
        "task_id": 581,
        "test_case_id": 7,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "5\n2 1\n2 3\n2 4\n2 5\n",
        "output": "7\n1 4 4\n1 5 5\n3 1 1\n3 2 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 530,
        "task_id": 581,
        "test_case_id": 8,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "5\n4 5\n4 1\n1 2\n2 3\n",
        "output": "10\n3 5 5\n3 4 4\n3 1 1\n3 2 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 531,
        "task_id": 581,
        "test_case_id": 9,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "5\n1 4\n4 3\n3 2\n2 5\n",
        "output": "10\n5 1 1\n5 4 4\n5 3 3\n5 2 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 532,
        "task_id": 581,
        "test_case_id": 10,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "6\n4 5\n4 1\n4 6\n4 2\n4 3\n",
        "output": "9\n1 5 5\n1 6 6\n1 3 3\n2 1 1\n2 4 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 533,
        "task_id": 581,
        "test_case_id": 11,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "6\n6 5\n6 2\n2 3\n5 4\n4 1\n",
        "output": "15\n3 1 1\n3 4 4\n3 5 5\n3 6 6\n3 2 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 534,
        "task_id": 581,
        "test_case_id": 12,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "6\n1 5\n5 4\n4 2\n2 6\n6 3\n",
        "output": "15\n3 1 1\n3 5 5\n3 4 4\n3 2 2\n3 6 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 535,
        "task_id": 581,
        "test_case_id": 13,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "7\n7 5\n7 3\n7 6\n7 4\n7 1\n7 2\n",
        "output": "11\n1 5 5\n1 3 3\n1 6 6\n1 4 4\n2 1 1\n2 7 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 536,
        "task_id": 581,
        "test_case_id": 14,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "7\n7 6\n7 5\n7 2\n7 1\n5 4\n5 3\n",
        "output": "15\n3 6 6\n3 2 2\n1 4 4\n3 1 1\n3 7 7\n3 5 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 537,
        "task_id": 581,
        "test_case_id": 15,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "7\n2 7\n7 6\n6 5\n5 4\n4 1\n1 3\n",
        "output": "21\n2 3 3\n2 1 1\n2 4 4\n2 5 5\n2 6 6\n2 7 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 538,
        "task_id": 581,
        "test_case_id": 16,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "8\n8 6\n8 7\n8 2\n8 5\n8 1\n8 4\n8 3\n",
        "output": "13\n1 6 6\n1 7 7\n1 5 5\n1 4 4\n1 3 3\n2 1 1\n2 8 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 539,
        "task_id": 581,
        "test_case_id": 17,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "8\n6 3\n3 7\n6 1\n1 2\n3 5\n5 4\n2 8\n",
        "output": "26\n8 7 7\n4 8 8\n4 2 2\n4 1 1\n4 6 6\n4 3 3\n4 5 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 540,
        "task_id": 581,
        "test_case_id": 18,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "8\n4 1\n1 3\n3 6\n6 2\n2 7\n7 5\n5 8\n",
        "output": "28\n8 4 4\n8 1 1\n8 3 3\n8 6 6\n8 2 2\n8 7 7\n8 5 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 541,
        "task_id": 581,
        "test_case_id": 19,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "9\n3 2\n3 1\n3 8\n3 5\n3 6\n3 9\n3 4\n3 7\n",
        "output": "15\n1 8 8\n1 5 5\n1 6 6\n1 9 9\n1 4 4\n1 7 7\n2 1 1\n2 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 542,
        "task_id": 581,
        "test_case_id": 20,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "9\n2 6\n6 1\n2 8\n6 7\n1 5\n7 3\n8 9\n5 4\n",
        "output": "30\n4 3 3\n4 7 7\n9 4 4\n9 5 5\n9 1 1\n9 6 6\n9 2 2\n9 8 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 543,
        "task_id": 581,
        "test_case_id": 21,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "9\n9 4\n4 6\n6 2\n2 1\n1 3\n3 5\n5 8\n8 7\n",
        "output": "36\n7 9 9\n7 4 4\n7 6 6\n7 2 2\n7 1 1\n7 3 3\n7 5 5\n7 8 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 544,
        "task_id": 581,
        "test_case_id": 22,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n3 2\n3 7\n3 6\n3 8\n3 1\n3 5\n3 9\n3 4\n3 10\n",
        "output": "17\n1 7 7\n1 6 6\n1 8 8\n1 5 5\n1 9 9\n1 4 4\n1 10 10\n2 1 1\n2 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 545,
        "task_id": 581,
        "test_case_id": 23,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n8 2\n8 10\n10 3\n2 4\n3 6\n8 1\n2 7\n10 9\n4 5\n",
        "output": "35\n5 9 9\n6 1 1\n6 7 7\n5 6 6\n5 3 3\n5 10 10\n5 8 8\n5 2 2\n5 4 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 546,
        "task_id": 581,
        "test_case_id": 24,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n7 10\n10 6\n6 4\n4 5\n5 8\n8 2\n2 1\n1 3\n3 9\n",
        "output": "45\n7 9 9\n7 3 3\n7 1 1\n7 2 2\n7 8 8\n7 5 5\n7 4 4\n7 6 6\n7 10 10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 547,
        "task_id": 581,
        "test_case_id": 25,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "4\n3 4\n4 1\n1 2\n",
        "output": "6\n3 2 2\n3 1 1\n3 4 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 548,
        "task_id": 581,
        "test_case_id": 26,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "5\n1 4\n4 2\n2 3\n3 5\n",
        "output": "10\n5 1 1\n5 4 4\n5 2 2\n5 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 549,
        "task_id": 581,
        "test_case_id": 27,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "6\n5 3\n3 6\n6 1\n1 4\n4 2\n",
        "output": "15\n5 2 2\n5 4 4\n5 1 1\n5 6 6\n5 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 550,
        "task_id": 581,
        "test_case_id": 28,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "7\n1 2\n2 3\n3 6\n6 7\n7 4\n4 5\n",
        "output": "21\n5 1 1\n5 2 2\n5 3 3\n5 6 6\n5 7 7\n5 4 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 551,
        "task_id": 581,
        "test_case_id": 29,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "8\n6 2\n2 1\n1 8\n8 5\n5 7\n7 3\n3 4\n",
        "output": "28\n4 6 6\n4 2 2\n4 1 1\n4 8 8\n4 5 5\n4 7 7\n4 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 552,
        "task_id": 581,
        "test_case_id": 30,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "9\n1 6\n6 4\n4 5\n5 9\n9 8\n8 7\n7 3\n3 2\n",
        "output": "36\n2 1 1\n2 6 6\n2 4 4\n2 5 5\n2 9 9\n2 8 8\n2 7 7\n2 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 553,
        "task_id": 581,
        "test_case_id": 31,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n5 1\n1 6\n6 2\n2 8\n8 3\n3 4\n4 10\n10 9\n9 7\n",
        "output": "45\n7 5 5\n7 1 1\n7 6 6\n7 2 2\n7 8 8\n7 3 3\n7 4 4\n7 10 10\n7 9 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 554,
        "task_id": 581,
        "test_case_id": 32,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "4\n3 4\n3 1\n3 2\n",
        "output": "5\n1 4 4\n2 1 1\n2 3 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 555,
        "task_id": 581,
        "test_case_id": 33,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "5\n1 4\n1 2\n1 3\n1 5\n",
        "output": "7\n3 4 4\n3 5 5\n2 3 3\n2 1 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 556,
        "task_id": 581,
        "test_case_id": 34,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "6\n5 3\n5 6\n5 1\n5 4\n5 2\n",
        "output": "9\n1 3 3\n1 6 6\n1 4 4\n2 1 1\n2 5 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 557,
        "task_id": 581,
        "test_case_id": 35,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "7\n1 2\n1 3\n1 6\n1 7\n1 4\n1 5\n",
        "output": "11\n3 6 6\n3 7 7\n3 4 4\n3 5 5\n2 3 3\n2 1 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 558,
        "task_id": 581,
        "test_case_id": 36,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "8\n6 2\n6 1\n6 8\n6 5\n6 7\n6 3\n6 4\n",
        "output": "13\n1 8 8\n1 5 5\n1 7 7\n1 3 3\n1 4 4\n2 1 1\n2 6 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 559,
        "task_id": 581,
        "test_case_id": 37,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "9\n1 6\n1 4\n1 5\n1 9\n1 8\n1 7\n1 3\n1 2\n",
        "output": "15\n3 6 6\n3 4 4\n3 5 5\n3 9 9\n3 8 8\n3 7 7\n2 3 3\n2 1 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 560,
        "task_id": 581,
        "test_case_id": 38,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n5 1\n5 6\n5 2\n5 8\n5 3\n5 4\n5 10\n5 9\n5 7\n",
        "output": "17\n1 6 6\n1 8 8\n1 3 3\n1 4 4\n1 10 10\n1 9 9\n1 7 7\n2 1 1\n2 5 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 561,
        "task_id": 581,
        "test_case_id": 39,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n4 10\n10 5\n5 1\n1 6\n6 8\n8 9\n9 2\n9 3\n9 7\n",
        "output": "42\n4 3 3\n4 7 7\n2 4 4\n2 10 10\n2 5 5\n2 1 1\n2 6 6\n2 8 8\n2 9 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 562,
        "task_id": 581,
        "test_case_id": 40,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n5 8\n8 4\n4 9\n9 6\n6 1\n6 2\n6 7\n6 3\n6 10\n",
        "output": "35\n5 2 2\n5 7 7\n5 3 3\n5 10 10\n5 1 1\n5 6 6\n5 9 9\n5 4 4\n5 8 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 563,
        "task_id": 581,
        "test_case_id": 41,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "10\n5 6\n6 7\n7 3\n7 8\n7 4\n7 2\n7 1\n7 10\n7 9\n",
        "output": "24\n5 3 3\n5 8 8\n5 4 4\n5 2 2\n5 10 10\n5 9 9\n5 1 1\n5 7 7\n5 6 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 564,
        "task_id": 893,
        "test_case_id": 1,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "1 4\n2 1 3 2\n1 2\n1 3\n3 4\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 565,
        "task_id": 893,
        "test_case_id": 2,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "0 3\n1 2 3\n1 2\n2 3\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 566,
        "task_id": 893,
        "test_case_id": 5,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "20 20\n1024 1003 1021 1020 1030 1026 1019 1028 1026 1008 1007 1011 1040 1033 1037 1039 1035 1010 1034 1018\n2 3\n9 10\n3 9\n6 7\n19 20\n5 14\n3 8\n4 6\n4 5\n11 17\n1 12\n5 15\n5 13\n5 16\n1 2\n3 4\n11 19\n4 18\n6 11\n",
        "output": "321\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 567,
        "task_id": 893,
        "test_case_id": 8,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "9 9\n17 23 33 17 19 35 32 32 35\n7 8\n2 7\n3 5\n1 2\n3 4\n2 9\n2 3\n1 6\n",
        "output": "13\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 568,
        "task_id": 893,
        "test_case_id": 11,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "0 12\n943 479 214 1646 151 565 846 1315 347 1766 1547 945\n3 8\n1 3\n3 4\n1 7\n2 5\n7 10\n2 9\n9 11\n1 2\n10 12\n1 6\n",
        "output": "12\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 569,
        "task_id": 893,
        "test_case_id": 12,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "0 20\n78 1918 620 127 1022 1498 33 908 403 508 155 588 505 1277 104 1970 1408 285 1304 998\n10 11\n9 10\n4 12\n1 6\n2 13\n1 2\n8 9\n6 7\n4 5\n4 8\n1 4\n19 20\n2 3\n9 14\n8 15\n11 18\n14 17\n13 16\n16 19\n",
        "output": "20\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 570,
        "task_id": 893,
        "test_case_id": 13,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "0 21\n688 744 568 726 814 204 732 87 590 367 813 339 148 412 913 361 617 471 120 123 717\n2 4\n2 12\n14 15\n3 5\n1 8\n1 6\n3 20\n8 21\n2 3\n2 14\n6 10\n13 18\n1 2\n6 19\n6 16\n10 13\n4 11\n6 7\n1 17\n7 9\n",
        "output": "21\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 571,
        "task_id": 893,
        "test_case_id": 14,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "0 22\n1656 1462 1355 1133 1809 1410 1032 1417 1373 1545 1643 1099 1327 1037 1031 1697 1356 1072 1335 1524 1523 1642\n8 14\n11 13\n14 21\n9 16\n1 2\n4 11\n2 4\n1 17\n3 7\n19 20\n3 5\n6 9\n6 8\n3 6\n7 15\n2 3\n16 18\n2 12\n1 10\n13 19\n18 22\n",
        "output": "22\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 572,
        "task_id": 893,
        "test_case_id": 16,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "13 13\n1903 1950 1423 1852 1919 1187 1091 1156 1075 1407 1377 1352 1361\n4 5\n1 2\n7 11\n5 8\n2 13\n6 12\n6 7\n7 10\n1 3\n1 4\n2 9\n1 6\n",
        "output": "13\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 573,
        "task_id": 893,
        "test_case_id": 24,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "9 9\n1273 1293 1412 1423 1270 1340 1242 1305 1264\n2 8\n1 4\n5 9\n1 3\n2 5\n4 7\n1 2\n2 6\n",
        "output": "10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 574,
        "task_id": 893,
        "test_case_id": 27,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "0 12\n1972 1982 1996 1994 1972 1991 1999 1984 1994 1995 1990 1999\n1 2\n3 7\n6 11\n1 8\n4 5\n2 3\n2 4\n9 10\n10 12\n7 9\n3 6\n",
        "output": "12\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 575,
        "task_id": 1421,
        "test_case_id": 1,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n",
        "output": "25",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 576,
        "task_id": 1421,
        "test_case_id": 2,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "4\n1 -5 1 1\n1 2\n1 4\n2 3\n",
        "output": "2",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 577,
        "task_id": 1421,
        "test_case_id": 5,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-1 2 -2 -3 -1 -1 0 -4 -5 -4\n4 6\n6 9\n1 2\n6 2\n7 8\n7 9\n5 10\n6 3\n10 1\n",
        "output": "-3",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 578,
        "task_id": 1421,
        "test_case_id": 12,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n1 -4 -4 0 1 -3 1 -2 -4 2\n7 1\n7 6\n5 6\n4 2\n2 6\n3 9\n5 9\n10 6\n8 5\n",
        "output": "2",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 579,
        "task_id": 2246,
        "test_case_id": 1,
        "question": "There are n cities and n - 1 roads in the Seven Kingdoms, each road connects two cities and we can reach any city from any other by the roads.\n\nTheon and Yara Greyjoy are on a horse in the first city, they are starting traveling through the roads. But the weather is foggy, so they can’t see where the horse brings them. When the horse reaches a city (including the first one), it goes to one of the cities connected to the current city. But it is a strange horse, it only goes to cities in which they weren't before. In each such city, the horse goes with equal probabilities and it stops when there are no such cities. \n\nLet the length of each road be 1. The journey starts in the city 1. What is the expected length (expected value of length) of their journey? You can read about expected (average) value by the link https://en.wikipedia.org/wiki/Expected_value.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 100000) — number of cities.\n\nThen n - 1 lines follow. The i-th line of these lines contains two integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the cities connected by the i-th road.\n\nIt is guaranteed that one can reach any city from any other by the roads.\n\n\n-----Output-----\n\nPrint a number — the expected length of their journey. The journey starts in the city 1.\n\nYour answer will be considered correct if its absolute or relative error does not exceed 10^{ - 6}.\n\nNamely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if $\\frac{|a - b|}{\\operatorname{max}(1, b)} \\leq 10^{-6}$.\n\n\n-----Examples-----\nInput\n4\n1 2\n1 3\n2 4\n\nOutput\n1.500000000000000\n\nInput\n5\n1 2\n1 3\n3 4\n2 5\n\nOutput\n2.000000000000000\n\n\n\n-----Note-----\n\nIn the first sample, their journey may end in cities 3 or 4 with equal probability. The distance to city 3 is 1 and to city 4 is 2, so the expected length is 1.5.\n\nIn the second sample, their journey may end in city 4 or 5. The distance to the both cities is 2, so the expected length is 2.",
        "solutions": "[\"from sys import stdin\\nfrom decimal import Decimal as D\\ninput = stdin.readline\\nn = int(input())\\nadj = [[] for i in range(n+1)]\\ntree = [[] for i in range(n+1)]\\nvisit = [-1]*(n+1)\\nlength = [-1]*(n+1)\\nfor i in range(n-1):\\n    a, b = map(int,input().split())\\n    adj[a].append(b)\\n    adj[b].append(a)\\nbfsord = []\\n\\nfrom collections import deque\\nQ = deque()\\nQ.append(1)\\nvisit[1] = 0\\nwhile len(Q):\\n    p = Q.popleft()\\n    bfsord.append(p)\\n    for q in adj[p]:\\n        if visit[q] != -1: continue\\n        visit[q] = visit[p]+1\\n        Q.append(q)\\n        tree[p].append(q)\\n\\nfor p in reversed(bfsord):\\n    if not tree[p]: length[p] = D(0)\\n    else: length[p] = D(1) + sum(length[q] for q in tree[p])/len(tree[p])\\nprint(length[1])\", \"import sys\\n\\ndef r():\\n    return list(map(int, input().split()))\\n\\nn = int(input())\\nedge = [r() for i in range(n-1)]\\n\\nadj = [[] for i in range(n+1)]\\nfor e in edge:\\n    adj[e[0]].append(e[1])\\n    adj[e[1]].append(e[0])\\n\\nprob = [0.0 for i in range(n+1)]\\nd = [0 for i in range(n+1)]\\nvisited = [False for i in range(n+1)]\\n\\nprob[1] = 1\\nans = 0.0\\n\\nst = [1]\\nvisited[1] = True\\nwhile st:\\n    u = st.pop()\\n    cnt = len(adj[u])\\n    if u != 1:\\n        cnt -= 1\\n    if cnt == 0:\\n        ans = ans + prob[u]*d[u]\\n    else:\\n        for v in adj[u]:\\n            if not visited[v]:\\n                visited[v] = True\\n                prob[v] = prob[u]*(1.0/cnt)\\n                d[v] = d[u]+1\\n                st.append(v)\\n                \\nprint(ans)\\n\\n\", \"from queue import *\\n\\nn = int(input())\\ng = [[] for x in range(n)]\\n\\nfor i in range(n-1):\\n  a, b = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  g[a].append(b)\\n  g[b].append(a)\\n\\nused = [0]*n\\n\\nanw = 0\\n\\ndef solve(v, d, r):\\n  nonlocal anw\\n  q = Queue()\\n  q.put((v, d, r))\\n  while not q.empty():\\n    v, d, r = q.get(v)\\n    used[v] = True\\n    for u in g[v]:\\n      if not used[u]:\\n        q.put((u, d+1, r*(len(g[v])-(v!=0))))\\n        #print(\\\"put\\\", u)\\n    #print(\\\"so at\\\", v, \\\"len\\\", len(g[v]))\\n    if v != 0 and len(g[v]) == 1:\\n      #print(\\\"At \\\", v, \\\"is\\\", d, r)\\n      anw += d/r\\n  \\nsolve(0, 0, 1)\\nprint(anw)\", \"def f():\\n    n = int(input())\\n\\n    if n == 1:\\n        print(0)\\n        return\\n\\n    road = {}\\n    visited = {}\\n    for i in range(n-1):\\n        u, v = input().split()\\n        if u in road:\\n            road[u].append(v)\\n        else:\\n            road[u] = [v]\\n        if v in road:\\n            road[v].append(u)\\n        else:\\n            road[v] = [u]\\n        visited[u] = False\\n        visited[v] = False\\n\\n    prob = {}\\n    length = {}\\n    res = []\\n\\n    queue = []\\n    for k in road['1']:\\n        prob[k] = 1.0 / len(road['1'])\\n        length[k] = 1\\n        queue.append(k)\\n        visited['1'] = True\\n\\n    while len(queue) > 0:\\n        cur = queue[0]\\n        del queue[0]\\n        dest = []\\n        for k in road[cur]:\\n            if not visited[k]:\\n                dest.append(k)\\n        if len(dest) == 0:\\n            res.append((cur, prob[cur], length[cur]))\\n            continue\\n        for k in dest:\\n            prob[k] = prob[cur] / len(dest)\\n            length[k] = length[cur] + 1\\n            queue.append(k)\\n            visited[cur] = True\\n\\n    val = 0.0\\n    for item in res:\\n        val += item[1] * item[2]\\n\\n    print(val)\\n\\nf()\\n\", \"from collections import deque\\nimport sys\\nfrom decimal import Decimal\\n\\nreadline = sys.stdin.readline\\nn = int(input())\\nedges = [[]*n for _ in [None]*n]\\n\\nfor _ in [None]*(n-1):\\n    a, b = map(int, readline().split())\\n    edges[a-1].append(b-1)\\n    edges[b-1].append(a-1)\\n\\nvisited = [False]*n\\nvisited[0] = True\\ncnt = 0\\ntotal = 0\\ndq = deque()\\nappend, pop = dq.append, dq.popleft\\nappend((0, 0, Decimal(1)))\\n\\nwhile dq:\\n    pos, l, multiple = pop()\\n    flag = True\\n    to = [x for x in edges[pos] if visited[x] == False]\\n\\n    if to:\\n        x = len(to)\\n        multiple /= Decimal(x)\\n        for v in to:\\n            visited[v] = True\\n            append((v, l+1, multiple))\\n    else:\\n        total += Decimal(l) * multiple\\n\\nprint(total)\", \"import sys\\nfrom collections import deque\\nread=lambda:sys.stdin.readline().rstrip()\\nreadi=lambda:int(sys.stdin.readline())\\nwriteln=lambda x:sys.stdout.write(str(x)+\\\"\\\\n\\\")\\nwrite=lambda x:sys.stdout.write(x)\\nN = readi()\\nif N == 1:\\n    writeln(0)\\n    return\\n\\nG = [set() for _ in range(N)]\\nfor _ in range(N-1):\\n    u, v = list(map(int, read().split()))\\n    G[u-1].add(v-1)\\n    G[v-1].add(u-1)\\n\\nvisited = [0]*N\\nprev = [0]*N\\nvisited[0] = 1\\nprev[0] = 1\\nq = deque([0])\\nlengths = []\\nev = 0\\nwhile q:\\n    cur = q.popleft()\\n    n_neighbor = len(G[cur])\\n    if cur == 0:\\n        divs = n_neighbor\\n    else:\\n        divs = n_neighbor - 1\\n    if n_neighbor == 1 and cur != 0:\\n        ev += (visited[cur] - 1)*prev[cur]\\n        continue\\n     \\n    for nxt in G[cur]:\\n        if not visited[nxt]:\\n            visited[nxt] = visited[cur] + 1\\n            prev[nxt] = (1 / divs)*prev[cur]\\n            q.append(nxt)\\n\\nprint('%.12f' % ev)\\n\", \"#!/usr/bin/env python\\n\\nimport sys\\n\\ndef explore(src, prob, length, adj_mat, gains, visited):\\n    nexts = set(adj_mat[src]) - visited\\n    if nexts:\\n        go_prob = 1 / len(nexts)\\n        length += 1\\n        for dst in nexts:\\n            visited.add(dst)\\n            explore(dst, prob * go_prob, length, adj_mat, gains, visited)\\n    else:\\n        gains[src] += prob * length\\n\\ndef main():\\n  n = int(sys.stdin.readline())\\n  adj_mat = [[] for __ in range(n)]\\n  node_exp = [0 for __ in range(n)]\\n  for __ in range(n-1):\\n      u, v = list(map(int, sys.stdin.readline().split()))\\n      adj_mat[u-1].append(v-1)\\n      adj_mat[v-1].append(u-1)\\n  stk = [0]\\n  visited= set([0])\\n  prob = [1] * n\\n  length = [0] * n\\n  while stk:\\n      nxt = stk.pop()\\n      nexts = set(adj_mat[nxt]) - visited\\n      if nexts:\\n          go_prob = 1 / len(nexts)\\n          for dst in nexts:\\n              visited.add(dst)\\n              stk.append(dst)\\n              prob[dst] *= prob[nxt] * go_prob\\n              length[dst] = length[nxt] + 1\\n      else:\\n        node_exp[nxt] += prob[nxt] * length[nxt]\\n\\n  #explore(0, 1, 0, adj_mat,node_exp, set([0]))\\n  print(\\\"{:.13f}\\\".format(sum(node_exp)))\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n    if n == 1:\\n        print(0)\\n        return\\n    x = [[] for i in range(n)]\\n    for i in range(n - 1):\\n        a, b = list(map(int, sys.stdin.readline().split()))\\n        x[a - 1].append(b - 1)\\n        x[b - 1].append(a - 1)\\n\\n    isl = [False] * n\\n    for i in range(1, n):\\n        if len(x[i]) == 1:\\n            isl[i] = True\\n\\n    u = [False] * n\\n    u[0] = True\\n    q = [(0, 0, 1 / len(x[0]))]\\n    a = 0\\n    while len(q) != 0:\\n        c, s, t = q.pop()\\n        for i in x[c]:\\n            if not u[i]:\\n                if isl[i]:\\n                    a += (s + 1) * t\\n                else:\\n                    q.append((i, s + 1, t / (len(x[i])-1)))\\n                u[i] = True\\n    print(a)\\n\\n\\nmain()\\n\", \"n = int(input())\\nr = [[]for _ in range(n+1)]\\n\\nfor _ in range(n-1):\\n\\tu,v = list(map(int,input().split()))\\n\\tu -= 1\\n\\tv -= 1\\n\\tr[u].append(v)\\n\\tr[v].append(u)\\n\\nif n == 1:\\n\\tprint(0)\\nelse:\\n\\tst = [(0,0,1/len(r[0]))]\\n\\tans = 0\\n\\trsum = 0\\n\\tvisit = [False] * n\\n\\tvisit[0] = True\\n\\twhile st:\\n\\t\\tv,t,m = st.pop()\\n\\t\\tfor l in r[v]:\\n\\t\\t\\tif not visit[l]:\\n\\t\\t\\t\\tif l != 0 and len(r[l]) == 1:\\n\\t\\t\\t\\t\\tans += (t+1)*m\\n\\t\\t\\t\\t\\trsum += 1\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\tst.append((l,t+1,m/(len(r[l])-1)))\\n\\t\\t\\t\\tvisit[l] = True\\n\\tprint(ans)\\n\", \"from collections import deque, Counter\\n\\ndef mean_of_ways(graph, n):\\n    leafs = []\\n    distances = [None] * n\\n    probabilities = [0] * n\\n    probabilities[0] = 1\\n    distances[0] = 0\\n    used = [False] * n\\n    used[0] = True\\n\\n    active_vertices = deque([0])\\n\\n    while active_vertices:\\n        where = active_vertices.popleft()\\n\\n        # remember what happens if we change place of used assignment\\n        is_leaf = True\\n\\n        leafs_count = ([not used[v] for v in graph[where]]).count(True)\\n        if leafs_count > 1:\\n            probabilities[where] /= leafs_count\\n\\n        for to in graph[where]:\\n            if not used[to]:\\n                active_vertices.append(to)\\n                distances[to] = distances[where] + 1\\n                probabilities[to] = probabilities[where]\\n                used[to] = True\\n                is_leaf = False\\n\\n        if is_leaf:\\n            leafs.append(where)\\n\\n    Mx = 0\\n    for i in leafs:\\n        Mx += probabilities[i] * distances[i]\\n    return Mx\\n\\ndef main():\\n    n = int(input())\\n\\n    graph = [list() for i in range(n)]\\n    for i in range(n-1):\\n        # It there are vertices in descending order we can don't check\\n        # used it or not by making edges one directional\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n    print(mean_of_ways(graph, n))\\n\\nmain()\\n\", \"\\\"\\\"\\\"\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout\\n\\n\\ndef main():\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n\\n    # iterative DFS\\n    stack = [(1, 0, 1)]\\n    while len(stack):\\n        u, plen, prb = stack.pop()\\n\\n        visited[u] = True\\n        nBranch = 0\\n        for v in g[u]:\\n            if not visited[v]: nBranch += 1\\n        \\n        if nBranch > 0: \\n            probability = prb * (1 / nBranch)\\n            for v in g[u]:\\n                if not visited[v]:\\n                    stack.append((v, plen+1, probability))\\n        \\n        if nBranch == 0:\\n            paths.append(plen * prb)\\n\\n        \\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\n# node, probablity, path\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\n# print(L)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"\\\"\\\"\\\"\\n    # recursive DFS gives RTE, implement iterative DFS\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout, setrecursionlimit\\nimport threading\\n\\n\\ng = None\\nvisited = None\\npaths = None\\n\\ndef dfs(u, plen, prb):\\n    nonlocal visited, paths\\n\\n    visited[u] = True\\n    \\n    nBranch = 0\\n    for v in g[u]:\\n        if not visited[v]: nBranch += 1\\n    \\n    if nBranch > 0: \\n        probability = prb * (1 / nBranch)\\n        for v in g[u]:\\n            #print(prb, nBranch)\\n            if not visited[v]:\\n                dfs(v, plen+1, probability)\\n\\n    if nBranch == 0:\\n        paths.append(plen * prb)\\n\\n\\ndef main():\\n    nonlocal g, visited, paths\\n\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n    #setrecursionlimit(n+10)\\n    dfs(1, 0, 1)\\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    setrecursionlimit(10**6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"n = int(input())\\n\\nd = { x : [] for x in range(1, n + 1)}#defaultdict(list)\\nfor x in range(n - 1):\\n    s,de = map(int, input().split())\\n    d[s].append(de)\\n    d[de].append(s)\\n\\ndef dfs():\\n    lst = [(0, 1.0, 1)]\\n    visited = set({1})\\n    ans = 0\\n\\n    while lst:\\n        depth,prob,source = lst.pop()\\n        \\n        for neigh in d[source]:\\n            if neigh not in visited:\\n                visited.add(neigh)\\n                lst.append((depth + 1, prob*(1/(len(d[source]) - (1 if depth != 0 else 0))), neigh))\\n        \\n        if depth != 0 and len(d[source]) == 1:\\n            ans += prob*depth\\n\\n    return ans\\n\\nprint(\\\"{:.8f}\\\".format(dfs()))\", \"n = int(input())\\n\\ntree = {x: [] for x in range(1, n+1)}\\n\\nfor _ in range(n-1):\\n    k,l = list(map(int, input().split()))\\n    tree[k].append(l)\\n    tree[l].append(k)\\n\\n#for i in range(99999):\\n#    tree[i+1].append(i+2)\\n#    tree[i+2].append(i+1)\\n\\nvisited = set()\\ns = 0\\n\\na = [(1,1,0)]\\n\\nwhile a:\\n    v,p,l = a.pop()\\n    visited.add(v) \\n    k = 0\\n    for vv in tree[v]:\\n        if vv not in visited:\\n            k += 1\\n    if k <= 0:\\n        s += p*l\\n    else:\\n        for vv in tree[v]:\\n            if vv not in visited:\\n                a.append((vv,p*1.0/k,l+1))\\n\\nprint(s)\", \"from collections import defaultdict\\nfrom collections import deque\\nfrom sys import stdin\\nlines = deque(line.strip() for line in stdin.readlines())\\n\\ndef nextline():\\n    return lines.popleft()\\n\\ndef types(cast):\\n    return tuple(int(x) for x in nextline().split())\\n\\ndef ints():\\n    return types(int)\\n\\ndef strs():\\n    return nextline().split()\\n\\ndef main():\\n    # lines will now contain all of the input's lines in a list\\n    n = int(nextline())\\n    graph = defaultdict(list)\\n    for _ in range(1, n):\\n        a, b = ints()\\n        graph[a].append(b)\\n        graph[b].append(a)\\n    stack = [(1, 1, 0)]\\n    visited = set()\\n    expected = 0\\n    while stack:\\n        city, denominator, length = stack.pop()\\n        visited.add(city)\\n        choices = [choice for choice in graph[city] if choice not in visited]\\n        if choices:\\n            for choice in choices:\\n                stack.append((choice, denominator * len(choices), length + 1))\\n        else:\\n            expected += length / denominator\\n    print(expected)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import queue\\nn = int(input())\\ndic = {}\\n\\nfor i in range(n+1):\\n    dic[i] = []\\nfor i in range(n-1):\\n    u,v = [ int(x) for x in input().split() ]\\n    dic[u].append(v)\\n    dic[v].append(u)\\n\\nweight = {1:(0,1)} # No.City:(length,weight)\\nq = queue.Queue()\\nq.put(1)\\nwhile(q.empty() == False):\\n    city = q.get()\\n    choice = len(dic[city])\\n    if city > 1:\\n        choice = choice - 1\\n    for to_city in dic[city]:\\n        if to_city not in weight:\\n            weight[to_city] = (weight[city][0]+1, weight[city][1]/choice)\\n            q.put(to_city)\\n    if(choice > 0):\\n        weight[city] = (weight[city][0], 0)\\n\\nsum = 0\\nfor city in weight:\\n    sum = sum + weight[city][0] * weight[city][1]\\n\\nprint(\\\"%.15f\\\" % sum)\\n\\n\", \"def __starting_point():\\n    n = int(input())\\n    \\n    edges = [[] for i in range(n+1)]\\n    edges[0] = None\\n    for i in range(n-1):\\n        n1,n2 = list(map(int, input().split()))\\n        edges[n1].append(n2)\\n        edges[n2].append(n1)\\n\\n    # stack = [(node, height, prob),...]\\n    stack = [(1,0,1.0)]\\n    sev = 0\\n    visited = [False for i in range(n+1)]\\n    \\n    while len(stack) > 0:\\n        (node, height, prob) = stack.pop()\\n        visited[node] = True\\n        children = len(edges[node]) - 1\\n        if node == 1:\\n            # since root has no parent\\n            children += 1\\n            \\n        for child in edges[node]:\\n            if visited[child]:\\n                continue\\n            stack.append((child, height+1, prob/children))\\n        \\n        if children == 0:\\n            sev += height * prob\\n    print(sev)\\n            \\n    \\n    \\n    \\n    \\n\\n\\n\\n__starting_point()\", \"class Node:\\n\\tdef __init__(self, index):\\n\\t\\tself.neighbours = []\\n\\t\\tself.index = index\\n\\t\\tself.prob = 1.0\\n\\t\\tself.vis = False\\n\\t\\tself.length = 0\\n\\n\\tdef addNode(self, city):\\n\\t\\tself.neighbours.append(city)\\n\\t\\tif self.index == 1:\\n\\t\\t\\tself.prob = 1.0 / len(self.neighbours)\\n\\t\\telse:\\n\\t\\t\\tl = len(self.neighbours)\\n\\t\\t\\tself.prob = 1.0  if l < 2 else (1.0 / (l - 1))\\n\\nn = int(input())\\nif n == 1:\\n\\tprint(0)\\n\\treturn\\n\\t\\ncities = {}\\n\\nfor i in range(n-1):\\n\\ta, b = [int(k) for k in input().split()]\\n\\t#print (\\\"test \\\", a, \\\" to \\\", b)\\n\\tif a not in cities:\\n\\t\\tcities[a] = Node(a)\\n\\tcities[a].addNode(b)\\n\\tif b not in cities:\\n\\t\\tcities[b] = Node(b)\\n\\tcities[b].addNode(a)\\n\\nif len(cities) == 2:\\n\\tprint(1)\\n\\treturn\\n\\n\\n# for i in range(1, n + 1, 1):\\n# \\tprint (\\\"city.index \\\", cities[i].index, ' roads ', cities[i].neighbours, ' prob ', cities[i].prob)\\n\\n\\n# deadends = []\\n# deadendsProb = []\\n\\n\\n# def Parse(city, prob, length, oldCity):\\n# \\t#print ('parse ', city.index)\\n# \\tnewprob = 1.0 if oldCity == 0  else cities[oldCity].prob\\n# \\t#print ('nnewProb ', newprob)\\n# \\tprob *= newprob\\n# \\t#print (city.index, ' len ', len(city.neighbours))\\n# \\tif len(city.neighbours) == 1 and oldCity != 0:\\n# \\t\\tdeadends.append(length)\\n# \\t\\tdeadendsProb.append(prob)\\n# \\telse:\\n# \\t\\t#print (city.neighbours)\\n# \\t\\tlength += 1\\n# \\t\\tfor c in city.neighbours:\\n# \\t\\t\\tif c != oldCity:\\n# \\t\\t\\t\\tParse(cities[c], prob, length, city.index )\\n\\n# Parse(cities[1], 1.0, 0, 0)\\n# #for i in range(len(deadends)):\\n# #\\tprint('len ', deadends[i], ' prob ', deadendsProb[i])\\n\\n# ans = sum(map(lambda l, p: l * p, deadends, deadendsProb))\\n# #print('ans', ans)\\n# print(ans)\\n\\n\\ndef inorder(city):\\n\\ts = []\\n\\ts.append(city)\\n\\t#print ('index ', city.index)\\n\\t#print ('neighbours ', city.neighbours)\\n\\twhile s:\\n\\t\\tcity = s.pop()\\n\\t\\tcity.vis = True\\n\\t\\tif city.neighbours:\\n\\t\\t\\tif city.index == 1 or len(city.neighbours) > 1:\\n\\t\\t\\t\\tfor c in city.neighbours:\\n\\t\\t\\t\\t\\tif not cities[c].vis:\\n\\t\\t\\t\\t\\t\\tcities[c].length = city.length + 1\\n\\t\\t\\t\\t\\t\\tcities[c].prob *= city.prob\\n\\t\\t\\t\\t\\t\\ts.append(cities[c])\\n\\t\\t\\telse:\\n\\t\\t\\t\\tyield (city.index, city.prob, city.length)\\n\\n\\n\\ntest = sum([city[1] * city[2] for city in inorder(cities[1])])\\nprint(test)\\n\\n\\n\\n# this is a tree\\n\", \"from sys import setrecursionlimit\\nimport threading\\n\\n\\nlength = 0\\nvisited = None\\ng = None\\n\\n\\ndef dfs(v, path_length, chance):\\n    nonlocal length\\n    visited.add(v)\\n    edges = 0\\n    for n in g[v]:\\n        if n not in visited:\\n            edges += 1\\n    for n in g[v]:\\n        if n not in visited:\\n            dfs(n, path_length + 1, chance * (1 / edges))\\n    if edges == 0:\\n        length += path_length * chance\\n\\n\\ndef main():\\n    nonlocal visited, g\\n    vert = int(input())\\n    g = [[] for _ in range(vert + 1)]\\n    visited = set()\\n    for _ in range(vert - 1):\\n        a, b = list(map(int, input().split()))\\n        g[a].append(b)\\n        g[b].append(a)\\n    dfs(1, 0, 1)\\n    print(length)\\n\\n\\ndef __starting_point():\\n    setrecursionlimit(10 ** 6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"n = int(input())\\ng = [[] for i in range(n+1)]\\nfor i in range(n-1):\\n    a,b = [int(x) for x in input().split()]\\n    g[a].append(b)\\n    g[b].append(a)\\n\\nd = [0 for i in range(n+1)]\\nv = [0 for i in range(n+1)]\\ne = [0 for i in range(n+1)]\\np = [0.0 for i in range(n+1)]\\np[1] = 1.0\\nen = 0\\nq = [1]\\nwhile len(q):\\n    c = q.pop(0)\\n    v[c] = 1\\n    a = 0\\n    for ne in g[c]:\\n        if not v[ne]:\\n            q.append(ne)\\n            d[ne] = d[c] + 1\\n            p[ne] = p[c]/len(g[c]) if c == 1 else p[c]/(len(g[c])-1)\\n            a += 1\\n    if a == 0:\\n        e[c] = 1\\n        en += 1\\nave = 0.0\\nfor i in range(n+1):\\n    if e[i]:\\n        ave += d[i]*p[i]\\nprint(ave)\\n\", \"n=int(input())\\nif n==1:\\n\\tprint(0)\\n\\treturn\\ngraph=[None]*n\\nfor i in range(n-1):\\n\\tu,v=map(int,input().split())\\n\\tu-=1\\n\\tv-=1\\n\\tif graph[u]==None:\\n\\t\\tgraph[u]=[v]\\n\\telse:\\n\\t\\tgraph[u].append(v)\\n\\tif graph[v]==None:\\n\\t\\tgraph[v]=[u]\\n\\telse:\\n\\t\\tgraph[v].append(u)\\n\\nis_checked=[False]*n\\nlens=[None]*n\\nlens[0]=0\\ndivs=[None]*n\\ndivs[0]=1\\nstack=[0]\\nans=0\\nwhile stack:\\n\\ttop=stack.pop()\\n\\tis_checked[top]=True\\n\\tvals=graph[top]\\n\\tfor i in range(len(vals)):\\n\\t\\tcur=vals[i]\\n\\t\\tif not is_checked[cur]:\\n\\t\\t\\tlens[cur]=lens[top]+1\\n\\t\\t\\tdivs[cur]=divs[top]*(len(vals)-(1 if top!=0 else 0))\\n\\t\\t\\tif len(graph[cur])==1:\\n\\t\\t\\t\\tans+=lens[cur]/divs[cur]\\n\\t\\t\\t\\tis_checked[cur]=None\\n\\t\\t\\t\\tlens[cur]=None\\n\\t\\t\\t\\tdivs[cur]=None\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tstack.append(cur)\\nprint('{:.8f}'.format(ans))\"]",
        "difficulty": "interview",
        "input": "4\n1 2\n1 3\n2 4\n",
        "output": "1.500000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/839/C"
    },
    {
        "id": 580,
        "task_id": 2246,
        "test_case_id": 2,
        "question": "There are n cities and n - 1 roads in the Seven Kingdoms, each road connects two cities and we can reach any city from any other by the roads.\n\nTheon and Yara Greyjoy are on a horse in the first city, they are starting traveling through the roads. But the weather is foggy, so they can’t see where the horse brings them. When the horse reaches a city (including the first one), it goes to one of the cities connected to the current city. But it is a strange horse, it only goes to cities in which they weren't before. In each such city, the horse goes with equal probabilities and it stops when there are no such cities. \n\nLet the length of each road be 1. The journey starts in the city 1. What is the expected length (expected value of length) of their journey? You can read about expected (average) value by the link https://en.wikipedia.org/wiki/Expected_value.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 100000) — number of cities.\n\nThen n - 1 lines follow. The i-th line of these lines contains two integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the cities connected by the i-th road.\n\nIt is guaranteed that one can reach any city from any other by the roads.\n\n\n-----Output-----\n\nPrint a number — the expected length of their journey. The journey starts in the city 1.\n\nYour answer will be considered correct if its absolute or relative error does not exceed 10^{ - 6}.\n\nNamely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if $\\frac{|a - b|}{\\operatorname{max}(1, b)} \\leq 10^{-6}$.\n\n\n-----Examples-----\nInput\n4\n1 2\n1 3\n2 4\n\nOutput\n1.500000000000000\n\nInput\n5\n1 2\n1 3\n3 4\n2 5\n\nOutput\n2.000000000000000\n\n\n\n-----Note-----\n\nIn the first sample, their journey may end in cities 3 or 4 with equal probability. The distance to city 3 is 1 and to city 4 is 2, so the expected length is 1.5.\n\nIn the second sample, their journey may end in city 4 or 5. The distance to the both cities is 2, so the expected length is 2.",
        "solutions": "[\"from sys import stdin\\nfrom decimal import Decimal as D\\ninput = stdin.readline\\nn = int(input())\\nadj = [[] for i in range(n+1)]\\ntree = [[] for i in range(n+1)]\\nvisit = [-1]*(n+1)\\nlength = [-1]*(n+1)\\nfor i in range(n-1):\\n    a, b = map(int,input().split())\\n    adj[a].append(b)\\n    adj[b].append(a)\\nbfsord = []\\n\\nfrom collections import deque\\nQ = deque()\\nQ.append(1)\\nvisit[1] = 0\\nwhile len(Q):\\n    p = Q.popleft()\\n    bfsord.append(p)\\n    for q in adj[p]:\\n        if visit[q] != -1: continue\\n        visit[q] = visit[p]+1\\n        Q.append(q)\\n        tree[p].append(q)\\n\\nfor p in reversed(bfsord):\\n    if not tree[p]: length[p] = D(0)\\n    else: length[p] = D(1) + sum(length[q] for q in tree[p])/len(tree[p])\\nprint(length[1])\", \"import sys\\n\\ndef r():\\n    return list(map(int, input().split()))\\n\\nn = int(input())\\nedge = [r() for i in range(n-1)]\\n\\nadj = [[] for i in range(n+1)]\\nfor e in edge:\\n    adj[e[0]].append(e[1])\\n    adj[e[1]].append(e[0])\\n\\nprob = [0.0 for i in range(n+1)]\\nd = [0 for i in range(n+1)]\\nvisited = [False for i in range(n+1)]\\n\\nprob[1] = 1\\nans = 0.0\\n\\nst = [1]\\nvisited[1] = True\\nwhile st:\\n    u = st.pop()\\n    cnt = len(adj[u])\\n    if u != 1:\\n        cnt -= 1\\n    if cnt == 0:\\n        ans = ans + prob[u]*d[u]\\n    else:\\n        for v in adj[u]:\\n            if not visited[v]:\\n                visited[v] = True\\n                prob[v] = prob[u]*(1.0/cnt)\\n                d[v] = d[u]+1\\n                st.append(v)\\n                \\nprint(ans)\\n\\n\", \"from queue import *\\n\\nn = int(input())\\ng = [[] for x in range(n)]\\n\\nfor i in range(n-1):\\n  a, b = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  g[a].append(b)\\n  g[b].append(a)\\n\\nused = [0]*n\\n\\nanw = 0\\n\\ndef solve(v, d, r):\\n  nonlocal anw\\n  q = Queue()\\n  q.put((v, d, r))\\n  while not q.empty():\\n    v, d, r = q.get(v)\\n    used[v] = True\\n    for u in g[v]:\\n      if not used[u]:\\n        q.put((u, d+1, r*(len(g[v])-(v!=0))))\\n        #print(\\\"put\\\", u)\\n    #print(\\\"so at\\\", v, \\\"len\\\", len(g[v]))\\n    if v != 0 and len(g[v]) == 1:\\n      #print(\\\"At \\\", v, \\\"is\\\", d, r)\\n      anw += d/r\\n  \\nsolve(0, 0, 1)\\nprint(anw)\", \"def f():\\n    n = int(input())\\n\\n    if n == 1:\\n        print(0)\\n        return\\n\\n    road = {}\\n    visited = {}\\n    for i in range(n-1):\\n        u, v = input().split()\\n        if u in road:\\n            road[u].append(v)\\n        else:\\n            road[u] = [v]\\n        if v in road:\\n            road[v].append(u)\\n        else:\\n            road[v] = [u]\\n        visited[u] = False\\n        visited[v] = False\\n\\n    prob = {}\\n    length = {}\\n    res = []\\n\\n    queue = []\\n    for k in road['1']:\\n        prob[k] = 1.0 / len(road['1'])\\n        length[k] = 1\\n        queue.append(k)\\n        visited['1'] = True\\n\\n    while len(queue) > 0:\\n        cur = queue[0]\\n        del queue[0]\\n        dest = []\\n        for k in road[cur]:\\n            if not visited[k]:\\n                dest.append(k)\\n        if len(dest) == 0:\\n            res.append((cur, prob[cur], length[cur]))\\n            continue\\n        for k in dest:\\n            prob[k] = prob[cur] / len(dest)\\n            length[k] = length[cur] + 1\\n            queue.append(k)\\n            visited[cur] = True\\n\\n    val = 0.0\\n    for item in res:\\n        val += item[1] * item[2]\\n\\n    print(val)\\n\\nf()\\n\", \"from collections import deque\\nimport sys\\nfrom decimal import Decimal\\n\\nreadline = sys.stdin.readline\\nn = int(input())\\nedges = [[]*n for _ in [None]*n]\\n\\nfor _ in [None]*(n-1):\\n    a, b = map(int, readline().split())\\n    edges[a-1].append(b-1)\\n    edges[b-1].append(a-1)\\n\\nvisited = [False]*n\\nvisited[0] = True\\ncnt = 0\\ntotal = 0\\ndq = deque()\\nappend, pop = dq.append, dq.popleft\\nappend((0, 0, Decimal(1)))\\n\\nwhile dq:\\n    pos, l, multiple = pop()\\n    flag = True\\n    to = [x for x in edges[pos] if visited[x] == False]\\n\\n    if to:\\n        x = len(to)\\n        multiple /= Decimal(x)\\n        for v in to:\\n            visited[v] = True\\n            append((v, l+1, multiple))\\n    else:\\n        total += Decimal(l) * multiple\\n\\nprint(total)\", \"import sys\\nfrom collections import deque\\nread=lambda:sys.stdin.readline().rstrip()\\nreadi=lambda:int(sys.stdin.readline())\\nwriteln=lambda x:sys.stdout.write(str(x)+\\\"\\\\n\\\")\\nwrite=lambda x:sys.stdout.write(x)\\nN = readi()\\nif N == 1:\\n    writeln(0)\\n    return\\n\\nG = [set() for _ in range(N)]\\nfor _ in range(N-1):\\n    u, v = list(map(int, read().split()))\\n    G[u-1].add(v-1)\\n    G[v-1].add(u-1)\\n\\nvisited = [0]*N\\nprev = [0]*N\\nvisited[0] = 1\\nprev[0] = 1\\nq = deque([0])\\nlengths = []\\nev = 0\\nwhile q:\\n    cur = q.popleft()\\n    n_neighbor = len(G[cur])\\n    if cur == 0:\\n        divs = n_neighbor\\n    else:\\n        divs = n_neighbor - 1\\n    if n_neighbor == 1 and cur != 0:\\n        ev += (visited[cur] - 1)*prev[cur]\\n        continue\\n     \\n    for nxt in G[cur]:\\n        if not visited[nxt]:\\n            visited[nxt] = visited[cur] + 1\\n            prev[nxt] = (1 / divs)*prev[cur]\\n            q.append(nxt)\\n\\nprint('%.12f' % ev)\\n\", \"#!/usr/bin/env python\\n\\nimport sys\\n\\ndef explore(src, prob, length, adj_mat, gains, visited):\\n    nexts = set(adj_mat[src]) - visited\\n    if nexts:\\n        go_prob = 1 / len(nexts)\\n        length += 1\\n        for dst in nexts:\\n            visited.add(dst)\\n            explore(dst, prob * go_prob, length, adj_mat, gains, visited)\\n    else:\\n        gains[src] += prob * length\\n\\ndef main():\\n  n = int(sys.stdin.readline())\\n  adj_mat = [[] for __ in range(n)]\\n  node_exp = [0 for __ in range(n)]\\n  for __ in range(n-1):\\n      u, v = list(map(int, sys.stdin.readline().split()))\\n      adj_mat[u-1].append(v-1)\\n      adj_mat[v-1].append(u-1)\\n  stk = [0]\\n  visited= set([0])\\n  prob = [1] * n\\n  length = [0] * n\\n  while stk:\\n      nxt = stk.pop()\\n      nexts = set(adj_mat[nxt]) - visited\\n      if nexts:\\n          go_prob = 1 / len(nexts)\\n          for dst in nexts:\\n              visited.add(dst)\\n              stk.append(dst)\\n              prob[dst] *= prob[nxt] * go_prob\\n              length[dst] = length[nxt] + 1\\n      else:\\n        node_exp[nxt] += prob[nxt] * length[nxt]\\n\\n  #explore(0, 1, 0, adj_mat,node_exp, set([0]))\\n  print(\\\"{:.13f}\\\".format(sum(node_exp)))\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n    if n == 1:\\n        print(0)\\n        return\\n    x = [[] for i in range(n)]\\n    for i in range(n - 1):\\n        a, b = list(map(int, sys.stdin.readline().split()))\\n        x[a - 1].append(b - 1)\\n        x[b - 1].append(a - 1)\\n\\n    isl = [False] * n\\n    for i in range(1, n):\\n        if len(x[i]) == 1:\\n            isl[i] = True\\n\\n    u = [False] * n\\n    u[0] = True\\n    q = [(0, 0, 1 / len(x[0]))]\\n    a = 0\\n    while len(q) != 0:\\n        c, s, t = q.pop()\\n        for i in x[c]:\\n            if not u[i]:\\n                if isl[i]:\\n                    a += (s + 1) * t\\n                else:\\n                    q.append((i, s + 1, t / (len(x[i])-1)))\\n                u[i] = True\\n    print(a)\\n\\n\\nmain()\\n\", \"n = int(input())\\nr = [[]for _ in range(n+1)]\\n\\nfor _ in range(n-1):\\n\\tu,v = list(map(int,input().split()))\\n\\tu -= 1\\n\\tv -= 1\\n\\tr[u].append(v)\\n\\tr[v].append(u)\\n\\nif n == 1:\\n\\tprint(0)\\nelse:\\n\\tst = [(0,0,1/len(r[0]))]\\n\\tans = 0\\n\\trsum = 0\\n\\tvisit = [False] * n\\n\\tvisit[0] = True\\n\\twhile st:\\n\\t\\tv,t,m = st.pop()\\n\\t\\tfor l in r[v]:\\n\\t\\t\\tif not visit[l]:\\n\\t\\t\\t\\tif l != 0 and len(r[l]) == 1:\\n\\t\\t\\t\\t\\tans += (t+1)*m\\n\\t\\t\\t\\t\\trsum += 1\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\tst.append((l,t+1,m/(len(r[l])-1)))\\n\\t\\t\\t\\tvisit[l] = True\\n\\tprint(ans)\\n\", \"from collections import deque, Counter\\n\\ndef mean_of_ways(graph, n):\\n    leafs = []\\n    distances = [None] * n\\n    probabilities = [0] * n\\n    probabilities[0] = 1\\n    distances[0] = 0\\n    used = [False] * n\\n    used[0] = True\\n\\n    active_vertices = deque([0])\\n\\n    while active_vertices:\\n        where = active_vertices.popleft()\\n\\n        # remember what happens if we change place of used assignment\\n        is_leaf = True\\n\\n        leafs_count = ([not used[v] for v in graph[where]]).count(True)\\n        if leafs_count > 1:\\n            probabilities[where] /= leafs_count\\n\\n        for to in graph[where]:\\n            if not used[to]:\\n                active_vertices.append(to)\\n                distances[to] = distances[where] + 1\\n                probabilities[to] = probabilities[where]\\n                used[to] = True\\n                is_leaf = False\\n\\n        if is_leaf:\\n            leafs.append(where)\\n\\n    Mx = 0\\n    for i in leafs:\\n        Mx += probabilities[i] * distances[i]\\n    return Mx\\n\\ndef main():\\n    n = int(input())\\n\\n    graph = [list() for i in range(n)]\\n    for i in range(n-1):\\n        # It there are vertices in descending order we can don't check\\n        # used it or not by making edges one directional\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n    print(mean_of_ways(graph, n))\\n\\nmain()\\n\", \"\\\"\\\"\\\"\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout\\n\\n\\ndef main():\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n\\n    # iterative DFS\\n    stack = [(1, 0, 1)]\\n    while len(stack):\\n        u, plen, prb = stack.pop()\\n\\n        visited[u] = True\\n        nBranch = 0\\n        for v in g[u]:\\n            if not visited[v]: nBranch += 1\\n        \\n        if nBranch > 0: \\n            probability = prb * (1 / nBranch)\\n            for v in g[u]:\\n                if not visited[v]:\\n                    stack.append((v, plen+1, probability))\\n        \\n        if nBranch == 0:\\n            paths.append(plen * prb)\\n\\n        \\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\n# node, probablity, path\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\n# print(L)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"\\\"\\\"\\\"\\n    # recursive DFS gives RTE, implement iterative DFS\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout, setrecursionlimit\\nimport threading\\n\\n\\ng = None\\nvisited = None\\npaths = None\\n\\ndef dfs(u, plen, prb):\\n    nonlocal visited, paths\\n\\n    visited[u] = True\\n    \\n    nBranch = 0\\n    for v in g[u]:\\n        if not visited[v]: nBranch += 1\\n    \\n    if nBranch > 0: \\n        probability = prb * (1 / nBranch)\\n        for v in g[u]:\\n            #print(prb, nBranch)\\n            if not visited[v]:\\n                dfs(v, plen+1, probability)\\n\\n    if nBranch == 0:\\n        paths.append(plen * prb)\\n\\n\\ndef main():\\n    nonlocal g, visited, paths\\n\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n    #setrecursionlimit(n+10)\\n    dfs(1, 0, 1)\\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    setrecursionlimit(10**6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"n = int(input())\\n\\nd = { x : [] for x in range(1, n + 1)}#defaultdict(list)\\nfor x in range(n - 1):\\n    s,de = map(int, input().split())\\n    d[s].append(de)\\n    d[de].append(s)\\n\\ndef dfs():\\n    lst = [(0, 1.0, 1)]\\n    visited = set({1})\\n    ans = 0\\n\\n    while lst:\\n        depth,prob,source = lst.pop()\\n        \\n        for neigh in d[source]:\\n            if neigh not in visited:\\n                visited.add(neigh)\\n                lst.append((depth + 1, prob*(1/(len(d[source]) - (1 if depth != 0 else 0))), neigh))\\n        \\n        if depth != 0 and len(d[source]) == 1:\\n            ans += prob*depth\\n\\n    return ans\\n\\nprint(\\\"{:.8f}\\\".format(dfs()))\", \"n = int(input())\\n\\ntree = {x: [] for x in range(1, n+1)}\\n\\nfor _ in range(n-1):\\n    k,l = list(map(int, input().split()))\\n    tree[k].append(l)\\n    tree[l].append(k)\\n\\n#for i in range(99999):\\n#    tree[i+1].append(i+2)\\n#    tree[i+2].append(i+1)\\n\\nvisited = set()\\ns = 0\\n\\na = [(1,1,0)]\\n\\nwhile a:\\n    v,p,l = a.pop()\\n    visited.add(v) \\n    k = 0\\n    for vv in tree[v]:\\n        if vv not in visited:\\n            k += 1\\n    if k <= 0:\\n        s += p*l\\n    else:\\n        for vv in tree[v]:\\n            if vv not in visited:\\n                a.append((vv,p*1.0/k,l+1))\\n\\nprint(s)\", \"from collections import defaultdict\\nfrom collections import deque\\nfrom sys import stdin\\nlines = deque(line.strip() for line in stdin.readlines())\\n\\ndef nextline():\\n    return lines.popleft()\\n\\ndef types(cast):\\n    return tuple(int(x) for x in nextline().split())\\n\\ndef ints():\\n    return types(int)\\n\\ndef strs():\\n    return nextline().split()\\n\\ndef main():\\n    # lines will now contain all of the input's lines in a list\\n    n = int(nextline())\\n    graph = defaultdict(list)\\n    for _ in range(1, n):\\n        a, b = ints()\\n        graph[a].append(b)\\n        graph[b].append(a)\\n    stack = [(1, 1, 0)]\\n    visited = set()\\n    expected = 0\\n    while stack:\\n        city, denominator, length = stack.pop()\\n        visited.add(city)\\n        choices = [choice for choice in graph[city] if choice not in visited]\\n        if choices:\\n            for choice in choices:\\n                stack.append((choice, denominator * len(choices), length + 1))\\n        else:\\n            expected += length / denominator\\n    print(expected)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import queue\\nn = int(input())\\ndic = {}\\n\\nfor i in range(n+1):\\n    dic[i] = []\\nfor i in range(n-1):\\n    u,v = [ int(x) for x in input().split() ]\\n    dic[u].append(v)\\n    dic[v].append(u)\\n\\nweight = {1:(0,1)} # No.City:(length,weight)\\nq = queue.Queue()\\nq.put(1)\\nwhile(q.empty() == False):\\n    city = q.get()\\n    choice = len(dic[city])\\n    if city > 1:\\n        choice = choice - 1\\n    for to_city in dic[city]:\\n        if to_city not in weight:\\n            weight[to_city] = (weight[city][0]+1, weight[city][1]/choice)\\n            q.put(to_city)\\n    if(choice > 0):\\n        weight[city] = (weight[city][0], 0)\\n\\nsum = 0\\nfor city in weight:\\n    sum = sum + weight[city][0] * weight[city][1]\\n\\nprint(\\\"%.15f\\\" % sum)\\n\\n\", \"def __starting_point():\\n    n = int(input())\\n    \\n    edges = [[] for i in range(n+1)]\\n    edges[0] = None\\n    for i in range(n-1):\\n        n1,n2 = list(map(int, input().split()))\\n        edges[n1].append(n2)\\n        edges[n2].append(n1)\\n\\n    # stack = [(node, height, prob),...]\\n    stack = [(1,0,1.0)]\\n    sev = 0\\n    visited = [False for i in range(n+1)]\\n    \\n    while len(stack) > 0:\\n        (node, height, prob) = stack.pop()\\n        visited[node] = True\\n        children = len(edges[node]) - 1\\n        if node == 1:\\n            # since root has no parent\\n            children += 1\\n            \\n        for child in edges[node]:\\n            if visited[child]:\\n                continue\\n            stack.append((child, height+1, prob/children))\\n        \\n        if children == 0:\\n            sev += height * prob\\n    print(sev)\\n            \\n    \\n    \\n    \\n    \\n\\n\\n\\n__starting_point()\", \"class Node:\\n\\tdef __init__(self, index):\\n\\t\\tself.neighbours = []\\n\\t\\tself.index = index\\n\\t\\tself.prob = 1.0\\n\\t\\tself.vis = False\\n\\t\\tself.length = 0\\n\\n\\tdef addNode(self, city):\\n\\t\\tself.neighbours.append(city)\\n\\t\\tif self.index == 1:\\n\\t\\t\\tself.prob = 1.0 / len(self.neighbours)\\n\\t\\telse:\\n\\t\\t\\tl = len(self.neighbours)\\n\\t\\t\\tself.prob = 1.0  if l < 2 else (1.0 / (l - 1))\\n\\nn = int(input())\\nif n == 1:\\n\\tprint(0)\\n\\treturn\\n\\t\\ncities = {}\\n\\nfor i in range(n-1):\\n\\ta, b = [int(k) for k in input().split()]\\n\\t#print (\\\"test \\\", a, \\\" to \\\", b)\\n\\tif a not in cities:\\n\\t\\tcities[a] = Node(a)\\n\\tcities[a].addNode(b)\\n\\tif b not in cities:\\n\\t\\tcities[b] = Node(b)\\n\\tcities[b].addNode(a)\\n\\nif len(cities) == 2:\\n\\tprint(1)\\n\\treturn\\n\\n\\n# for i in range(1, n + 1, 1):\\n# \\tprint (\\\"city.index \\\", cities[i].index, ' roads ', cities[i].neighbours, ' prob ', cities[i].prob)\\n\\n\\n# deadends = []\\n# deadendsProb = []\\n\\n\\n# def Parse(city, prob, length, oldCity):\\n# \\t#print ('parse ', city.index)\\n# \\tnewprob = 1.0 if oldCity == 0  else cities[oldCity].prob\\n# \\t#print ('nnewProb ', newprob)\\n# \\tprob *= newprob\\n# \\t#print (city.index, ' len ', len(city.neighbours))\\n# \\tif len(city.neighbours) == 1 and oldCity != 0:\\n# \\t\\tdeadends.append(length)\\n# \\t\\tdeadendsProb.append(prob)\\n# \\telse:\\n# \\t\\t#print (city.neighbours)\\n# \\t\\tlength += 1\\n# \\t\\tfor c in city.neighbours:\\n# \\t\\t\\tif c != oldCity:\\n# \\t\\t\\t\\tParse(cities[c], prob, length, city.index )\\n\\n# Parse(cities[1], 1.0, 0, 0)\\n# #for i in range(len(deadends)):\\n# #\\tprint('len ', deadends[i], ' prob ', deadendsProb[i])\\n\\n# ans = sum(map(lambda l, p: l * p, deadends, deadendsProb))\\n# #print('ans', ans)\\n# print(ans)\\n\\n\\ndef inorder(city):\\n\\ts = []\\n\\ts.append(city)\\n\\t#print ('index ', city.index)\\n\\t#print ('neighbours ', city.neighbours)\\n\\twhile s:\\n\\t\\tcity = s.pop()\\n\\t\\tcity.vis = True\\n\\t\\tif city.neighbours:\\n\\t\\t\\tif city.index == 1 or len(city.neighbours) > 1:\\n\\t\\t\\t\\tfor c in city.neighbours:\\n\\t\\t\\t\\t\\tif not cities[c].vis:\\n\\t\\t\\t\\t\\t\\tcities[c].length = city.length + 1\\n\\t\\t\\t\\t\\t\\tcities[c].prob *= city.prob\\n\\t\\t\\t\\t\\t\\ts.append(cities[c])\\n\\t\\t\\telse:\\n\\t\\t\\t\\tyield (city.index, city.prob, city.length)\\n\\n\\n\\ntest = sum([city[1] * city[2] for city in inorder(cities[1])])\\nprint(test)\\n\\n\\n\\n# this is a tree\\n\", \"from sys import setrecursionlimit\\nimport threading\\n\\n\\nlength = 0\\nvisited = None\\ng = None\\n\\n\\ndef dfs(v, path_length, chance):\\n    nonlocal length\\n    visited.add(v)\\n    edges = 0\\n    for n in g[v]:\\n        if n not in visited:\\n            edges += 1\\n    for n in g[v]:\\n        if n not in visited:\\n            dfs(n, path_length + 1, chance * (1 / edges))\\n    if edges == 0:\\n        length += path_length * chance\\n\\n\\ndef main():\\n    nonlocal visited, g\\n    vert = int(input())\\n    g = [[] for _ in range(vert + 1)]\\n    visited = set()\\n    for _ in range(vert - 1):\\n        a, b = list(map(int, input().split()))\\n        g[a].append(b)\\n        g[b].append(a)\\n    dfs(1, 0, 1)\\n    print(length)\\n\\n\\ndef __starting_point():\\n    setrecursionlimit(10 ** 6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"n = int(input())\\ng = [[] for i in range(n+1)]\\nfor i in range(n-1):\\n    a,b = [int(x) for x in input().split()]\\n    g[a].append(b)\\n    g[b].append(a)\\n\\nd = [0 for i in range(n+1)]\\nv = [0 for i in range(n+1)]\\ne = [0 for i in range(n+1)]\\np = [0.0 for i in range(n+1)]\\np[1] = 1.0\\nen = 0\\nq = [1]\\nwhile len(q):\\n    c = q.pop(0)\\n    v[c] = 1\\n    a = 0\\n    for ne in g[c]:\\n        if not v[ne]:\\n            q.append(ne)\\n            d[ne] = d[c] + 1\\n            p[ne] = p[c]/len(g[c]) if c == 1 else p[c]/(len(g[c])-1)\\n            a += 1\\n    if a == 0:\\n        e[c] = 1\\n        en += 1\\nave = 0.0\\nfor i in range(n+1):\\n    if e[i]:\\n        ave += d[i]*p[i]\\nprint(ave)\\n\", \"n=int(input())\\nif n==1:\\n\\tprint(0)\\n\\treturn\\ngraph=[None]*n\\nfor i in range(n-1):\\n\\tu,v=map(int,input().split())\\n\\tu-=1\\n\\tv-=1\\n\\tif graph[u]==None:\\n\\t\\tgraph[u]=[v]\\n\\telse:\\n\\t\\tgraph[u].append(v)\\n\\tif graph[v]==None:\\n\\t\\tgraph[v]=[u]\\n\\telse:\\n\\t\\tgraph[v].append(u)\\n\\nis_checked=[False]*n\\nlens=[None]*n\\nlens[0]=0\\ndivs=[None]*n\\ndivs[0]=1\\nstack=[0]\\nans=0\\nwhile stack:\\n\\ttop=stack.pop()\\n\\tis_checked[top]=True\\n\\tvals=graph[top]\\n\\tfor i in range(len(vals)):\\n\\t\\tcur=vals[i]\\n\\t\\tif not is_checked[cur]:\\n\\t\\t\\tlens[cur]=lens[top]+1\\n\\t\\t\\tdivs[cur]=divs[top]*(len(vals)-(1 if top!=0 else 0))\\n\\t\\t\\tif len(graph[cur])==1:\\n\\t\\t\\t\\tans+=lens[cur]/divs[cur]\\n\\t\\t\\t\\tis_checked[cur]=None\\n\\t\\t\\t\\tlens[cur]=None\\n\\t\\t\\t\\tdivs[cur]=None\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tstack.append(cur)\\nprint('{:.8f}'.format(ans))\"]",
        "difficulty": "interview",
        "input": "5\n1 2\n1 3\n3 4\n2 5\n",
        "output": "2.000000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/839/C"
    },
    {
        "id": 581,
        "task_id": 2246,
        "test_case_id": 3,
        "question": "There are n cities and n - 1 roads in the Seven Kingdoms, each road connects two cities and we can reach any city from any other by the roads.\n\nTheon and Yara Greyjoy are on a horse in the first city, they are starting traveling through the roads. But the weather is foggy, so they can’t see where the horse brings them. When the horse reaches a city (including the first one), it goes to one of the cities connected to the current city. But it is a strange horse, it only goes to cities in which they weren't before. In each such city, the horse goes with equal probabilities and it stops when there are no such cities. \n\nLet the length of each road be 1. The journey starts in the city 1. What is the expected length (expected value of length) of their journey? You can read about expected (average) value by the link https://en.wikipedia.org/wiki/Expected_value.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 100000) — number of cities.\n\nThen n - 1 lines follow. The i-th line of these lines contains two integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the cities connected by the i-th road.\n\nIt is guaranteed that one can reach any city from any other by the roads.\n\n\n-----Output-----\n\nPrint a number — the expected length of their journey. The journey starts in the city 1.\n\nYour answer will be considered correct if its absolute or relative error does not exceed 10^{ - 6}.\n\nNamely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if $\\frac{|a - b|}{\\operatorname{max}(1, b)} \\leq 10^{-6}$.\n\n\n-----Examples-----\nInput\n4\n1 2\n1 3\n2 4\n\nOutput\n1.500000000000000\n\nInput\n5\n1 2\n1 3\n3 4\n2 5\n\nOutput\n2.000000000000000\n\n\n\n-----Note-----\n\nIn the first sample, their journey may end in cities 3 or 4 with equal probability. The distance to city 3 is 1 and to city 4 is 2, so the expected length is 1.5.\n\nIn the second sample, their journey may end in city 4 or 5. The distance to the both cities is 2, so the expected length is 2.",
        "solutions": "[\"from sys import stdin\\nfrom decimal import Decimal as D\\ninput = stdin.readline\\nn = int(input())\\nadj = [[] for i in range(n+1)]\\ntree = [[] for i in range(n+1)]\\nvisit = [-1]*(n+1)\\nlength = [-1]*(n+1)\\nfor i in range(n-1):\\n    a, b = map(int,input().split())\\n    adj[a].append(b)\\n    adj[b].append(a)\\nbfsord = []\\n\\nfrom collections import deque\\nQ = deque()\\nQ.append(1)\\nvisit[1] = 0\\nwhile len(Q):\\n    p = Q.popleft()\\n    bfsord.append(p)\\n    for q in adj[p]:\\n        if visit[q] != -1: continue\\n        visit[q] = visit[p]+1\\n        Q.append(q)\\n        tree[p].append(q)\\n\\nfor p in reversed(bfsord):\\n    if not tree[p]: length[p] = D(0)\\n    else: length[p] = D(1) + sum(length[q] for q in tree[p])/len(tree[p])\\nprint(length[1])\", \"import sys\\n\\ndef r():\\n    return list(map(int, input().split()))\\n\\nn = int(input())\\nedge = [r() for i in range(n-1)]\\n\\nadj = [[] for i in range(n+1)]\\nfor e in edge:\\n    adj[e[0]].append(e[1])\\n    adj[e[1]].append(e[0])\\n\\nprob = [0.0 for i in range(n+1)]\\nd = [0 for i in range(n+1)]\\nvisited = [False for i in range(n+1)]\\n\\nprob[1] = 1\\nans = 0.0\\n\\nst = [1]\\nvisited[1] = True\\nwhile st:\\n    u = st.pop()\\n    cnt = len(adj[u])\\n    if u != 1:\\n        cnt -= 1\\n    if cnt == 0:\\n        ans = ans + prob[u]*d[u]\\n    else:\\n        for v in adj[u]:\\n            if not visited[v]:\\n                visited[v] = True\\n                prob[v] = prob[u]*(1.0/cnt)\\n                d[v] = d[u]+1\\n                st.append(v)\\n                \\nprint(ans)\\n\\n\", \"from queue import *\\n\\nn = int(input())\\ng = [[] for x in range(n)]\\n\\nfor i in range(n-1):\\n  a, b = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  g[a].append(b)\\n  g[b].append(a)\\n\\nused = [0]*n\\n\\nanw = 0\\n\\ndef solve(v, d, r):\\n  nonlocal anw\\n  q = Queue()\\n  q.put((v, d, r))\\n  while not q.empty():\\n    v, d, r = q.get(v)\\n    used[v] = True\\n    for u in g[v]:\\n      if not used[u]:\\n        q.put((u, d+1, r*(len(g[v])-(v!=0))))\\n        #print(\\\"put\\\", u)\\n    #print(\\\"so at\\\", v, \\\"len\\\", len(g[v]))\\n    if v != 0 and len(g[v]) == 1:\\n      #print(\\\"At \\\", v, \\\"is\\\", d, r)\\n      anw += d/r\\n  \\nsolve(0, 0, 1)\\nprint(anw)\", \"def f():\\n    n = int(input())\\n\\n    if n == 1:\\n        print(0)\\n        return\\n\\n    road = {}\\n    visited = {}\\n    for i in range(n-1):\\n        u, v = input().split()\\n        if u in road:\\n            road[u].append(v)\\n        else:\\n            road[u] = [v]\\n        if v in road:\\n            road[v].append(u)\\n        else:\\n            road[v] = [u]\\n        visited[u] = False\\n        visited[v] = False\\n\\n    prob = {}\\n    length = {}\\n    res = []\\n\\n    queue = []\\n    for k in road['1']:\\n        prob[k] = 1.0 / len(road['1'])\\n        length[k] = 1\\n        queue.append(k)\\n        visited['1'] = True\\n\\n    while len(queue) > 0:\\n        cur = queue[0]\\n        del queue[0]\\n        dest = []\\n        for k in road[cur]:\\n            if not visited[k]:\\n                dest.append(k)\\n        if len(dest) == 0:\\n            res.append((cur, prob[cur], length[cur]))\\n            continue\\n        for k in dest:\\n            prob[k] = prob[cur] / len(dest)\\n            length[k] = length[cur] + 1\\n            queue.append(k)\\n            visited[cur] = True\\n\\n    val = 0.0\\n    for item in res:\\n        val += item[1] * item[2]\\n\\n    print(val)\\n\\nf()\\n\", \"from collections import deque\\nimport sys\\nfrom decimal import Decimal\\n\\nreadline = sys.stdin.readline\\nn = int(input())\\nedges = [[]*n for _ in [None]*n]\\n\\nfor _ in [None]*(n-1):\\n    a, b = map(int, readline().split())\\n    edges[a-1].append(b-1)\\n    edges[b-1].append(a-1)\\n\\nvisited = [False]*n\\nvisited[0] = True\\ncnt = 0\\ntotal = 0\\ndq = deque()\\nappend, pop = dq.append, dq.popleft\\nappend((0, 0, Decimal(1)))\\n\\nwhile dq:\\n    pos, l, multiple = pop()\\n    flag = True\\n    to = [x for x in edges[pos] if visited[x] == False]\\n\\n    if to:\\n        x = len(to)\\n        multiple /= Decimal(x)\\n        for v in to:\\n            visited[v] = True\\n            append((v, l+1, multiple))\\n    else:\\n        total += Decimal(l) * multiple\\n\\nprint(total)\", \"import sys\\nfrom collections import deque\\nread=lambda:sys.stdin.readline().rstrip()\\nreadi=lambda:int(sys.stdin.readline())\\nwriteln=lambda x:sys.stdout.write(str(x)+\\\"\\\\n\\\")\\nwrite=lambda x:sys.stdout.write(x)\\nN = readi()\\nif N == 1:\\n    writeln(0)\\n    return\\n\\nG = [set() for _ in range(N)]\\nfor _ in range(N-1):\\n    u, v = list(map(int, read().split()))\\n    G[u-1].add(v-1)\\n    G[v-1].add(u-1)\\n\\nvisited = [0]*N\\nprev = [0]*N\\nvisited[0] = 1\\nprev[0] = 1\\nq = deque([0])\\nlengths = []\\nev = 0\\nwhile q:\\n    cur = q.popleft()\\n    n_neighbor = len(G[cur])\\n    if cur == 0:\\n        divs = n_neighbor\\n    else:\\n        divs = n_neighbor - 1\\n    if n_neighbor == 1 and cur != 0:\\n        ev += (visited[cur] - 1)*prev[cur]\\n        continue\\n     \\n    for nxt in G[cur]:\\n        if not visited[nxt]:\\n            visited[nxt] = visited[cur] + 1\\n            prev[nxt] = (1 / divs)*prev[cur]\\n            q.append(nxt)\\n\\nprint('%.12f' % ev)\\n\", \"#!/usr/bin/env python\\n\\nimport sys\\n\\ndef explore(src, prob, length, adj_mat, gains, visited):\\n    nexts = set(adj_mat[src]) - visited\\n    if nexts:\\n        go_prob = 1 / len(nexts)\\n        length += 1\\n        for dst in nexts:\\n            visited.add(dst)\\n            explore(dst, prob * go_prob, length, adj_mat, gains, visited)\\n    else:\\n        gains[src] += prob * length\\n\\ndef main():\\n  n = int(sys.stdin.readline())\\n  adj_mat = [[] for __ in range(n)]\\n  node_exp = [0 for __ in range(n)]\\n  for __ in range(n-1):\\n      u, v = list(map(int, sys.stdin.readline().split()))\\n      adj_mat[u-1].append(v-1)\\n      adj_mat[v-1].append(u-1)\\n  stk = [0]\\n  visited= set([0])\\n  prob = [1] * n\\n  length = [0] * n\\n  while stk:\\n      nxt = stk.pop()\\n      nexts = set(adj_mat[nxt]) - visited\\n      if nexts:\\n          go_prob = 1 / len(nexts)\\n          for dst in nexts:\\n              visited.add(dst)\\n              stk.append(dst)\\n              prob[dst] *= prob[nxt] * go_prob\\n              length[dst] = length[nxt] + 1\\n      else:\\n        node_exp[nxt] += prob[nxt] * length[nxt]\\n\\n  #explore(0, 1, 0, adj_mat,node_exp, set([0]))\\n  print(\\\"{:.13f}\\\".format(sum(node_exp)))\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n    if n == 1:\\n        print(0)\\n        return\\n    x = [[] for i in range(n)]\\n    for i in range(n - 1):\\n        a, b = list(map(int, sys.stdin.readline().split()))\\n        x[a - 1].append(b - 1)\\n        x[b - 1].append(a - 1)\\n\\n    isl = [False] * n\\n    for i in range(1, n):\\n        if len(x[i]) == 1:\\n            isl[i] = True\\n\\n    u = [False] * n\\n    u[0] = True\\n    q = [(0, 0, 1 / len(x[0]))]\\n    a = 0\\n    while len(q) != 0:\\n        c, s, t = q.pop()\\n        for i in x[c]:\\n            if not u[i]:\\n                if isl[i]:\\n                    a += (s + 1) * t\\n                else:\\n                    q.append((i, s + 1, t / (len(x[i])-1)))\\n                u[i] = True\\n    print(a)\\n\\n\\nmain()\\n\", \"n = int(input())\\nr = [[]for _ in range(n+1)]\\n\\nfor _ in range(n-1):\\n\\tu,v = list(map(int,input().split()))\\n\\tu -= 1\\n\\tv -= 1\\n\\tr[u].append(v)\\n\\tr[v].append(u)\\n\\nif n == 1:\\n\\tprint(0)\\nelse:\\n\\tst = [(0,0,1/len(r[0]))]\\n\\tans = 0\\n\\trsum = 0\\n\\tvisit = [False] * n\\n\\tvisit[0] = True\\n\\twhile st:\\n\\t\\tv,t,m = st.pop()\\n\\t\\tfor l in r[v]:\\n\\t\\t\\tif not visit[l]:\\n\\t\\t\\t\\tif l != 0 and len(r[l]) == 1:\\n\\t\\t\\t\\t\\tans += (t+1)*m\\n\\t\\t\\t\\t\\trsum += 1\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\tst.append((l,t+1,m/(len(r[l])-1)))\\n\\t\\t\\t\\tvisit[l] = True\\n\\tprint(ans)\\n\", \"from collections import deque, Counter\\n\\ndef mean_of_ways(graph, n):\\n    leafs = []\\n    distances = [None] * n\\n    probabilities = [0] * n\\n    probabilities[0] = 1\\n    distances[0] = 0\\n    used = [False] * n\\n    used[0] = True\\n\\n    active_vertices = deque([0])\\n\\n    while active_vertices:\\n        where = active_vertices.popleft()\\n\\n        # remember what happens if we change place of used assignment\\n        is_leaf = True\\n\\n        leafs_count = ([not used[v] for v in graph[where]]).count(True)\\n        if leafs_count > 1:\\n            probabilities[where] /= leafs_count\\n\\n        for to in graph[where]:\\n            if not used[to]:\\n                active_vertices.append(to)\\n                distances[to] = distances[where] + 1\\n                probabilities[to] = probabilities[where]\\n                used[to] = True\\n                is_leaf = False\\n\\n        if is_leaf:\\n            leafs.append(where)\\n\\n    Mx = 0\\n    for i in leafs:\\n        Mx += probabilities[i] * distances[i]\\n    return Mx\\n\\ndef main():\\n    n = int(input())\\n\\n    graph = [list() for i in range(n)]\\n    for i in range(n-1):\\n        # It there are vertices in descending order we can don't check\\n        # used it or not by making edges one directional\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n    print(mean_of_ways(graph, n))\\n\\nmain()\\n\", \"\\\"\\\"\\\"\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout\\n\\n\\ndef main():\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n\\n    # iterative DFS\\n    stack = [(1, 0, 1)]\\n    while len(stack):\\n        u, plen, prb = stack.pop()\\n\\n        visited[u] = True\\n        nBranch = 0\\n        for v in g[u]:\\n            if not visited[v]: nBranch += 1\\n        \\n        if nBranch > 0: \\n            probability = prb * (1 / nBranch)\\n            for v in g[u]:\\n                if not visited[v]:\\n                    stack.append((v, plen+1, probability))\\n        \\n        if nBranch == 0:\\n            paths.append(plen * prb)\\n\\n        \\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\n# node, probablity, path\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\n# print(L)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"\\\"\\\"\\\"\\n    # recursive DFS gives RTE, implement iterative DFS\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout, setrecursionlimit\\nimport threading\\n\\n\\ng = None\\nvisited = None\\npaths = None\\n\\ndef dfs(u, plen, prb):\\n    nonlocal visited, paths\\n\\n    visited[u] = True\\n    \\n    nBranch = 0\\n    for v in g[u]:\\n        if not visited[v]: nBranch += 1\\n    \\n    if nBranch > 0: \\n        probability = prb * (1 / nBranch)\\n        for v in g[u]:\\n            #print(prb, nBranch)\\n            if not visited[v]:\\n                dfs(v, plen+1, probability)\\n\\n    if nBranch == 0:\\n        paths.append(plen * prb)\\n\\n\\ndef main():\\n    nonlocal g, visited, paths\\n\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n    #setrecursionlimit(n+10)\\n    dfs(1, 0, 1)\\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    setrecursionlimit(10**6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"n = int(input())\\n\\nd = { x : [] for x in range(1, n + 1)}#defaultdict(list)\\nfor x in range(n - 1):\\n    s,de = map(int, input().split())\\n    d[s].append(de)\\n    d[de].append(s)\\n\\ndef dfs():\\n    lst = [(0, 1.0, 1)]\\n    visited = set({1})\\n    ans = 0\\n\\n    while lst:\\n        depth,prob,source = lst.pop()\\n        \\n        for neigh in d[source]:\\n            if neigh not in visited:\\n                visited.add(neigh)\\n                lst.append((depth + 1, prob*(1/(len(d[source]) - (1 if depth != 0 else 0))), neigh))\\n        \\n        if depth != 0 and len(d[source]) == 1:\\n            ans += prob*depth\\n\\n    return ans\\n\\nprint(\\\"{:.8f}\\\".format(dfs()))\", \"n = int(input())\\n\\ntree = {x: [] for x in range(1, n+1)}\\n\\nfor _ in range(n-1):\\n    k,l = list(map(int, input().split()))\\n    tree[k].append(l)\\n    tree[l].append(k)\\n\\n#for i in range(99999):\\n#    tree[i+1].append(i+2)\\n#    tree[i+2].append(i+1)\\n\\nvisited = set()\\ns = 0\\n\\na = [(1,1,0)]\\n\\nwhile a:\\n    v,p,l = a.pop()\\n    visited.add(v) \\n    k = 0\\n    for vv in tree[v]:\\n        if vv not in visited:\\n            k += 1\\n    if k <= 0:\\n        s += p*l\\n    else:\\n        for vv in tree[v]:\\n            if vv not in visited:\\n                a.append((vv,p*1.0/k,l+1))\\n\\nprint(s)\", \"from collections import defaultdict\\nfrom collections import deque\\nfrom sys import stdin\\nlines = deque(line.strip() for line in stdin.readlines())\\n\\ndef nextline():\\n    return lines.popleft()\\n\\ndef types(cast):\\n    return tuple(int(x) for x in nextline().split())\\n\\ndef ints():\\n    return types(int)\\n\\ndef strs():\\n    return nextline().split()\\n\\ndef main():\\n    # lines will now contain all of the input's lines in a list\\n    n = int(nextline())\\n    graph = defaultdict(list)\\n    for _ in range(1, n):\\n        a, b = ints()\\n        graph[a].append(b)\\n        graph[b].append(a)\\n    stack = [(1, 1, 0)]\\n    visited = set()\\n    expected = 0\\n    while stack:\\n        city, denominator, length = stack.pop()\\n        visited.add(city)\\n        choices = [choice for choice in graph[city] if choice not in visited]\\n        if choices:\\n            for choice in choices:\\n                stack.append((choice, denominator * len(choices), length + 1))\\n        else:\\n            expected += length / denominator\\n    print(expected)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import queue\\nn = int(input())\\ndic = {}\\n\\nfor i in range(n+1):\\n    dic[i] = []\\nfor i in range(n-1):\\n    u,v = [ int(x) for x in input().split() ]\\n    dic[u].append(v)\\n    dic[v].append(u)\\n\\nweight = {1:(0,1)} # No.City:(length,weight)\\nq = queue.Queue()\\nq.put(1)\\nwhile(q.empty() == False):\\n    city = q.get()\\n    choice = len(dic[city])\\n    if city > 1:\\n        choice = choice - 1\\n    for to_city in dic[city]:\\n        if to_city not in weight:\\n            weight[to_city] = (weight[city][0]+1, weight[city][1]/choice)\\n            q.put(to_city)\\n    if(choice > 0):\\n        weight[city] = (weight[city][0], 0)\\n\\nsum = 0\\nfor city in weight:\\n    sum = sum + weight[city][0] * weight[city][1]\\n\\nprint(\\\"%.15f\\\" % sum)\\n\\n\", \"def __starting_point():\\n    n = int(input())\\n    \\n    edges = [[] for i in range(n+1)]\\n    edges[0] = None\\n    for i in range(n-1):\\n        n1,n2 = list(map(int, input().split()))\\n        edges[n1].append(n2)\\n        edges[n2].append(n1)\\n\\n    # stack = [(node, height, prob),...]\\n    stack = [(1,0,1.0)]\\n    sev = 0\\n    visited = [False for i in range(n+1)]\\n    \\n    while len(stack) > 0:\\n        (node, height, prob) = stack.pop()\\n        visited[node] = True\\n        children = len(edges[node]) - 1\\n        if node == 1:\\n            # since root has no parent\\n            children += 1\\n            \\n        for child in edges[node]:\\n            if visited[child]:\\n                continue\\n            stack.append((child, height+1, prob/children))\\n        \\n        if children == 0:\\n            sev += height * prob\\n    print(sev)\\n            \\n    \\n    \\n    \\n    \\n\\n\\n\\n__starting_point()\", \"class Node:\\n\\tdef __init__(self, index):\\n\\t\\tself.neighbours = []\\n\\t\\tself.index = index\\n\\t\\tself.prob = 1.0\\n\\t\\tself.vis = False\\n\\t\\tself.length = 0\\n\\n\\tdef addNode(self, city):\\n\\t\\tself.neighbours.append(city)\\n\\t\\tif self.index == 1:\\n\\t\\t\\tself.prob = 1.0 / len(self.neighbours)\\n\\t\\telse:\\n\\t\\t\\tl = len(self.neighbours)\\n\\t\\t\\tself.prob = 1.0  if l < 2 else (1.0 / (l - 1))\\n\\nn = int(input())\\nif n == 1:\\n\\tprint(0)\\n\\treturn\\n\\t\\ncities = {}\\n\\nfor i in range(n-1):\\n\\ta, b = [int(k) for k in input().split()]\\n\\t#print (\\\"test \\\", a, \\\" to \\\", b)\\n\\tif a not in cities:\\n\\t\\tcities[a] = Node(a)\\n\\tcities[a].addNode(b)\\n\\tif b not in cities:\\n\\t\\tcities[b] = Node(b)\\n\\tcities[b].addNode(a)\\n\\nif len(cities) == 2:\\n\\tprint(1)\\n\\treturn\\n\\n\\n# for i in range(1, n + 1, 1):\\n# \\tprint (\\\"city.index \\\", cities[i].index, ' roads ', cities[i].neighbours, ' prob ', cities[i].prob)\\n\\n\\n# deadends = []\\n# deadendsProb = []\\n\\n\\n# def Parse(city, prob, length, oldCity):\\n# \\t#print ('parse ', city.index)\\n# \\tnewprob = 1.0 if oldCity == 0  else cities[oldCity].prob\\n# \\t#print ('nnewProb ', newprob)\\n# \\tprob *= newprob\\n# \\t#print (city.index, ' len ', len(city.neighbours))\\n# \\tif len(city.neighbours) == 1 and oldCity != 0:\\n# \\t\\tdeadends.append(length)\\n# \\t\\tdeadendsProb.append(prob)\\n# \\telse:\\n# \\t\\t#print (city.neighbours)\\n# \\t\\tlength += 1\\n# \\t\\tfor c in city.neighbours:\\n# \\t\\t\\tif c != oldCity:\\n# \\t\\t\\t\\tParse(cities[c], prob, length, city.index )\\n\\n# Parse(cities[1], 1.0, 0, 0)\\n# #for i in range(len(deadends)):\\n# #\\tprint('len ', deadends[i], ' prob ', deadendsProb[i])\\n\\n# ans = sum(map(lambda l, p: l * p, deadends, deadendsProb))\\n# #print('ans', ans)\\n# print(ans)\\n\\n\\ndef inorder(city):\\n\\ts = []\\n\\ts.append(city)\\n\\t#print ('index ', city.index)\\n\\t#print ('neighbours ', city.neighbours)\\n\\twhile s:\\n\\t\\tcity = s.pop()\\n\\t\\tcity.vis = True\\n\\t\\tif city.neighbours:\\n\\t\\t\\tif city.index == 1 or len(city.neighbours) > 1:\\n\\t\\t\\t\\tfor c in city.neighbours:\\n\\t\\t\\t\\t\\tif not cities[c].vis:\\n\\t\\t\\t\\t\\t\\tcities[c].length = city.length + 1\\n\\t\\t\\t\\t\\t\\tcities[c].prob *= city.prob\\n\\t\\t\\t\\t\\t\\ts.append(cities[c])\\n\\t\\t\\telse:\\n\\t\\t\\t\\tyield (city.index, city.prob, city.length)\\n\\n\\n\\ntest = sum([city[1] * city[2] for city in inorder(cities[1])])\\nprint(test)\\n\\n\\n\\n# this is a tree\\n\", \"from sys import setrecursionlimit\\nimport threading\\n\\n\\nlength = 0\\nvisited = None\\ng = None\\n\\n\\ndef dfs(v, path_length, chance):\\n    nonlocal length\\n    visited.add(v)\\n    edges = 0\\n    for n in g[v]:\\n        if n not in visited:\\n            edges += 1\\n    for n in g[v]:\\n        if n not in visited:\\n            dfs(n, path_length + 1, chance * (1 / edges))\\n    if edges == 0:\\n        length += path_length * chance\\n\\n\\ndef main():\\n    nonlocal visited, g\\n    vert = int(input())\\n    g = [[] for _ in range(vert + 1)]\\n    visited = set()\\n    for _ in range(vert - 1):\\n        a, b = list(map(int, input().split()))\\n        g[a].append(b)\\n        g[b].append(a)\\n    dfs(1, 0, 1)\\n    print(length)\\n\\n\\ndef __starting_point():\\n    setrecursionlimit(10 ** 6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"n = int(input())\\ng = [[] for i in range(n+1)]\\nfor i in range(n-1):\\n    a,b = [int(x) for x in input().split()]\\n    g[a].append(b)\\n    g[b].append(a)\\n\\nd = [0 for i in range(n+1)]\\nv = [0 for i in range(n+1)]\\ne = [0 for i in range(n+1)]\\np = [0.0 for i in range(n+1)]\\np[1] = 1.0\\nen = 0\\nq = [1]\\nwhile len(q):\\n    c = q.pop(0)\\n    v[c] = 1\\n    a = 0\\n    for ne in g[c]:\\n        if not v[ne]:\\n            q.append(ne)\\n            d[ne] = d[c] + 1\\n            p[ne] = p[c]/len(g[c]) if c == 1 else p[c]/(len(g[c])-1)\\n            a += 1\\n    if a == 0:\\n        e[c] = 1\\n        en += 1\\nave = 0.0\\nfor i in range(n+1):\\n    if e[i]:\\n        ave += d[i]*p[i]\\nprint(ave)\\n\", \"n=int(input())\\nif n==1:\\n\\tprint(0)\\n\\treturn\\ngraph=[None]*n\\nfor i in range(n-1):\\n\\tu,v=map(int,input().split())\\n\\tu-=1\\n\\tv-=1\\n\\tif graph[u]==None:\\n\\t\\tgraph[u]=[v]\\n\\telse:\\n\\t\\tgraph[u].append(v)\\n\\tif graph[v]==None:\\n\\t\\tgraph[v]=[u]\\n\\telse:\\n\\t\\tgraph[v].append(u)\\n\\nis_checked=[False]*n\\nlens=[None]*n\\nlens[0]=0\\ndivs=[None]*n\\ndivs[0]=1\\nstack=[0]\\nans=0\\nwhile stack:\\n\\ttop=stack.pop()\\n\\tis_checked[top]=True\\n\\tvals=graph[top]\\n\\tfor i in range(len(vals)):\\n\\t\\tcur=vals[i]\\n\\t\\tif not is_checked[cur]:\\n\\t\\t\\tlens[cur]=lens[top]+1\\n\\t\\t\\tdivs[cur]=divs[top]*(len(vals)-(1 if top!=0 else 0))\\n\\t\\t\\tif len(graph[cur])==1:\\n\\t\\t\\t\\tans+=lens[cur]/divs[cur]\\n\\t\\t\\t\\tis_checked[cur]=None\\n\\t\\t\\t\\tlens[cur]=None\\n\\t\\t\\t\\tdivs[cur]=None\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tstack.append(cur)\\nprint('{:.8f}'.format(ans))\"]",
        "difficulty": "interview",
        "input": "70\n1 25\n57 1\n18 1\n65 1\n38 1\n1 41\n1 5\n1 69\n1 3\n31 1\n1 8\n1 9\n53 1\n70 1\n45 1\n1 24\n1 42\n1 30\n1 12\n1 37\n64 1\n1 28\n1 58\n1 22\n11 1\n1 4\n1 27\n1 16\n1 21\n54 1\n1 51\n1 43\n29 1\n56 1\n1 39\n32 1\n1 15\n1 17\n1 19\n1 40\n36 1\n48 1\n63 1\n1 7\n1 47\n1 13\n1 46\n60 1\n1 6\n23 1\n20 1\n1 52\n2 1\n26 1\n1 59\n1 66\n10 1\n1 62\n1 68\n1 55\n50 1\n33 1\n44 1\n1 34\n1 35\n1 61\n14 1\n67 1\n49 1\n",
        "output": "1.000000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/839/C"
    },
    {
        "id": 582,
        "task_id": 2246,
        "test_case_id": 4,
        "question": "There are n cities and n - 1 roads in the Seven Kingdoms, each road connects two cities and we can reach any city from any other by the roads.\n\nTheon and Yara Greyjoy are on a horse in the first city, they are starting traveling through the roads. But the weather is foggy, so they can’t see where the horse brings them. When the horse reaches a city (including the first one), it goes to one of the cities connected to the current city. But it is a strange horse, it only goes to cities in which they weren't before. In each such city, the horse goes with equal probabilities and it stops when there are no such cities. \n\nLet the length of each road be 1. The journey starts in the city 1. What is the expected length (expected value of length) of their journey? You can read about expected (average) value by the link https://en.wikipedia.org/wiki/Expected_value.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 100000) — number of cities.\n\nThen n - 1 lines follow. The i-th line of these lines contains two integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the cities connected by the i-th road.\n\nIt is guaranteed that one can reach any city from any other by the roads.\n\n\n-----Output-----\n\nPrint a number — the expected length of their journey. The journey starts in the city 1.\n\nYour answer will be considered correct if its absolute or relative error does not exceed 10^{ - 6}.\n\nNamely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if $\\frac{|a - b|}{\\operatorname{max}(1, b)} \\leq 10^{-6}$.\n\n\n-----Examples-----\nInput\n4\n1 2\n1 3\n2 4\n\nOutput\n1.500000000000000\n\nInput\n5\n1 2\n1 3\n3 4\n2 5\n\nOutput\n2.000000000000000\n\n\n\n-----Note-----\n\nIn the first sample, their journey may end in cities 3 or 4 with equal probability. The distance to city 3 is 1 and to city 4 is 2, so the expected length is 1.5.\n\nIn the second sample, their journey may end in city 4 or 5. The distance to the both cities is 2, so the expected length is 2.",
        "solutions": "[\"from sys import stdin\\nfrom decimal import Decimal as D\\ninput = stdin.readline\\nn = int(input())\\nadj = [[] for i in range(n+1)]\\ntree = [[] for i in range(n+1)]\\nvisit = [-1]*(n+1)\\nlength = [-1]*(n+1)\\nfor i in range(n-1):\\n    a, b = map(int,input().split())\\n    adj[a].append(b)\\n    adj[b].append(a)\\nbfsord = []\\n\\nfrom collections import deque\\nQ = deque()\\nQ.append(1)\\nvisit[1] = 0\\nwhile len(Q):\\n    p = Q.popleft()\\n    bfsord.append(p)\\n    for q in adj[p]:\\n        if visit[q] != -1: continue\\n        visit[q] = visit[p]+1\\n        Q.append(q)\\n        tree[p].append(q)\\n\\nfor p in reversed(bfsord):\\n    if not tree[p]: length[p] = D(0)\\n    else: length[p] = D(1) + sum(length[q] for q in tree[p])/len(tree[p])\\nprint(length[1])\", \"import sys\\n\\ndef r():\\n    return list(map(int, input().split()))\\n\\nn = int(input())\\nedge = [r() for i in range(n-1)]\\n\\nadj = [[] for i in range(n+1)]\\nfor e in edge:\\n    adj[e[0]].append(e[1])\\n    adj[e[1]].append(e[0])\\n\\nprob = [0.0 for i in range(n+1)]\\nd = [0 for i in range(n+1)]\\nvisited = [False for i in range(n+1)]\\n\\nprob[1] = 1\\nans = 0.0\\n\\nst = [1]\\nvisited[1] = True\\nwhile st:\\n    u = st.pop()\\n    cnt = len(adj[u])\\n    if u != 1:\\n        cnt -= 1\\n    if cnt == 0:\\n        ans = ans + prob[u]*d[u]\\n    else:\\n        for v in adj[u]:\\n            if not visited[v]:\\n                visited[v] = True\\n                prob[v] = prob[u]*(1.0/cnt)\\n                d[v] = d[u]+1\\n                st.append(v)\\n                \\nprint(ans)\\n\\n\", \"from queue import *\\n\\nn = int(input())\\ng = [[] for x in range(n)]\\n\\nfor i in range(n-1):\\n  a, b = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  g[a].append(b)\\n  g[b].append(a)\\n\\nused = [0]*n\\n\\nanw = 0\\n\\ndef solve(v, d, r):\\n  nonlocal anw\\n  q = Queue()\\n  q.put((v, d, r))\\n  while not q.empty():\\n    v, d, r = q.get(v)\\n    used[v] = True\\n    for u in g[v]:\\n      if not used[u]:\\n        q.put((u, d+1, r*(len(g[v])-(v!=0))))\\n        #print(\\\"put\\\", u)\\n    #print(\\\"so at\\\", v, \\\"len\\\", len(g[v]))\\n    if v != 0 and len(g[v]) == 1:\\n      #print(\\\"At \\\", v, \\\"is\\\", d, r)\\n      anw += d/r\\n  \\nsolve(0, 0, 1)\\nprint(anw)\", \"def f():\\n    n = int(input())\\n\\n    if n == 1:\\n        print(0)\\n        return\\n\\n    road = {}\\n    visited = {}\\n    for i in range(n-1):\\n        u, v = input().split()\\n        if u in road:\\n            road[u].append(v)\\n        else:\\n            road[u] = [v]\\n        if v in road:\\n            road[v].append(u)\\n        else:\\n            road[v] = [u]\\n        visited[u] = False\\n        visited[v] = False\\n\\n    prob = {}\\n    length = {}\\n    res = []\\n\\n    queue = []\\n    for k in road['1']:\\n        prob[k] = 1.0 / len(road['1'])\\n        length[k] = 1\\n        queue.append(k)\\n        visited['1'] = True\\n\\n    while len(queue) > 0:\\n        cur = queue[0]\\n        del queue[0]\\n        dest = []\\n        for k in road[cur]:\\n            if not visited[k]:\\n                dest.append(k)\\n        if len(dest) == 0:\\n            res.append((cur, prob[cur], length[cur]))\\n            continue\\n        for k in dest:\\n            prob[k] = prob[cur] / len(dest)\\n            length[k] = length[cur] + 1\\n            queue.append(k)\\n            visited[cur] = True\\n\\n    val = 0.0\\n    for item in res:\\n        val += item[1] * item[2]\\n\\n    print(val)\\n\\nf()\\n\", \"from collections import deque\\nimport sys\\nfrom decimal import Decimal\\n\\nreadline = sys.stdin.readline\\nn = int(input())\\nedges = [[]*n for _ in [None]*n]\\n\\nfor _ in [None]*(n-1):\\n    a, b = map(int, readline().split())\\n    edges[a-1].append(b-1)\\n    edges[b-1].append(a-1)\\n\\nvisited = [False]*n\\nvisited[0] = True\\ncnt = 0\\ntotal = 0\\ndq = deque()\\nappend, pop = dq.append, dq.popleft\\nappend((0, 0, Decimal(1)))\\n\\nwhile dq:\\n    pos, l, multiple = pop()\\n    flag = True\\n    to = [x for x in edges[pos] if visited[x] == False]\\n\\n    if to:\\n        x = len(to)\\n        multiple /= Decimal(x)\\n        for v in to:\\n            visited[v] = True\\n            append((v, l+1, multiple))\\n    else:\\n        total += Decimal(l) * multiple\\n\\nprint(total)\", \"import sys\\nfrom collections import deque\\nread=lambda:sys.stdin.readline().rstrip()\\nreadi=lambda:int(sys.stdin.readline())\\nwriteln=lambda x:sys.stdout.write(str(x)+\\\"\\\\n\\\")\\nwrite=lambda x:sys.stdout.write(x)\\nN = readi()\\nif N == 1:\\n    writeln(0)\\n    return\\n\\nG = [set() for _ in range(N)]\\nfor _ in range(N-1):\\n    u, v = list(map(int, read().split()))\\n    G[u-1].add(v-1)\\n    G[v-1].add(u-1)\\n\\nvisited = [0]*N\\nprev = [0]*N\\nvisited[0] = 1\\nprev[0] = 1\\nq = deque([0])\\nlengths = []\\nev = 0\\nwhile q:\\n    cur = q.popleft()\\n    n_neighbor = len(G[cur])\\n    if cur == 0:\\n        divs = n_neighbor\\n    else:\\n        divs = n_neighbor - 1\\n    if n_neighbor == 1 and cur != 0:\\n        ev += (visited[cur] - 1)*prev[cur]\\n        continue\\n     \\n    for nxt in G[cur]:\\n        if not visited[nxt]:\\n            visited[nxt] = visited[cur] + 1\\n            prev[nxt] = (1 / divs)*prev[cur]\\n            q.append(nxt)\\n\\nprint('%.12f' % ev)\\n\", \"#!/usr/bin/env python\\n\\nimport sys\\n\\ndef explore(src, prob, length, adj_mat, gains, visited):\\n    nexts = set(adj_mat[src]) - visited\\n    if nexts:\\n        go_prob = 1 / len(nexts)\\n        length += 1\\n        for dst in nexts:\\n            visited.add(dst)\\n            explore(dst, prob * go_prob, length, adj_mat, gains, visited)\\n    else:\\n        gains[src] += prob * length\\n\\ndef main():\\n  n = int(sys.stdin.readline())\\n  adj_mat = [[] for __ in range(n)]\\n  node_exp = [0 for __ in range(n)]\\n  for __ in range(n-1):\\n      u, v = list(map(int, sys.stdin.readline().split()))\\n      adj_mat[u-1].append(v-1)\\n      adj_mat[v-1].append(u-1)\\n  stk = [0]\\n  visited= set([0])\\n  prob = [1] * n\\n  length = [0] * n\\n  while stk:\\n      nxt = stk.pop()\\n      nexts = set(adj_mat[nxt]) - visited\\n      if nexts:\\n          go_prob = 1 / len(nexts)\\n          for dst in nexts:\\n              visited.add(dst)\\n              stk.append(dst)\\n              prob[dst] *= prob[nxt] * go_prob\\n              length[dst] = length[nxt] + 1\\n      else:\\n        node_exp[nxt] += prob[nxt] * length[nxt]\\n\\n  #explore(0, 1, 0, adj_mat,node_exp, set([0]))\\n  print(\\\"{:.13f}\\\".format(sum(node_exp)))\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n    if n == 1:\\n        print(0)\\n        return\\n    x = [[] for i in range(n)]\\n    for i in range(n - 1):\\n        a, b = list(map(int, sys.stdin.readline().split()))\\n        x[a - 1].append(b - 1)\\n        x[b - 1].append(a - 1)\\n\\n    isl = [False] * n\\n    for i in range(1, n):\\n        if len(x[i]) == 1:\\n            isl[i] = True\\n\\n    u = [False] * n\\n    u[0] = True\\n    q = [(0, 0, 1 / len(x[0]))]\\n    a = 0\\n    while len(q) != 0:\\n        c, s, t = q.pop()\\n        for i in x[c]:\\n            if not u[i]:\\n                if isl[i]:\\n                    a += (s + 1) * t\\n                else:\\n                    q.append((i, s + 1, t / (len(x[i])-1)))\\n                u[i] = True\\n    print(a)\\n\\n\\nmain()\\n\", \"n = int(input())\\nr = [[]for _ in range(n+1)]\\n\\nfor _ in range(n-1):\\n\\tu,v = list(map(int,input().split()))\\n\\tu -= 1\\n\\tv -= 1\\n\\tr[u].append(v)\\n\\tr[v].append(u)\\n\\nif n == 1:\\n\\tprint(0)\\nelse:\\n\\tst = [(0,0,1/len(r[0]))]\\n\\tans = 0\\n\\trsum = 0\\n\\tvisit = [False] * n\\n\\tvisit[0] = True\\n\\twhile st:\\n\\t\\tv,t,m = st.pop()\\n\\t\\tfor l in r[v]:\\n\\t\\t\\tif not visit[l]:\\n\\t\\t\\t\\tif l != 0 and len(r[l]) == 1:\\n\\t\\t\\t\\t\\tans += (t+1)*m\\n\\t\\t\\t\\t\\trsum += 1\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\tst.append((l,t+1,m/(len(r[l])-1)))\\n\\t\\t\\t\\tvisit[l] = True\\n\\tprint(ans)\\n\", \"from collections import deque, Counter\\n\\ndef mean_of_ways(graph, n):\\n    leafs = []\\n    distances = [None] * n\\n    probabilities = [0] * n\\n    probabilities[0] = 1\\n    distances[0] = 0\\n    used = [False] * n\\n    used[0] = True\\n\\n    active_vertices = deque([0])\\n\\n    while active_vertices:\\n        where = active_vertices.popleft()\\n\\n        # remember what happens if we change place of used assignment\\n        is_leaf = True\\n\\n        leafs_count = ([not used[v] for v in graph[where]]).count(True)\\n        if leafs_count > 1:\\n            probabilities[where] /= leafs_count\\n\\n        for to in graph[where]:\\n            if not used[to]:\\n                active_vertices.append(to)\\n                distances[to] = distances[where] + 1\\n                probabilities[to] = probabilities[where]\\n                used[to] = True\\n                is_leaf = False\\n\\n        if is_leaf:\\n            leafs.append(where)\\n\\n    Mx = 0\\n    for i in leafs:\\n        Mx += probabilities[i] * distances[i]\\n    return Mx\\n\\ndef main():\\n    n = int(input())\\n\\n    graph = [list() for i in range(n)]\\n    for i in range(n-1):\\n        # It there are vertices in descending order we can don't check\\n        # used it or not by making edges one directional\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n    print(mean_of_ways(graph, n))\\n\\nmain()\\n\", \"\\\"\\\"\\\"\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout\\n\\n\\ndef main():\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n\\n    # iterative DFS\\n    stack = [(1, 0, 1)]\\n    while len(stack):\\n        u, plen, prb = stack.pop()\\n\\n        visited[u] = True\\n        nBranch = 0\\n        for v in g[u]:\\n            if not visited[v]: nBranch += 1\\n        \\n        if nBranch > 0: \\n            probability = prb * (1 / nBranch)\\n            for v in g[u]:\\n                if not visited[v]:\\n                    stack.append((v, plen+1, probability))\\n        \\n        if nBranch == 0:\\n            paths.append(plen * prb)\\n\\n        \\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\n# node, probablity, path\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\n# print(L)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"\\\"\\\"\\\"\\n    # recursive DFS gives RTE, implement iterative DFS\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout, setrecursionlimit\\nimport threading\\n\\n\\ng = None\\nvisited = None\\npaths = None\\n\\ndef dfs(u, plen, prb):\\n    nonlocal visited, paths\\n\\n    visited[u] = True\\n    \\n    nBranch = 0\\n    for v in g[u]:\\n        if not visited[v]: nBranch += 1\\n    \\n    if nBranch > 0: \\n        probability = prb * (1 / nBranch)\\n        for v in g[u]:\\n            #print(prb, nBranch)\\n            if not visited[v]:\\n                dfs(v, plen+1, probability)\\n\\n    if nBranch == 0:\\n        paths.append(plen * prb)\\n\\n\\ndef main():\\n    nonlocal g, visited, paths\\n\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n    #setrecursionlimit(n+10)\\n    dfs(1, 0, 1)\\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    setrecursionlimit(10**6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"n = int(input())\\n\\nd = { x : [] for x in range(1, n + 1)}#defaultdict(list)\\nfor x in range(n - 1):\\n    s,de = map(int, input().split())\\n    d[s].append(de)\\n    d[de].append(s)\\n\\ndef dfs():\\n    lst = [(0, 1.0, 1)]\\n    visited = set({1})\\n    ans = 0\\n\\n    while lst:\\n        depth,prob,source = lst.pop()\\n        \\n        for neigh in d[source]:\\n            if neigh not in visited:\\n                visited.add(neigh)\\n                lst.append((depth + 1, prob*(1/(len(d[source]) - (1 if depth != 0 else 0))), neigh))\\n        \\n        if depth != 0 and len(d[source]) == 1:\\n            ans += prob*depth\\n\\n    return ans\\n\\nprint(\\\"{:.8f}\\\".format(dfs()))\", \"n = int(input())\\n\\ntree = {x: [] for x in range(1, n+1)}\\n\\nfor _ in range(n-1):\\n    k,l = list(map(int, input().split()))\\n    tree[k].append(l)\\n    tree[l].append(k)\\n\\n#for i in range(99999):\\n#    tree[i+1].append(i+2)\\n#    tree[i+2].append(i+1)\\n\\nvisited = set()\\ns = 0\\n\\na = [(1,1,0)]\\n\\nwhile a:\\n    v,p,l = a.pop()\\n    visited.add(v) \\n    k = 0\\n    for vv in tree[v]:\\n        if vv not in visited:\\n            k += 1\\n    if k <= 0:\\n        s += p*l\\n    else:\\n        for vv in tree[v]:\\n            if vv not in visited:\\n                a.append((vv,p*1.0/k,l+1))\\n\\nprint(s)\", \"from collections import defaultdict\\nfrom collections import deque\\nfrom sys import stdin\\nlines = deque(line.strip() for line in stdin.readlines())\\n\\ndef nextline():\\n    return lines.popleft()\\n\\ndef types(cast):\\n    return tuple(int(x) for x in nextline().split())\\n\\ndef ints():\\n    return types(int)\\n\\ndef strs():\\n    return nextline().split()\\n\\ndef main():\\n    # lines will now contain all of the input's lines in a list\\n    n = int(nextline())\\n    graph = defaultdict(list)\\n    for _ in range(1, n):\\n        a, b = ints()\\n        graph[a].append(b)\\n        graph[b].append(a)\\n    stack = [(1, 1, 0)]\\n    visited = set()\\n    expected = 0\\n    while stack:\\n        city, denominator, length = stack.pop()\\n        visited.add(city)\\n        choices = [choice for choice in graph[city] if choice not in visited]\\n        if choices:\\n            for choice in choices:\\n                stack.append((choice, denominator * len(choices), length + 1))\\n        else:\\n            expected += length / denominator\\n    print(expected)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import queue\\nn = int(input())\\ndic = {}\\n\\nfor i in range(n+1):\\n    dic[i] = []\\nfor i in range(n-1):\\n    u,v = [ int(x) for x in input().split() ]\\n    dic[u].append(v)\\n    dic[v].append(u)\\n\\nweight = {1:(0,1)} # No.City:(length,weight)\\nq = queue.Queue()\\nq.put(1)\\nwhile(q.empty() == False):\\n    city = q.get()\\n    choice = len(dic[city])\\n    if city > 1:\\n        choice = choice - 1\\n    for to_city in dic[city]:\\n        if to_city not in weight:\\n            weight[to_city] = (weight[city][0]+1, weight[city][1]/choice)\\n            q.put(to_city)\\n    if(choice > 0):\\n        weight[city] = (weight[city][0], 0)\\n\\nsum = 0\\nfor city in weight:\\n    sum = sum + weight[city][0] * weight[city][1]\\n\\nprint(\\\"%.15f\\\" % sum)\\n\\n\", \"def __starting_point():\\n    n = int(input())\\n    \\n    edges = [[] for i in range(n+1)]\\n    edges[0] = None\\n    for i in range(n-1):\\n        n1,n2 = list(map(int, input().split()))\\n        edges[n1].append(n2)\\n        edges[n2].append(n1)\\n\\n    # stack = [(node, height, prob),...]\\n    stack = [(1,0,1.0)]\\n    sev = 0\\n    visited = [False for i in range(n+1)]\\n    \\n    while len(stack) > 0:\\n        (node, height, prob) = stack.pop()\\n        visited[node] = True\\n        children = len(edges[node]) - 1\\n        if node == 1:\\n            # since root has no parent\\n            children += 1\\n            \\n        for child in edges[node]:\\n            if visited[child]:\\n                continue\\n            stack.append((child, height+1, prob/children))\\n        \\n        if children == 0:\\n            sev += height * prob\\n    print(sev)\\n            \\n    \\n    \\n    \\n    \\n\\n\\n\\n__starting_point()\", \"class Node:\\n\\tdef __init__(self, index):\\n\\t\\tself.neighbours = []\\n\\t\\tself.index = index\\n\\t\\tself.prob = 1.0\\n\\t\\tself.vis = False\\n\\t\\tself.length = 0\\n\\n\\tdef addNode(self, city):\\n\\t\\tself.neighbours.append(city)\\n\\t\\tif self.index == 1:\\n\\t\\t\\tself.prob = 1.0 / len(self.neighbours)\\n\\t\\telse:\\n\\t\\t\\tl = len(self.neighbours)\\n\\t\\t\\tself.prob = 1.0  if l < 2 else (1.0 / (l - 1))\\n\\nn = int(input())\\nif n == 1:\\n\\tprint(0)\\n\\treturn\\n\\t\\ncities = {}\\n\\nfor i in range(n-1):\\n\\ta, b = [int(k) for k in input().split()]\\n\\t#print (\\\"test \\\", a, \\\" to \\\", b)\\n\\tif a not in cities:\\n\\t\\tcities[a] = Node(a)\\n\\tcities[a].addNode(b)\\n\\tif b not in cities:\\n\\t\\tcities[b] = Node(b)\\n\\tcities[b].addNode(a)\\n\\nif len(cities) == 2:\\n\\tprint(1)\\n\\treturn\\n\\n\\n# for i in range(1, n + 1, 1):\\n# \\tprint (\\\"city.index \\\", cities[i].index, ' roads ', cities[i].neighbours, ' prob ', cities[i].prob)\\n\\n\\n# deadends = []\\n# deadendsProb = []\\n\\n\\n# def Parse(city, prob, length, oldCity):\\n# \\t#print ('parse ', city.index)\\n# \\tnewprob = 1.0 if oldCity == 0  else cities[oldCity].prob\\n# \\t#print ('nnewProb ', newprob)\\n# \\tprob *= newprob\\n# \\t#print (city.index, ' len ', len(city.neighbours))\\n# \\tif len(city.neighbours) == 1 and oldCity != 0:\\n# \\t\\tdeadends.append(length)\\n# \\t\\tdeadendsProb.append(prob)\\n# \\telse:\\n# \\t\\t#print (city.neighbours)\\n# \\t\\tlength += 1\\n# \\t\\tfor c in city.neighbours:\\n# \\t\\t\\tif c != oldCity:\\n# \\t\\t\\t\\tParse(cities[c], prob, length, city.index )\\n\\n# Parse(cities[1], 1.0, 0, 0)\\n# #for i in range(len(deadends)):\\n# #\\tprint('len ', deadends[i], ' prob ', deadendsProb[i])\\n\\n# ans = sum(map(lambda l, p: l * p, deadends, deadendsProb))\\n# #print('ans', ans)\\n# print(ans)\\n\\n\\ndef inorder(city):\\n\\ts = []\\n\\ts.append(city)\\n\\t#print ('index ', city.index)\\n\\t#print ('neighbours ', city.neighbours)\\n\\twhile s:\\n\\t\\tcity = s.pop()\\n\\t\\tcity.vis = True\\n\\t\\tif city.neighbours:\\n\\t\\t\\tif city.index == 1 or len(city.neighbours) > 1:\\n\\t\\t\\t\\tfor c in city.neighbours:\\n\\t\\t\\t\\t\\tif not cities[c].vis:\\n\\t\\t\\t\\t\\t\\tcities[c].length = city.length + 1\\n\\t\\t\\t\\t\\t\\tcities[c].prob *= city.prob\\n\\t\\t\\t\\t\\t\\ts.append(cities[c])\\n\\t\\t\\telse:\\n\\t\\t\\t\\tyield (city.index, city.prob, city.length)\\n\\n\\n\\ntest = sum([city[1] * city[2] for city in inorder(cities[1])])\\nprint(test)\\n\\n\\n\\n# this is a tree\\n\", \"from sys import setrecursionlimit\\nimport threading\\n\\n\\nlength = 0\\nvisited = None\\ng = None\\n\\n\\ndef dfs(v, path_length, chance):\\n    nonlocal length\\n    visited.add(v)\\n    edges = 0\\n    for n in g[v]:\\n        if n not in visited:\\n            edges += 1\\n    for n in g[v]:\\n        if n not in visited:\\n            dfs(n, path_length + 1, chance * (1 / edges))\\n    if edges == 0:\\n        length += path_length * chance\\n\\n\\ndef main():\\n    nonlocal visited, g\\n    vert = int(input())\\n    g = [[] for _ in range(vert + 1)]\\n    visited = set()\\n    for _ in range(vert - 1):\\n        a, b = list(map(int, input().split()))\\n        g[a].append(b)\\n        g[b].append(a)\\n    dfs(1, 0, 1)\\n    print(length)\\n\\n\\ndef __starting_point():\\n    setrecursionlimit(10 ** 6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"n = int(input())\\ng = [[] for i in range(n+1)]\\nfor i in range(n-1):\\n    a,b = [int(x) for x in input().split()]\\n    g[a].append(b)\\n    g[b].append(a)\\n\\nd = [0 for i in range(n+1)]\\nv = [0 for i in range(n+1)]\\ne = [0 for i in range(n+1)]\\np = [0.0 for i in range(n+1)]\\np[1] = 1.0\\nen = 0\\nq = [1]\\nwhile len(q):\\n    c = q.pop(0)\\n    v[c] = 1\\n    a = 0\\n    for ne in g[c]:\\n        if not v[ne]:\\n            q.append(ne)\\n            d[ne] = d[c] + 1\\n            p[ne] = p[c]/len(g[c]) if c == 1 else p[c]/(len(g[c])-1)\\n            a += 1\\n    if a == 0:\\n        e[c] = 1\\n        en += 1\\nave = 0.0\\nfor i in range(n+1):\\n    if e[i]:\\n        ave += d[i]*p[i]\\nprint(ave)\\n\", \"n=int(input())\\nif n==1:\\n\\tprint(0)\\n\\treturn\\ngraph=[None]*n\\nfor i in range(n-1):\\n\\tu,v=map(int,input().split())\\n\\tu-=1\\n\\tv-=1\\n\\tif graph[u]==None:\\n\\t\\tgraph[u]=[v]\\n\\telse:\\n\\t\\tgraph[u].append(v)\\n\\tif graph[v]==None:\\n\\t\\tgraph[v]=[u]\\n\\telse:\\n\\t\\tgraph[v].append(u)\\n\\nis_checked=[False]*n\\nlens=[None]*n\\nlens[0]=0\\ndivs=[None]*n\\ndivs[0]=1\\nstack=[0]\\nans=0\\nwhile stack:\\n\\ttop=stack.pop()\\n\\tis_checked[top]=True\\n\\tvals=graph[top]\\n\\tfor i in range(len(vals)):\\n\\t\\tcur=vals[i]\\n\\t\\tif not is_checked[cur]:\\n\\t\\t\\tlens[cur]=lens[top]+1\\n\\t\\t\\tdivs[cur]=divs[top]*(len(vals)-(1 if top!=0 else 0))\\n\\t\\t\\tif len(graph[cur])==1:\\n\\t\\t\\t\\tans+=lens[cur]/divs[cur]\\n\\t\\t\\t\\tis_checked[cur]=None\\n\\t\\t\\t\\tlens[cur]=None\\n\\t\\t\\t\\tdivs[cur]=None\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tstack.append(cur)\\nprint('{:.8f}'.format(ans))\"]",
        "difficulty": "interview",
        "input": "10\n8 6\n9 10\n8 7\n1 4\n1 8\n9 5\n9 8\n2 5\n3 1\n",
        "output": "1.500000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/839/C"
    },
    {
        "id": 583,
        "task_id": 3469,
        "test_case_id": 1,
        "question": "As a Hunter, you will undoubtedly face difficult obstacles during your journey. At such time, a Hunter is expected to model the situation using Mathematical models, and apply probability and statistics knowledge to overcome the situation.\n\nThus, the third round of the Hunter Exam is based on the following game with coins:\n - Before the game starts, Gon selects a string $g$ and Killua selects a string $k$. The two strings must contain only characters ‘H’ and ‘T’.\n - A game master will flip a coin an infinite number of times. After each flip, the result (either ‘H’ or ‘T’ — representing head or tail) is appended into a string $s$. The string $s$ is empty at the beginning of the game.\n - After some coin flip:\n - If both $g$ and $k$ become a substring of $s$, the game ends in a draw.\n - If only $g$ becomes a substring of $s$, Gon wins, and the game ends.\n - If only $k$ becomes a substring of $s$, Killua wins, and the game ends.\n - Gon and Killua only have finite amount of time. They will stop the game in a draw after $10^{100}$ turns.\n\nPlease calculate the probability that Gon wins.\n\n-----Input-----\nThe input contains exactly three lines:\n - The first line contains the string $g$.\n - The second line contains the string $k$.\n - The third line contains a real number $p$ with exactly one digit after the decimal point — the probability that a coin flip will result in head $(0 < p < 1)$.\n\nThe length of $g$ and $k$ are at most $20$. $g$ and $k$ are non-empty, and contain only characters ‘H’ and ‘T’.\n\n-----Output-----\nThe output must contain a single number — the probability that Gon wins.\n\nYour answer will be considered correct if its relative or absolute error doesn’t exceed $10^{-6}$.\n\nNamely: let’s assume that your answer is $a$, and the answer of the jury is $b$. The checker program will consider your answer correct, if $\\frac{|a-b|}{max(1,b)} \\leq 10^{-6}$.\n\n-----Examples-----\nSample Input:\nH\nT\n0.5\nSample Output:\n0.5",
        "solutions": "",
        "difficulty": "competition",
        "input": "H\nT\n0.5\n",
        "output": "0.5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/easyprobability"
    },
    {
        "id": 584,
        "task_id": 3469,
        "test_case_id": 2,
        "question": "As a Hunter, you will undoubtedly face difficult obstacles during your journey. At such time, a Hunter is expected to model the situation using Mathematical models, and apply probability and statistics knowledge to overcome the situation.\n\nThus, the third round of the Hunter Exam is based on the following game with coins:\n - Before the game starts, Gon selects a string $g$ and Killua selects a string $k$. The two strings must contain only characters ‘H’ and ‘T’.\n - A game master will flip a coin an infinite number of times. After each flip, the result (either ‘H’ or ‘T’ — representing head or tail) is appended into a string $s$. The string $s$ is empty at the beginning of the game.\n - After some coin flip:\n - If both $g$ and $k$ become a substring of $s$, the game ends in a draw.\n - If only $g$ becomes a substring of $s$, Gon wins, and the game ends.\n - If only $k$ becomes a substring of $s$, Killua wins, and the game ends.\n - Gon and Killua only have finite amount of time. They will stop the game in a draw after $10^{100}$ turns.\n\nPlease calculate the probability that Gon wins.\n\n-----Input-----\nThe input contains exactly three lines:\n - The first line contains the string $g$.\n - The second line contains the string $k$.\n - The third line contains a real number $p$ with exactly one digit after the decimal point — the probability that a coin flip will result in head $(0 < p < 1)$.\n\nThe length of $g$ and $k$ are at most $20$. $g$ and $k$ are non-empty, and contain only characters ‘H’ and ‘T’.\n\n-----Output-----\nThe output must contain a single number — the probability that Gon wins.\n\nYour answer will be considered correct if its relative or absolute error doesn’t exceed $10^{-6}$.\n\nNamely: let’s assume that your answer is $a$, and the answer of the jury is $b$. The checker program will consider your answer correct, if $\\frac{|a-b|}{max(1,b)} \\leq 10^{-6}$.\n\n-----Examples-----\nSample Input:\nH\nT\n0.5\nSample Output:\n0.5",
        "solutions": "",
        "difficulty": "competition",
        "input": "HH\nTH\n0.5\n",
        "output": "0.25\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/easyprobability"
    },
    {
        "id": 585,
        "task_id": 581,
        "test_case_id": 3,
        "question": "You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. \n\nInitial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. \n\nCalculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!\n\n\n-----Input-----\n\nThe first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. \n\nNext n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer number — maximal possible answer. \n\nIn the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. \n\nSee the examples for better understanding.\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n3\n2 3 3\n2 1 1\n\nInput\n5\n1 2\n1 3\n2 4\n2 5\n\nOutput\n9\n3 5 5\n4 3 3\n4 1 1\n4 2 2",
        "solutions": "[\"import sys\\nfrom collections import deque as dq\\nn = int(input())\\n\\nind = 0\\ninp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]\\n\\ncoupl = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b = inp[ind],inp[ind+1]\\n    ind+=2\\n    coupl[a].append(b)\\n    coupl[b].append(a)\\n\\nQ = dq()\\nfound = [False]*n\\nmaster = 0\\nfound[master] = True\\n\\ndia1 = 0\\nQ.append(master)\\nwhile Q:\\n    node = Q.popleft()\\n    dia1 = node\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append(nei)\\n\\n\\ndia2 = 0\\nQ.append((dia1,0))\\ndist1 = [0]*n\\nfound = [False]*n\\nfound[dia1] = True\\nwhile Q:\\n    node,d = Q.popleft()\\n    dia2 = node\\n    dist1[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nQ = []\\nQ.append((dia2,0))\\ndist2 = [0]*n\\nfound = [False]*n\\nfound[dia2] = True\\nwhile Q:\\n    node,d = Q.pop()\\n    dist2[node]=d\\n    for nei in coupl[node]:\\n        if not found[nei]:\\n            found[nei] = True\\n            Q.append((nei,d+1))\\n\\nneigs = [0]*n\\n\\nleaves = []\\nfor i in range(n):\\n    if i != dia1 and i != dia2 and len(coupl[i])==1:\\n        leaves.append(i)\\n    neigs[i]=len(coupl[i])\\npoints = 0\\nlista = []\\n\\nwhile leaves:\\n    node = leaves.pop()\\n    if dist1[node]<dist2[node]:\\n        lista.append((dia2,node,node))\\n        points += dist2[node]\\n    else:\\n        lista.append((dia1,node,node))\\n        points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nleaves.append(dia2)\\nwhile leaves:\\n    node = leaves.pop()\\n    lista.append((dia1,node,node))\\n    points += dist1[node]\\n    for nei in coupl[node]:\\n        neigs[nei]-=1\\n        if neigs[nei]==1:\\n            leaves.append(nei)\\nprint(points)\\nfor l in lista:\\n    a,b,c = l\\n    print(a+1,b+1,c+1)\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in list(tree_edges.items()):\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n\\n    edges = list(map(int, sys.stdin.read().split()))\\n    tree_edges = dict()\\n    for i in range(n):\\n        tree_edges[i + 1] = set()\\n\\n    for i in range(0, len(edges) - 1, 2):\\n        tree_edges[edges[i]].add(edges[i + 1])\\n        tree_edges[edges[i + 1]].add(edges[i])\\n\\n    init_distants = [-1] * (n + 1)\\n\\n    queue = [1]\\n    init_distants[1] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if init_distants[next_vertex] == -1:\\n                    init_distants[next_vertex] = init_distants[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    head = init_distants.index(max(init_distants))\\n    distants_from_head = [-1] * (n + 1)\\n    queue = [head]\\n\\n    distants_from_head[head] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_head[next_vertex] == -1:\\n                    distants_from_head[next_vertex] = distants_from_head[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    tail = distants_from_head.index(max(distants_from_head))\\n    distants_from_tail = [-1] * (n + 1)\\n    queue = [tail]\\n\\n    distants_from_tail[tail] = 0\\n\\n    while queue:\\n        next_queue = []\\n        for process in queue:\\n            for next_vertex in tree_edges[process]:\\n                if distants_from_tail[next_vertex] == -1:\\n                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1\\n                    next_queue.append(next_vertex)\\n        queue = next_queue\\n\\n    path_len_sum = 0\\n    removal_history = list()\\n\\n    process_queue = []\\n\\n    for vertex, adj in tree_edges.items():\\n        if len(adj) == 1:\\n            process_queue.append(vertex)\\n\\n    while process_queue:\\n        next_queue = []\\n\\n        for leaf in process_queue:\\n\\n            if leaf == head or leaf == tail:\\n                continue\\n\\n            if distants_from_tail[leaf] > distants_from_head[leaf]:\\n                path_len_sum += distants_from_tail[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, tail))\\n            else:\\n                path_len_sum += distants_from_head[leaf]\\n                new_leaves = []\\n\\n                for w in tree_edges[leaf]:\\n                    tree_edges[w].remove(leaf)\\n                    if len(tree_edges[w]) == 1:\\n                        new_leaves.append(w)\\n                next_queue.extend(new_leaves)\\n                removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n        process_queue = next_queue\\n\\n    process_queue = [tail]\\n\\n    while process_queue:\\n        leaf = process_queue[0]\\n\\n        if leaf == head:\\n            continue\\n\\n        path_len_sum += distants_from_head[leaf]\\n        new_leaves = []\\n\\n        for w in tree_edges[leaf]:\\n            tree_edges[w].remove(leaf)\\n            if len(tree_edges[w]) == 1:\\n                new_leaves.append(w)\\n        process_queue = new_leaves\\n        removal_history.append(\\\"{0} {1} {0}\\\".format(leaf, head))\\n\\n    print(str(path_len_sum))\\n    sys.stdout.write(\\\"\\\\n\\\".join(removal_history))\\n    sys.stdout.write(\\\"\\\\n\\\")\\n\\n\\nmain()\"]",
        "difficulty": "interview",
        "input": "2\n1 2\n",
        "output": "1\n2 1 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/911/F"
    },
    {
        "id": 586,
        "task_id": 697,
        "test_case_id": 3,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "2 2\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 587,
        "task_id": 697,
        "test_case_id": 4,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "2000 2000\n",
        "output": "674532367\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 588,
        "task_id": 697,
        "test_case_id": 6,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "11 2\n",
        "output": "716\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 589,
        "task_id": 697,
        "test_case_id": 8,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "5 13\n",
        "output": "4048\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 590,
        "task_id": 697,
        "test_case_id": 9,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "60 59\n",
        "output": "271173738\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 591,
        "task_id": 697,
        "test_case_id": 10,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "27 16\n",
        "output": "886006554\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 592,
        "task_id": 697,
        "test_case_id": 11,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1134 1092\n",
        "output": "134680101\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 593,
        "task_id": 697,
        "test_case_id": 12,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "756 1061\n",
        "output": "72270489\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 594,
        "task_id": 697,
        "test_case_id": 13,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "953 1797\n",
        "output": "557692333\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 595,
        "task_id": 697,
        "test_case_id": 14,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "76 850\n",
        "output": "103566263\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 596,
        "task_id": 697,
        "test_case_id": 15,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "24 1508\n",
        "output": "540543518\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 597,
        "task_id": 697,
        "test_case_id": 16,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1087 1050\n",
        "output": "973930225\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 598,
        "task_id": 697,
        "test_case_id": 17,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "149 821\n",
        "output": "64450770\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 599,
        "task_id": 697,
        "test_case_id": 18,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "983 666\n",
        "output": "917123830\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 600,
        "task_id": 697,
        "test_case_id": 19,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "45 1323\n",
        "output": "357852234\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 601,
        "task_id": 697,
        "test_case_id": 20,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1994 1981\n",
        "output": "596939902\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 602,
        "task_id": 697,
        "test_case_id": 21,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1942 1523\n",
        "output": "89088577\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 603,
        "task_id": 697,
        "test_case_id": 22,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1891 1294\n",
        "output": "696966158\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 604,
        "task_id": 697,
        "test_case_id": 23,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1132 1727\n",
        "output": "878164775\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 605,
        "task_id": 697,
        "test_case_id": 24,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1080 383\n",
        "output": "161999131\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 606,
        "task_id": 697,
        "test_case_id": 25,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1028 1040\n",
        "output": "119840364\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 607,
        "task_id": 697,
        "test_case_id": 26,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "976 1698\n",
        "output": "621383232\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 608,
        "task_id": 697,
        "test_case_id": 27,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "38 656\n",
        "output": "814958661\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 609,
        "task_id": 697,
        "test_case_id": 28,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "872 1313\n",
        "output": "261808476\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 610,
        "task_id": 697,
        "test_case_id": 29,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1935 856\n",
        "output": "707458926\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 611,
        "task_id": 697,
        "test_case_id": 30,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1883 1513\n",
        "output": "265215482\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 612,
        "task_id": 697,
        "test_case_id": 33,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1991 1992\n",
        "output": "518738831\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 613,
        "task_id": 697,
        "test_case_id": 34,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1935 1977\n",
        "output": "16604630\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 614,
        "task_id": 697,
        "test_case_id": 35,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1990 2000\n",
        "output": "516468539\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 615,
        "task_id": 697,
        "test_case_id": 36,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1915 1915\n",
        "output": "534527105\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 616,
        "task_id": 893,
        "test_case_id": 3,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n",
        "output": "41\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 617,
        "task_id": 893,
        "test_case_id": 4,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "7 25\n113 106 118 108 106 102 106 104 107 120 114 120 112 100 113 118 112 118 113 102 110 105 118 114 101\n13 16\n16 23\n10 19\n6 9\n17 20\n8 12\n9 13\n8 24\n8 14\n17 22\n1 17\n1 5\n18 21\n1 8\n2 4\n2 3\n5 15\n2 10\n7 18\n3 25\n4 11\n3 6\n1 2\n4 7\n",
        "output": "61\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 618,
        "task_id": 893,
        "test_case_id": 6,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "5 3\n15 7 9\n1 2\n2 3\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 619,
        "task_id": 893,
        "test_case_id": 9,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "6 17\n1239 1243 1236 1235 1240 1245 1258 1245 1239 1244 1241 1251 1245 1250 1259 1245 1259\n8 16\n7 11\n4 8\n1 2\n7 9\n3 4\n3 15\n11 12\n10 17\n1 5\n3 14\n5 6\n9 10\n5 13\n4 7\n1 3\n",
        "output": "36\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 620,
        "task_id": 893,
        "test_case_id": 10,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "8 19\n1983 1991 1992 1985 1980 1990 1989 1985 1998 2000 1994 1984 1981 1996 1996 2000 2000 1992 1986\n9 12\n1 2\n1 10\n12 16\n4 5\n2 3\n13 18\n4 7\n11 15\n2 6\n10 19\n5 14\n4 17\n2 8\n3 4\n9 11\n11 13\n8 9\n",
        "output": "45\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 621,
        "task_id": 893,
        "test_case_id": 15,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "10 20\n1500 958 622 62 224 951 1600 1465 1230 1965 1940 1032 914 1501 1719 1134 1756 130 330 1826\n7 15\n6 10\n1 9\n5 8\n9 18\n1 16\n2 20\n9 14\n7 13\n8 11\n1 2\n1 6\n2 3\n7 17\n2 5\n1 4\n14 19\n5 7\n4 12\n",
        "output": "20\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 622,
        "task_id": 893,
        "test_case_id": 17,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "100 17\n1848 1816 1632 1591 1239 1799 1429 1867 1265 1770 1492 1812 1753 1548 1712 1780 1618\n12 15\n2 3\n7 16\n1 2\n1 10\n6 9\n5 11\n14 17\n6 8\n6 14\n9 12\n4 6\n3 7\n1 4\n1 5\n8 13\n",
        "output": "23\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 623,
        "task_id": 893,
        "test_case_id": 18,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "285 8\n529 1024 507 126 1765 1260 1837 251\n2 4\n2 7\n4 8\n1 3\n1 5\n1 2\n5 6\n",
        "output": "10\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 624,
        "task_id": 893,
        "test_case_id": 19,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "530 21\n6 559 930 239 252 949 641 700 99 477 525 654 796 68 497 492 940 496 10 749 590\n3 11\n2 5\n12 13\n17 18\n1 8\n1 2\n5 19\n3 17\n2 3\n2 4\n4 20\n8 10\n2 7\n5 9\n3 14\n11 16\n5 6\n14 21\n4 15\n3 12\n",
        "output": "134\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 625,
        "task_id": 893,
        "test_case_id": 20,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "1110 28\n913 1686 784 243 1546 1700 1383 1859 1322 198 1883 793 687 1719 1365 277 1887 1675 1659 1616 1325 1937 732 1789 1078 1408 736 1402\n4 10\n4 16\n2 7\n10 18\n10 14\n7 9\n2 15\n7 11\n8 13\n9 25\n15 26\n1 3\n4 8\n3 4\n1 5\n7 23\n26 28\n12 19\n7 17\n1 2\n3 6\n2 12\n15 27\n16 20\n1 24\n15 21\n9 22\n",
        "output": "6374\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 626,
        "task_id": 893,
        "test_case_id": 21,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "777 24\n1087 729 976 1558 1397 1137 1041 576 1693 541 1144 682 1577 1843 339 703 195 18 1145 818 145 484 237 1315\n3 13\n18 19\n8 12\n2 4\n1 15\n5 7\n11 17\n18 23\n1 22\n1 2\n3 9\n12 18\n8 10\n6 8\n13 21\n10 11\n1 5\n4 6\n14 20\n2 16\n1 24\n2 3\n6 14\n",
        "output": "97\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 627,
        "task_id": 893,
        "test_case_id": 22,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "5 9\n1164 1166 1167 1153 1153 1153 1155 1156 1140\n4 5\n6 9\n6 7\n2 6\n1 3\n1 2\n1 8\n3 4\n",
        "output": "14\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 628,
        "task_id": 893,
        "test_case_id": 23,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "11 25\n380 387 381 390 386 384 378 389 390 390 389 385 379 387 390 381 390 386 384 379 379 384 379 388 383\n3 25\n16 18\n7 17\n6 10\n1 13\n5 7\n2 19\n5 12\n1 9\n2 4\n5 16\n3 15\n1 11\n8 24\n14 23\n4 5\n6 14\n5 6\n7 8\n3 22\n2 3\n6 20\n1 2\n6 21\n",
        "output": "25223\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 629,
        "task_id": 893,
        "test_case_id": 25,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "300 34\n777 497 1099 1221 1255 733 1119 533 1130 822 1000 1272 1104 575 1012 1137 1125 733 1036 823 845 923 1271 949 709 766 935 1226 1088 765 1269 475 1020 977\n5 18\n5 8\n1 20\n2 25\n4 19\n11 34\n6 9\n14 23\n21 22\n12 30\n7 11\n3 12\n18 21\n1 4\n2 6\n1 2\n11 15\n2 31\n4 13\n25 28\n1 3\n23 24\n1 17\n4 5\n15 29\n9 10\n11 33\n1 32\n4 14\n8 16\n2 7\n4 27\n15 26\n",
        "output": "86\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 630,
        "task_id": 893,
        "test_case_id": 26,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "18 29\n18 2 24 10 8 10 19 12 16 2 2 23 15 17 29 13 10 14 21 8 2 13 23 29 20 3 18 16 22\n11 23\n10 19\n14 22\n14 17\n25 26\n7 25\n7 11\n6 13\n1 3\n12 28\n1 2\n8 18\n6 8\n9 12\n2 9\n4 14\n1 20\n6 15\n4 10\n5 6\n21 27\n2 16\n7 21\n1 5\n19 29\n6 7\n9 24\n1 4\n",
        "output": "13297\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 631,
        "task_id": 893,
        "test_case_id": 28,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "13 5\n125 118 129 146 106\n3 4\n1 3\n1 2\n4 5\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 632,
        "task_id": 893,
        "test_case_id": 30,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "17 25\n32 39 34 47 13 44 46 44 24 28 12 22 33 13 47 27 23 16 35 10 37 29 39 35 10\n4 6\n7 12\n9 15\n2 5\n4 8\n4 17\n6 21\n22 23\n21 22\n6 10\n8 9\n1 14\n1 4\n11 13\n1 24\n1 2\n6 18\n7 16\n6 25\n8 11\n17 19\n10 20\n2 3\n4 7\n",
        "output": "125\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 633,
        "task_id": 1246,
        "test_case_id": 1,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "2\ninsert 3\ngetMin 4\n",
        "output": "4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 634,
        "task_id": 1246,
        "test_case_id": 2,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n",
        "output": "6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 635,
        "task_id": 1246,
        "test_case_id": 12,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "6\ninsert -799688192\ngetMin 491561656\nremoveMin\ninsert -805250162\ninsert -945439443\nremoveMin\n",
        "output": "8\ninsert -799688192\nremoveMin\ninsert 491561656\ngetMin 491561656\nremoveMin\ninsert -805250162\ninsert -945439443\nremoveMin\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 636,
        "task_id": 1246,
        "test_case_id": 13,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "9\ninsert 3\ninsert 4\ninsert 5\nremoveMin\ngetMin 3\nremoveMin\ngetMin 4\nremoveMin\ngetMin 5\n",
        "output": "10\ninsert 3\ninsert 4\ninsert 5\nremoveMin\ninsert 3\ngetMin 3\nremoveMin\ngetMin 4\nremoveMin\ngetMin 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 637,
        "task_id": 1246,
        "test_case_id": 14,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "9\ninsert 3\ninsert 4\ninsert 5\nremoveMin\ngetMin 5\nremoveMin\ngetMin 4\nremoveMin\ngetMin 3\n",
        "output": "12\ninsert 3\ninsert 4\ninsert 5\nremoveMin\nremoveMin\ngetMin 5\nremoveMin\ninsert 4\ngetMin 4\nremoveMin\ninsert 3\ngetMin 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 638,
        "task_id": 1246,
        "test_case_id": 16,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "3\ninsert 3\ninsert 4\ngetMin 4\n",
        "output": "4\ninsert 3\ninsert 4\nremoveMin\ngetMin 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 639,
        "task_id": 1246,
        "test_case_id": 17,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "3\ninsert 1\ninsert 2\ngetMin 2\n",
        "output": "4\ninsert 1\ninsert 2\nremoveMin\ngetMin 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 640,
        "task_id": 1246,
        "test_case_id": 18,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "3\ninsert -1\ninsert 0\ngetMin 0\n",
        "output": "4\ninsert -1\ninsert 0\nremoveMin\ngetMin 0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 641,
        "task_id": 1246,
        "test_case_id": 21,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "3\ninsert 1\ninsert 0\ngetMin 1\n",
        "output": "4\ninsert 1\ninsert 0\nremoveMin\ngetMin 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 642,
        "task_id": 1388,
        "test_case_id": 14,
        "question": "Ashish has a tree consisting of $n$ nodes numbered $1$ to $n$ rooted at node $1$. The $i$-th node in the tree has a cost $a_i$, and binary digit $b_i$ is written in it. He wants to have binary digit $c_i$ written in the $i$-th node in the end.\n\nTo achieve this, he can perform the following operation any number of times:   Select any $k$ nodes from the subtree of any node $u$, and shuffle the digits in these nodes as he wishes, incurring a cost of $k \\cdot a_u$. Here, he can choose $k$ ranging from $1$ to the size of the subtree of $u$. \n\nHe wants to perform the operations in such a way that every node finally has the digit corresponding to its target.\n\nHelp him find the minimum total cost he needs to spend so that after all the operations, every node $u$ has digit $c_u$ written in it, or determine that it is impossible.\n\n\n-----Input-----\n\nFirst line contains a single integer $n$ $(1 \\le n \\le 2 \\cdot 10^5)$ denoting the number of nodes in the tree.\n\n$i$-th line of the next $n$ lines contains 3 space-separated integers $a_i$, $b_i$, $c_i$ $(1 \\leq a_i \\leq 10^9, 0 \\leq b_i, c_i \\leq 1)$  — the cost of the $i$-th node, its initial digit and its goal digit.\n\nEach of the next $n - 1$ lines contain two integers $u$, $v$ $(1 \\leq u, v \\leq n, \\text{ } u \\ne v)$, meaning that there is an edge between nodes $u$ and $v$ in the tree.\n\n\n-----Output-----\n\nPrint the minimum total cost to make every node reach its target digit, and $-1$ if it is impossible.\n\n\n-----Examples-----\nInput\n5\n1 0 1\n20 1 0\n300 0 1\n4000 0 0\n50000 1 0\n1 2\n2 3\n2 4\n1 5\n\nOutput\n4\nInput\n5\n10000 0 1\n2000 1 0\n300 0 1\n40 0 0\n1 1 0\n1 2\n2 3\n2 4\n1 5\n\nOutput\n24000\nInput\n2\n109 0 1\n205 0 1\n1 2\n\nOutput\n-1\n\n\n-----Note-----\n\nThe tree corresponding to samples $1$ and $2$ are: [Image]\n\nIn sample $1$, we can choose node $1$ and $k = 4$ for a cost of $4 \\cdot 1$ = $4$ and select nodes ${1, 2, 3, 5}$, shuffle their digits and get the desired digits in every node.\n\nIn sample $2$, we can choose node $1$ and $k = 2$ for a cost of $10000 \\cdot 2$, select nodes ${1, 5}$ and exchange their digits, and similarly, choose node $2$ and $k = 2$ for a cost of $2000 \\cdot 2$, select nodes ${2, 3}$ and exchange their digits to get the desired digits in every node.\n\nIn sample $3$, it is impossible to get the desired digits, because there is no node with digit $1$ initially.",
        "solutions": "[\"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nfrom collections import deque\\nN = int(input())\\nC, Y = [], []\\nfor _ in range(N):\\n    a, b, c = list(map(int, input().split()))\\n    C.append(a)\\n    Y.append(c - b)\\n\\nif sum(Y):\\n    print(-1)\\n    return\\n\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = list(map(int, input().split()))\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\nP = [-1] * N\\nQ = deque([0])\\nR = []\\nwhile Q:\\n    i = deque.popleft(Q)\\n    R.append(i)\\n    for a in X[i]:\\n        if a != P[i]:\\n            P[a] = i\\n            X[a].remove(i)\\n            deque.append(Q, a)\\n\\nfor i in R[1:]:\\n    C[i] = min(C[i], C[P[i]])\\n\\nans = 0\\nfor i in R[1:][::-1]:\\n    if Y[i] * Y[P[i]] < 0:\\n        ans += C[P[i]] * min(abs(Y[i]), abs(Y[P[i]]))\\n    Y[P[i]] += Y[i]\\n\\nprint(ans * 2)\\n\", \"import sys\\ninput = sys.stdin.readline\\nn = int(input())\\ncbc = [list(map(int,input().split())) for i in range(n)]\\nab = [list(map(int,input().split())) for i in range(n-1)]\\ngraph = [[] for i in range(n+1)]\\nif n == 1:\\n  if cbc[0][1] != cbc[0][2]:\\n    print(-1)\\n  else:\\n    print(0)\\n  return\\ndeg = [0]*(n+1)\\nfor a,b in ab:\\n  graph[a].append(b)\\n  graph[b].append(a)\\n  deg[a] += 1\\n  deg[b] += 1\\ndeg[1] += 1\\nstack = [1]\\npar = [0]*(n+1)\\npar[1] = -1\\nleaf = []\\nwhile stack:\\n  x = stack.pop()\\n  if x != 1 and len(graph[x]) == 1:\\n    leaf.append(x)\\n  for y in graph[x]:\\n    if par[y]:\\n      continue\\n    par[y] = x\\n    cbc[y-1][0] = min(cbc[y-1][0],cbc[x-1][0])\\n    stack.append(y)\\ndp = [[0,0] for i in range(n+1)]\\nans = 0\\nwhile leaf:\\n  x = leaf.pop()\\n  p = par[x]\\n  if cbc[x-1][1] != cbc[x-1][2]:\\n    if cbc[x-1][1] == 1:\\n      dp[x][0] += 1\\n    else:\\n      dp[x][1] += 1\\n  if min(dp[x][0],dp[x][1]):\\n    if dp[x][0] > dp[x][1]:\\n      dp[x][0] -= dp[x][1]\\n      ans += cbc[x-1][0]*dp[x][1]*2\\n      dp[x][1] = 0\\n    else:\\n      dp[x][1] -= dp[x][0]\\n      ans += cbc[x-1][0]*dp[x][0]*2\\n      dp[x][0] = 0\\n  dp[p][0] += dp[x][0]\\n  dp[p][1] += dp[x][1]\\n  deg[p] -= 1\\n  if deg[p] == 1:\\n    leaf.append(p)\\nif dp[1][0] != dp[1][1]:\\n  print(-1)\\nelse:\\n  print(ans)\", \"#!usr/bin/env python3\\nfrom collections import defaultdict, deque\\nfrom heapq import heappush, heappop\\nfrom itertools import permutations, accumulate\\nimport sys\\nimport math\\nimport bisect\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef LS():return [list(x) for x in sys.stdin.readline().split()]\\ndef S():\\n    res = list(sys.stdin.readline())\\n    if res[-1] == \\\"\\\\n\\\":\\n        return res[:-1]\\n    return res\\ndef IR(n):\\n    return [I() for i in range(n)]\\ndef LIR(n):\\n    return [LI() for i in range(n)]\\ndef SR(n):\\n    return [S() for i in range(n)]\\ndef LSR(n):\\n    return [LS() for i in range(n)]\\n\\nsys.setrecursionlimit(10000000)\\nmod = 1000000007\\ndef solve():\\n    n = I()\\n    a = []\\n    b = []\\n    c = []\\n    for _ in range(n):\\n        x,y,z = LI()\\n        a.append(x)\\n        b.append(y)\\n        c.append(z)\\n    v = [[] for i in range(n)]\\n    for _ in range(n-1):\\n        x,y = LI()\\n        x -= 1\\n        y -= 1\\n        v[x].append(y)\\n        v[y].append(x)\\n    if b.count(1) != c.count(1):\\n        print(-1)\\n        return\\n    q = deque([0])\\n    q2 = deque()\\n    d = [1]*n\\n    d[0] = 0\\n    ans = 0\\n    p = [[0]*2 for i in range(n)]\\n    while q:\\n        x = q.popleft()\\n        ax = a[x]\\n        if b[x] != c[x]:\\n            p[x][b[x]] = 1\\n        for y in v[x]:\\n            if d[y]:\\n                d[y] = 0\\n                if ax < a[y]:\\n                    a[y] = ax\\n                q.append(y)\\n                q2.append((x,y))\\n    while q2:\\n        x,y = q2.pop()\\n        p[x][0] += p[y][0]\\n        p[x][1] += p[y][1]\\n        m = min(p[x])\\n        ans += m*a[x]\\n        p[x][0] -= m\\n        p[x][1] -= m\\n    print(ans*2)\\n    return\\n\\n#Solve\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "3\n57 1 0\n6 0 1\n76 1 1\n1 3\n3 2\n",
        "output": "114",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1363/E"
    },
    {
        "id": 643,
        "task_id": 1421,
        "test_case_id": 4,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n3 2 -4 2 0 3 3 3 3 4\n10 8\n4 2\n4 9\n3 5\n5 2\n7 4\n2 6\n1 8\n10 9\n",
        "output": "6",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 644,
        "task_id": 1421,
        "test_case_id": 6,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n4 -4 2 5 -1 3 -1 1 4 5\n1 8\n7 1\n4 1\n9 6\n1 2\n5 10\n10 1\n9 3\n1 9\n",
        "output": "14",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 645,
        "task_id": 1421,
        "test_case_id": 7,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n29 -2 39 1 98 98 82 56 5 -2\n3 1\n7 9\n8 9\n7 3\n4 2\n5 10\n6 8\n10 6\n5 4\n",
        "output": "Impossible",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 646,
        "task_id": 1421,
        "test_case_id": 8,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-3 0 1 -3 2 1 2 5 3 1\n6 2\n10 3\n10 5\n2 7\n3 4\n8 2\n8 10\n4 9\n1 9\n",
        "output": "10",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 647,
        "task_id": 1421,
        "test_case_id": 9,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-5 0 1 -2 2 1 2 1 -1 -3\n10 4\n10 5\n4 1\n3 5\n2 8\n6 7\n9 7\n8 7\n6 3\n",
        "output": "0",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 648,
        "task_id": 1421,
        "test_case_id": 10,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n2 -4 5 4 -1 -5 -1 -5 3 -1\n8 6\n8 7\n10 2\n6 3\n5 2\n3 9\n10 1\n5 4\n9 4\n",
        "output": "Impossible",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 649,
        "task_id": 1421,
        "test_case_id": 11,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-1 2 -5 -5 3 1 -4 0 3 -5\n2 10\n2 6\n8 4\n9 2\n10 7\n1 7\n9 5\n8 3\n9 4\n",
        "output": "4",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 650,
        "task_id": 1421,
        "test_case_id": 13,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-4 -1 -5 -1 -3 -1 -2 -2 -5 -3\n7 9\n3 6\n1 4\n7 2\n3 10\n8 9\n4 10\n3 5\n6 8\n",
        "output": "-4",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 651,
        "task_id": 1421,
        "test_case_id": 14,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-21 -17 -16 -45 -93 -77 -73 -12 -81 -33\n9 5\n8 6\n1 10\n9 4\n3 2\n10 4\n3 8\n7 2\n5 6\n",
        "output": "Impossible",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 652,
        "task_id": 1421,
        "test_case_id": 15,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-5 -2 -4 -1 -4 -5 -1 -4 -1 -3\n8 6\n7 2\n1 2\n10 4\n9 3\n6 10\n7 9\n5 4\n5 3\n",
        "output": "Impossible",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 653,
        "task_id": 1421,
        "test_case_id": 16,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n-1000000000 -1000000000 -1000000000 -1000000000 -1000000000 -1000000000 -1000000000 -1000000000 -1000000000 -1000000000\n7 8\n5 10\n8 6\n1 5\n7 9\n3 9\n2 10\n2 6\n4 3\n",
        "output": "Impossible",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 654,
        "task_id": 1421,
        "test_case_id": 17,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "10\n1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000\n7 8\n8 3\n3 10\n2 10\n2 4\n9 6\n7 5\n6 5\n9 1\n",
        "output": "Impossible",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 655,
        "task_id": 1421,
        "test_case_id": 18,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "3\n1 -1000000000 -1000000000\n1 2\n1 3\n",
        "output": "-2000000000",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 656,
        "task_id": 1665,
        "test_case_id": 1,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "3\n1 2\n1 3\n",
        "output": "0\n1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 657,
        "task_id": 1665,
        "test_case_id": 2,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "6\n1 2\n1 3\n2 4\n2 5\n5 6\n",
        "output": "0\n3\n1\n2\n4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 658,
        "task_id": 1665,
        "test_case_id": 3,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "7\n3 2\n5 7\n4 2\n7 6\n6 3\n1 6\n",
        "output": "3\n4\n5\n0\n1\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 659,
        "task_id": 1665,
        "test_case_id": 4,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "7\n5 6\n2 6\n6 4\n6 1\n6 3\n6 7\n",
        "output": "0\n1\n2\n3\n4\n5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 660,
        "task_id": 1665,
        "test_case_id": 5,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "7\n2 1\n4 7\n2 3\n6 4\n7 3\n1 5\n",
        "output": "2\n0\n3\n4\n1\n5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 661,
        "task_id": 1665,
        "test_case_id": 6,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "7\n6 4\n3 7\n3 5\n1 3\n4 2\n7 4\n",
        "output": "0\n3\n4\n5\n1\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 662,
        "task_id": 1665,
        "test_case_id": 7,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "4\n2 3\n2 4\n2 1\n",
        "output": "0\n1\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 663,
        "task_id": 1665,
        "test_case_id": 8,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "5\n1 3\n4 3\n2 5\n3 2\n",
        "output": "0\n1\n3\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 664,
        "task_id": 1665,
        "test_case_id": 9,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "2\n1 2\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 665,
        "task_id": 1665,
        "test_case_id": 10,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "2\n2 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 666,
        "task_id": 1665,
        "test_case_id": 11,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "6\n1 2\n2 3\n3 4\n3 5\n3 6\n",
        "output": "4\n0\n1\n2\n3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 667,
        "task_id": 1665,
        "test_case_id": 12,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "13\n1 4\n2 4\n3 4\n4 5\n5 11\n11 12\n12 13\n13 6\n6 7\n7 8\n7 9\n7 10\n",
        "output": "4\n5\n6\n7\n8\n9\n10\n11\n0\n1\n2\n3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 668,
        "task_id": 1665,
        "test_case_id": 13,
        "question": "You are given a tree consisting of $n$ nodes. You want to write some labels on the tree's edges such that the following conditions hold:\n\n  Every label is an integer between $0$ and $n-2$ inclusive.  All the written labels are distinct.  The largest value among $MEX(u,v)$ over all pairs of nodes $(u,v)$ is as small as possible. \n\nHere, $MEX(u,v)$ denotes the smallest non-negative integer that isn't written on any edge on the unique simple path from node $u$ to node $v$.\n\n\n-----Input-----\n\nThe first line contains the integer $n$ ($2 \\le n \\le 10^5$) — the number of nodes in the tree.\n\nEach of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \\le u,v \\le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.\n\n\n-----Output-----\n\nOutput $n-1$ integers. The $i^{th}$ of them will be the number written on the $i^{th}$ edge (in the input order).\n\n\n-----Examples-----\nInput\n3\n1 2\n1 3\n\nOutput\n0\n1\n\nInput\n6\n1 2\n1 3\n2 4\n2 5\n5 6\n\nOutput\n0\n3\n2\n4\n1\n\n\n-----Note-----\n\nThe tree from the second sample:\n\n[Image]",
        "solutions": "[\"#!/usr/bin/env python\\n#-*- coding:utf-8 -*-\\n\\n# Code by H~$~C\\n\\nfrom sys import stdin\\ninput = stdin.readline\\nimport math\\n\\nn = int(input())\\nG = [[] for i in range(n + 1)]\\nfor i in range(n - 1):\\n  u, v = map(int, input().split())\\n  G[u].append([v, i])\\n  G[v].append([u, i])\\n\\nans = [-1] * (n + 1)\\n\\nfor u in range(1, n + 1):\\n  if (len(G[u]) >= 3):\\n    for j in range(len(G[u])):\\n      ans[G[u][j][1]] = j\\n    cnt = len(G[u])\\n    for j in range(n - 1):\\n      if (ans[j] == -1):\\n        ans[j] = cnt\\n        cnt += 1\\n    for j in range(n - 1):\\n      print(ans[j])\\n    return\\n\\nfor i in range(n - 1):\\n  print(i)\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\nD=dict()\\nDEG=[0]*(n+1)\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n    D[x,y]=i\\n    D[y,x]=i\\n    DEG[x]+=1\\n    DEG[y]+=1\\n\\nANS=[-1]*(n-1)\\n\\nLIST=[]\\nfor i in range(n+1):\\n    if DEG[i]==1:\\n        LIST.append(i)\\n\\nif len(LIST)==2:\\n    for i in range(n-1):\\n        print(i)\\n\\nelse:\\n    now=0\\n    for l in LIST:\\n        k=E[l][0]\\n        ANS[D[k,l]]=now\\n        now+=1\\n\\n    #print(ANS)\\n\\n    for i in range(n-1):\\n        if ANS[i]==-1:\\n            ANS[i]=now\\n            now+=1\\n\\n    for a in ANS:\\n        print(a)\\n\\n\\n\", \"from bisect import bisect_left as bl, bisect_right as br, insort\\nimport sys\\nimport heapq\\nfrom math import *\\nfrom collections import defaultdict as dd, deque\\ndef data(): return sys.stdin.readline().strip()\\ndef mdata(): return map(int, data().split())\\n#sys.setrecursionlimit(100000)\\n\\nn=int(input())\\nt=[dd(int) for i in range(n)]\\nl=[]\\nfor i in range(n-1):\\n    u,v=mdata()\\n    t[u-1][v-1]=0\\n    t[v-1][u-1]=0\\n    l.append([u-1,v-1])\\ns=0\\ne=n-2\\nfor i in range(n-1):\\n    a,b=l[i]\\n    if len(t[a])==1 or len(t[b])==1:\\n        print(s)\\n        s+=1\\n    else:\\n        print(e)\\n        e-=1\", \"import sys\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor i in range(n-1):\\n\\tu, v = map(int, input().split())\\n\\tadj[u-1].append((v-1, i))\\n\\tadj[v-1].append((u-1, i))\\n\\nif max([len(x) for x in adj]) <= 2:\\n\\tprint(*list(range(n-1)), sep=\\\"\\\\n\\\")\\nelse:\\n\\tans = [-1 for _ in range(n-1)]\\n\\tfor i in range(n):\\n\\t\\tif len(adj[i]) > 2:\\n\\t\\t\\tans[adj[i][0][1]] = 0\\n\\t\\t\\tans[adj[i][1][1]] = 1\\n\\t\\t\\tans[adj[i][2][1]] = 2\\n\\t\\t\\tbreak\\n\\ttmp = 3\\n\\tfor i in range(n-1):\\n\\t\\tif ans[i] < 0:\\n\\t\\t\\tans[i] = tmp\\n\\t\\t\\ttmp += 1\\n\\tprint(*ans, sep=\\\"\\\\n\\\")\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\ntree = [[] for i in range(n)]\\nmemo = {}\\nfor i in range(n - 1):\\n    a, b = info[i]\\n    a -= 1\\n    b -= 1\\n    if a > b:\\n        a, b = b, a\\n    tree[a].append(b)\\n    tree[b].append(a)\\n    memo[a * 1000000 + b] = i\\nans = [-1] * (n - 1)\\npos = 0\\nfor i in range(n):\\n    if len(tree[i]) == 1:\\n        a, b = i, tree[i][0]\\n        if a > b:\\n            a, b = b, a\\n        ans[memo[a * 1000000 + b]] = pos\\n        pos += 1\\nfor i in range(n - 1):\\n    if ans[i] == -1:\\n        ans[i] = pos\\n        pos += 1\\nif n == 2:\\n    print(0)\\n    return\\nfor i in ans:\\n    print(i)\", \"import sys\\ninput=sys.stdin.readline\\nn=int(input())\\nL=[[]for i in range(n)]\\nM=dict()\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    M[i]=(a,b)\\n    L[a-1].append(b)\\n    L[b-1].append(a)\\nmxnode=-1\\nmxdeg=0\\nfor j in range(n):\\n    if len(L[j])>mxdeg:\\n        mxdeg=len(L[j])\\n        mxnode=j+1\\nmxpt=0\\nnormpt=0\\nmxedge=list(range(mxdeg))\\nnoredg=list(range(mxdeg,n))\\nfor j in range(n-1):\\n    if mxnode in M[j]:\\n        print(mxedge[mxpt])\\n        mxpt+=1\\n    else:\\n        print(noredg[normpt])\\n        normpt+=1\"]",
        "difficulty": "interview",
        "input": "11\n2 3\n3 4\n4 7\n7 8\n1 3\n3 5\n4 10\n4 11\n6 7\n7 9\n",
        "output": "4\n5\n0\n1\n6\n7\n8\n9\n2\n3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1325/C"
    },
    {
        "id": 669,
        "task_id": 2002,
        "test_case_id": 1,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n",
        "output": "42\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 670,
        "task_id": 2002,
        "test_case_id": 2,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n",
        "output": "30\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 671,
        "task_id": 2002,
        "test_case_id": 3,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "2\n0 0\n2 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 672,
        "task_id": 2002,
        "test_case_id": 4,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "2\n123456789234 987654321432\n1 2\n",
        "output": "111102907\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 673,
        "task_id": 2002,
        "test_case_id": 5,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "2\n987987987987 987987987987\n2 1\n",
        "output": "963943220\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 674,
        "task_id": 2002,
        "test_case_id": 6,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "32\n402528994560 0 0 0 0 0 0 932646223872 893192888700 0 813583026900 0 0 0 0 143521875000 0 177570054144 186624000000 0 517655600000 202145625000 341007975000 0 116252718750 0 148561875000 0 304819200000 248474688000 0 103125000000\n29 25\n20 24\n8 21\n23 3\n32 14\n29 30\n31 24\n28 12\n7 10\n18 1\n11 7\n29 5\n6 8\n8 12\n2 1\n2 15\n26 15\n11 13\n16 12\n12 1\n31 28\n9 11\n21 30\n27 13\n23 1\n17 16\n32 12\n18 22\n1 11\n8 19\n11 4\n",
        "output": "662903569\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 675,
        "task_id": 2002,
        "test_case_id": 7,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "4\n6 10 15 0\n1 4\n2 4\n3 4\n",
        "output": "67\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 676,
        "task_id": 2002,
        "test_case_id": 8,
        "question": "Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached $100$ million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him?\n\nYou're given a tree — a connected undirected graph consisting of $n$ vertices connected by $n - 1$ edges. The tree is rooted at vertex $1$. A vertex $u$ is called an ancestor of $v$ if it lies on the shortest path between the root and $v$. In particular, a vertex is an ancestor of itself.\n\nEach vertex $v$ is assigned its beauty $x_v$ — a non-negative integer not larger than $10^{12}$. This allows us to define the beauty of a path. Let $u$ be an ancestor of $v$. Then we define the beauty $f(u, v)$ as the greatest common divisor of the beauties of all vertices on the shortest path between $u$ and $v$. Formally, if $u=t_1, t_2, t_3, \\dots, t_k=v$ are the vertices on the shortest path between $u$ and $v$, then $f(u, v) = \\gcd(x_{t_1}, x_{t_2}, \\dots, x_{t_k})$. Here, $\\gcd$ denotes the greatest common divisor of a set of numbers. In particular, $f(u, u) = \\gcd(x_u) = x_u$.\n\nYour task is to find the sum\n\n$$ \\sum_{u\\text{ is an ancestor of }v} f(u, v). $$\n\nAs the result might be too large, please output it modulo $10^9 + 7$.\n\nNote that for each $y$, $\\gcd(0, y) = \\gcd(y, 0) = y$. In particular, $\\gcd(0, 0) = 0$.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($2 \\le n \\le 100\\,000$) — the number of vertices in the tree.\n\nThe following line contains $n$ integers $x_1, x_2, \\dots, x_n$ ($0 \\le x_i \\le 10^{12}$). The value $x_v$ denotes the beauty of vertex $v$.\n\nThe following $n - 1$ lines describe the edges of the tree. Each of them contains two integers $a, b$ ($1 \\le a, b \\le n$, $a \\neq b$) — the vertices connected by a single edge.\n\n\n-----Output-----\n\nOutput the sum of the beauties on all paths $(u, v)$ such that $u$ is ancestor of $v$. This sum should be printed modulo $10^9 + 7$.\n\n\n-----Examples-----\nInput\n5\n4 5 6 0 8\n1 2\n1 3\n1 4\n4 5\n\nOutput\n42\n\nInput\n7\n0 2 3 0 0 0 0\n1 2\n1 3\n2 4\n2 5\n3 6\n3 7\n\nOutput\n30\n\n\n\n-----Note-----\n\nThe following figure shows all $10$ possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to $42$: [Image]",
        "solutions": "[\"n = int(input())\\nbeauty = list(map(int, input().strip().split()))\\ntree = [[] for i in range(n)]\\nmod = 1000000007\\nused = [False for i in range(n)]\\ndef gcd(a,b):\\n    mn = min(a,b)\\n    mx = max(a,b)\\n    if mn == 0:\\n        return mx\\n    md = mx%mn\\n    if md == 0:\\n        return mn\\n    else:\\n        return gcd(mn, md)\\nfor i in range(n-1):\\n    a,b = map(int, input().strip().split())\\n    tree[a-1].append(b-1)\\n    tree[b-1].append(a-1)\\nsegment_vals = [{} for i in range(n)]\\nans = beauty[0]\\nsegment_vals[0][beauty[0]] = 1\\ncur_nodes = [0]\\nused[0] = True\\nwhile 1:\\n    new_nodes = []\\n    for node in cur_nodes:\\n        for potential_new in tree[node]:\\n            if used[potential_new] == False:\\n                used[potential_new] = True\\n                new_nodes.append(potential_new)\\n                new_beauty = beauty[potential_new]\\n                segment_vals[potential_new][new_beauty] = 1\\n                for g in segment_vals[node].keys():\\n                    segment_gcd = gcd(new_beauty,g)\\n                    segment_vals[potential_new][segment_gcd] = segment_vals[potential_new].get(segment_gcd,0) + segment_vals[node][g]\\n                for k in segment_vals[potential_new].keys():\\n                    ans += k*segment_vals[potential_new][k]\\n                    ans = ans % mod\\n    if len(new_nodes) == 0:\\n        break\\n    else:\\n        cur_nodes = new_nodes\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "8\n1000000000000 0 0 1000000000000 0 0 999999999999 1000000000000\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n7 8\n",
        "output": "999867015\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1210/C"
    },
    {
        "id": 677,
        "task_id": 2050,
        "test_case_id": 1,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "1 1\n",
        "output": "5\n1 3 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 678,
        "task_id": 2050,
        "test_case_id": 2,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "2 2\n",
        "output": "22\n2 6 8 10\n14 18 20 22\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 679,
        "task_id": 2050,
        "test_case_id": 3,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "7 7\n",
        "output": "287\n7 21 28 35\n49 63 70 77\n91 105 112 119\n133 147 154 161\n175 189 196 203\n217 231 238 245\n259 273 280 287\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 680,
        "task_id": 2050,
        "test_case_id": 4,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "13 7\n",
        "output": "539\n7 21 28 35\n49 63 70 77\n91 105 112 119\n133 147 154 161\n175 189 196 203\n217 231 238 245\n259 273 280 287\n301 315 322 329\n343 357 364 371\n385 399 406 413\n427 441 448 455\n469 483 490 497\n511 525 532 539\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 681,
        "task_id": 2050,
        "test_case_id": 5,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "15 27\n",
        "output": "2403\n27 81 108 135\n189 243 270 297\n351 405 432 459\n513 567 594 621\n675 729 756 783\n837 891 918 945\n999 1053 1080 1107\n1161 1215 1242 1269\n1323 1377 1404 1431\n1485 1539 1566 1593\n1647 1701 1728 1755\n1809 1863 1890 1917\n1971 2025 2052 2079\n2133 2187 2214 2241\n2295 2349 2376 2403\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 682,
        "task_id": 2050,
        "test_case_id": 6,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "19 21\n",
        "output": "2373\n21 63 84 105\n147 189 210 231\n273 315 336 357\n399 441 462 483\n525 567 588 609\n651 693 714 735\n777 819 840 861\n903 945 966 987\n1029 1071 1092 1113\n1155 1197 1218 1239\n1281 1323 1344 1365\n1407 1449 1470 1491\n1533 1575 1596 1617\n1659 1701 1722 1743\n1785 1827 1848 1869\n1911 1953 1974 1995\n2037 2079 2100 2121\n2163 2205 2226 2247\n2289 2331 2352 2373\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 683,
        "task_id": 2050,
        "test_case_id": 7,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "1 100\n",
        "output": "500\n100 300 400 500\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 684,
        "task_id": 2050,
        "test_case_id": 8,
        "question": "Dreamoon likes to play with sets, integers and $gcd$. $\\operatorname{gcd}(a, b)$ is defined as the largest positive integer that divides both a and b.\n\nLet S be a set of exactly four distinct integers greater than 0. Define S to be of rank k if and only if for all pairs of distinct elements s_{i}, s_{j} from S, $\\operatorname{gcd}(s_{i}, s_{j}) = k$.\n\nGiven k and n, Dreamoon wants to make up n sets of rank k using integers from 1 to m such that no integer is used in two different sets (of course you can leave some integers without use). Calculate the minimum m that makes it possible and print one possible solution.\n\n\n-----Input-----\n\nThe single line of the input contains two space separated integers n, k (1 ≤ n ≤ 10 000, 1 ≤ k ≤ 100).\n\n\n-----Output-----\n\nOn the first line print a single integer — the minimal possible m. \n\nOn each of the next n lines print four space separated integers representing the i-th set.\n\nNeither the order of the sets nor the order of integers within a set is important. If there are multiple possible solutions with minimal m, print any one of them.\n\n\n-----Examples-----\nInput\n1 1\n\nOutput\n5\n1 2 3 5\n\nInput\n2 2\n\nOutput\n22\n2 4 6 22\n14 18 10 16\n\n\n\n-----Note-----\n\nFor the first example it's easy to see that set {1, 2, 3, 4} isn't a valid set of rank 1 since $\\operatorname{gcd}(2,4) = 2 \\neq 1$.",
        "solutions": "[\"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"k = 0\\ni = 1\\na = []\\nwhile k <10000:\\n\\tb = []\\n\\tb.append(i)\\n\\tb.append(i+1)\\n\\tb.append(i+2)\\n\\tb.append(i+4)\\n\\ti += 6\\n\\ta.append(b)\\n\\tk += 1\\nn, k = tuple(map(int, input().split()))\\nprint(a[n-1][3]*k)\\nfor i in range(0, n):\\n\\tprint(a[i][0]*k, a[i][1]*k, a[i][2]*k, a[i][3]*k)\", \"'''\\nCreated on Oct 12, 2014\\n\\n@author: Ismael\\n'''\\n#import time\\n\\nfrom fractions import gcd\\n\\ndef checkSet(setK,newVal):\\n    for v in setK:\\n        if(gcd(v,newVal) != 1):\\n            return False\\n    return True\\n\\ndef solve(n,k):\\n    j = 1\\n    sets = []\\n    for _ in range(n):\\n        setK = set()\\n        while(len(setK) < 4):\\n            if(checkSet(setK,j) and not(len(setK) == 0 and j%3==0)):\\n                setK.add(j)\\n            j += 1\\n        sets.append(setK)\\n    maxV = 0\\n    for setK in sets:\\n        maxV = max(maxV,max(setK))\\n    print(maxV*k)\\n    for setK in sets:\\n        print(' '.join([str(x*k) for x in setK]))\\n    \\n#t = time.clock()\\nn,k = list(map(int,input().split()))\\nsolve(n,k)\\n#print(time.clock()-t)\\n\", \"load = [int(i) for i in input().split()]\\nn = load[0]\\nk = load[1]\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    buff = \\\"\\\"\\n    buff += str(k * (6 * i - 5)) + ' '\\n    buff += str(k * (6 * i - 3)) + ' '\\n    buff += str(k * (6 * i - 2)) + ' '\\n    buff += str(k * (6 * i - 1)) + ' '\\n    print(buff)\\n\", \"a, b = list(map(int, input().split(' ')))\\nprint((6*a-1)*b)\\nfor i in range(a):\\n    print((6*i+1)*b, (6*i+2)*b, (6*i+3)*b, (6*i+5)*b)\\n\", \"#input\\nn,k=map(int,input().split())\\n\\n#variables\\n\\n#main\\nprint((6*n-1)*k)\\n\\nfor i in range(n):\\n\\tprint(str((6*i+1)*k)+' '+str((6*i+2)*k)+' '+str((6*i+3)*k)+' '+str((6*i+5)*k))\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n\\tprint(k * m, end = ' ')\\n\\tfor j in range(3):\\n\\t\\tprint(k * now, end = ' ')\\n\\t\\tnow += 2\\n\\tm += 6\\n\\tprint()\", \"n, k = list(map(int, input().split()))\\n\\nprint((6*n-1)*k)\\n\\nprint((\\\"\\\\n\\\".join(\\n   \\\"%i %i %i %i\\\" % tuple(k*(6*i+j) for j in (1,2,3,5)) for i in range(n)\\n)))\\n\", \"n, k = map(int, input().split())\\nprint(k * (2 * (3 * n) - 1))\\nnow = 1\\nm = 2\\nfor i in range(n):\\n    print(k * m, end = ' ')\\n    for j in range(3):\\n        print(k * now, end = ' ')\\n        now += 2\\n    m += 6\\n    print()\", \"n, k = list(map(int, input().split()))\\nprint((n * 6 - 1) * k)\\nfor i in range(n):\\n    m = i * 6\\n    print((m + 1) * k, (m + 2) * k, (m + 3) * k, (m + 5) * k)    \\n\", \"n, k = map(int, input().split())\\nprint((6 * n - 1) * k)\\nfor i in range(1, 6 * n, 6): print(i * k, (i + 2) * k, (i + 3) * k, (i + 4) * k)\", \"n, k = map(int, input().split())\\nm = 6 * n - 1\\nprint(m * k)\\nfor i in range(n):\\n    for j in [1, 2, 3, 5]:\\n        print(k * (6 * i + j), end = ' ')\\n    print()\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"import sys\\n\\ndef solve():\\n    n, k, = rv()\\n    res = list()\\n    cur = 1\\n    for i in range(n):\\n        while cur % 2 == 0: cur+=1\\n        res.append(((cur) * k, (cur + 1) * k, (cur + 2) * k, (cur + 4) * k))\\n        cur += 5\\n    print(res[-1][-1])\\n    print('\\\\n'.join((' '.join(map(str, l))) for l in res))\\n\\n#12 14 16 20\\n# 2 * 2 * 3   2 * 7   2 * 2 * 2 * 2   2 * 3 * 3\\n# 1 2 3 5\\n# 7 8 9 11\\n# 13\\n# 19\\n# 25\\n# 31\\n\\ndef prt(l): return print(' '.join(l))\\ndef rv(): return map(int, input().split())\\ndef rl(n): return [list(map(int, input().split())) for _ in range(n)]\\nif sys.hexversion == 50594544 : sys.stdin = open(\\\"test.txt\\\")\\nsolve()\\n\\n\\n\", \"n, k = list(map(int,input().split()))\\nprint((6*n-1)*k)\\nfor i in range(0, n): print((6*i+1)*k,(6*i+2)*k,(6*i+3)*k,(6*i+5)*k)\", \"n, k = map(int, input().split())\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n): print(k * (6 * i + 1), k * (6 * i + 2), k * (6 * i + 3), k * (6 * i + 5))\", \"l = input().split(\\\" \\\")\\nn = int(l[0])\\nk = int(l[1])\\nprint((6*n-1)*k)\\nfor i in range(n):\\n    print(str((6*i+1)*k)+\\\" \\\"+str((6*i+2)*k)+\\\" \\\"+str((6*i+3)*k)+\\\" \\\"+str((6*i+5)*k))\", \"n, k = list(map(int, input().split()))\\nprint(k * (6 * (n - 1) + 5))\\nfor i in range(0, n):\\n    t = [6 * i + 1, 6 * i + 2, 6 * i + 3, 6 * i + 5]\\n    print((' '.join([str(m * k) for m in t])));\\n\\n\", \"n, k = map(int, input().split())\\nprint(k*(6*n-1))\\nfor i in range (n):\\n    print(k*(6*i+1), k * (6*i+3), k*(6*i+4), k * (6*i+5))\", \"x, y = map(int, input().split())\\nz = []\\nprint((6 * x - 1) * y)\\nfor i in range(0, x):\\n    z.append((6 * i + 1) * y); z.append((6 * i + 3) * y); z.append((6 * i + 4) * y); z.append((6 * i + 5) * y)\\n    print(*z)\\n    z.clear()\", \"inp = input().split(' ')\\ndef result(sets,maximum_divisor):\\n    output = list()\\n    output.append(str(int(maximum_divisor) * (6 * int(sets) - 1)))\\n    for i in range(int(sets)):\\n        output.append(str(int(maximum_divisor)* (6 * int(i) + 1)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 3)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 4)) + ' ' + str(int(maximum_divisor) * (6 * int(i) + 5)))\\n    return output\\nfor i in result(inp[0],inp[1]):\\n    print(i)\\n\", \"n,k = [int(i) for i in input().split()]\\nprint((6*n-1)*k)\\nfor a in range(n):\\n    val = 6*a\\n    print((val+1)*k,(val+2)*k,(val+3)*k,(val+5)*k)\\n\", \"n, k = [int(x) for x in input().split()]\\nprint(k*((n-1)*6+5))\\nfor i in range(n):\\n    print(k*(6*i+1),k*(6*i+3),k*(6*i+5),k*(6*i+2))\", \"n, k = map(int, input().split())\\nprint(k * (6 * n - 1))\\nfor i in range(1, n + 1):\\n    x = (i - 1) * 6 + 1\\n    print(k * x, end = ' ')\\n    print(k * (x + 1), end = ' ')\\n    print(k * (x + 2), end = ' ')\\n    print(k * (x + 4))\"]",
        "difficulty": "interview",
        "input": "1 10\n",
        "output": "50\n10 30 40 50\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/476/D"
    },
    {
        "id": 685,
        "task_id": 2263,
        "test_case_id": 1,
        "question": "New Year is coming in Tree World! In this world, as the name implies, there are n cities connected by n - 1 roads, and for any two distinct cities there always exists a path between them. The cities are numbered by integers from 1 to n, and the roads are numbered by integers from 1 to n - 1. Let's define d(u, v) as total length of roads on the path between city u and city v.\n\nAs an annual event, people in Tree World repairs exactly one road per year. As a result, the length of one road decreases. It is already known that in the i-th year, the length of the r_{i}-th road is going to become w_{i}, which is shorter than its length before. Assume that the current year is year 1.\n\nThree Santas are planning to give presents annually to all the children in Tree World. In order to do that, they need some preparation, so they are going to choose three distinct cities c_1, c_2, c_3 and make exactly one warehouse in each city. The k-th (1 ≤ k ≤ 3) Santa will take charge of the warehouse in city c_{k}.\n\nIt is really boring for the three Santas to keep a warehouse alone. So, they decided to build an only-for-Santa network! The cost needed to build this network equals to d(c_1, c_2) + d(c_2, c_3) + d(c_3, c_1) dollars. Santas are too busy to find the best place, so they decided to choose c_1, c_2, c_3 randomly uniformly over all triples of distinct numbers from 1 to n. Santas would like to know the expected value of the cost needed to build the network.\n\nHowever, as mentioned, each year, the length of exactly one road decreases. So, the Santas want to calculate the expected after each length change. Help them to calculate the value.\n\n\n-----Input-----\n\nThe first line contains an integer n (3 ≤ n ≤ 10^5) — the number of cities in Tree World.\n\nNext n - 1 lines describe the roads. The i-th line of them (1 ≤ i ≤ n - 1) contains three space-separated integers a_{i}, b_{i}, l_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}, 1 ≤ l_{i} ≤ 10^3), denoting that the i-th road connects cities a_{i} and b_{i}, and the length of i-th road is l_{i}.\n\nThe next line contains an integer q (1 ≤ q ≤ 10^5) — the number of road length changes.\n\nNext q lines describe the length changes. The j-th line of them (1 ≤ j ≤ q) contains two space-separated integers r_{j}, w_{j} (1 ≤ r_{j} ≤ n - 1, 1 ≤ w_{j} ≤ 10^3). It means that in the j-th repair, the length of the r_{j}-th road becomes w_{j}. It is guaranteed that w_{j} is smaller than the current length of the r_{j}-th road. The same road can be repaired several times.\n\n\n-----Output-----\n\nOutput q numbers. For each given change, print a line containing the expected cost needed to build the network in Tree World. The answer will be considered correct if its absolute and relative error doesn't exceed 10^{ - 6}.\n\n\n-----Examples-----\nInput\n3\n2 3 5\n1 3 3\n5\n1 4\n2 2\n1 2\n2 1\n1 1\n\nOutput\n14.0000000000\n12.0000000000\n8.0000000000\n6.0000000000\n4.0000000000\n\nInput\n6\n1 5 3\n5 3 2\n6 1 7\n1 4 4\n5 2 3\n5\n1 2\n2 1\n3 5\n4 1\n5 2\n\nOutput\n19.6000000000\n18.6000000000\n16.6000000000\n13.6000000000\n12.6000000000\n\n\n\n-----Note-----\n\nConsider the first sample. There are 6 triples: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). Because n = 3, the cost needed to build the network is always d(1, 2) + d(2, 3) + d(3, 1) for all the triples. So, the expected cost equals to d(1, 2) + d(2, 3) + d(3, 1).",
        "solutions": "[\"\\nfrom queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return list(map(int, input().split(' ')))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n\\tu, v, c = readarray()\\n\\tu, v = u - 1, v - 1\\n\\tcost.append(c)\\n\\tgraph[u].append((v, i))\\n\\tgraph[v].append((u, i))\\n\\t\\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n\\tv = q[qt]\\n\\tqt += 1\\n\\t\\n\\torder.append(v)\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif used[to]:\\n\\t\\t\\tcontinue\\n\\t\\tused[to] = 1\\n\\t\\tq[qh] = to\\n\\t\\tqh += 1\\n\\t\\t\\n\\n\\t\\t\\norder.reverse()\\n\\t\\t\\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n\\tsz[v] = 1\\n\\tfor (to, e) in graph[v]:\\n\\t\\tsz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n\\tsz[v] = 1\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif to != p:\\n\\t\\t\\tdfs(to, v)\\n\\t\\t\\tsz[v] += sz[to]\\n\\t\\t\\t\\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n\\tfor (to, e) in graph[v]:\\n\\t\\tx = min(sz[v], sz[to])\\n\\t\\tedgeMult[e] = x\\n\\t\\tdistanceSum += 1.0 * cost[e] * x * (n - x)\\n\\t\\t\\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n\\tx, y = readarray()\\n\\tx -= 1\\n\\t\\n\\tdistanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\tcost[x] = y\\n\\tdistanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\t\\n\\tans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\\n\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n#coo\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"import sys\\nsys.setrecursionlimit(1500)\\n\\nMAX = 100005;\\ng = [[] for _ in range(MAX)]\\nvis = [False] * MAX\\ndp = [0] * MAX\\nprod = [0] * MAX\\n\\nedges = []\\norder = []\\ndef dfs(st):\\n    stack = []\\n    stack.append((st, -1))\\n    vis[st] = True\\n    while stack:\\n        st, parent = stack.pop()\\n        order.append(st)\\n        vis[st] = True\\n        if (st == parent): continue;\\n        for i in g[st]:\\n            if (vis[i[0]]): continue;\\n            stack.append((i[0], st))\\n\\nn = int(input())\\nfor i in range(n-1):\\n    a, b, w = list(map(int, sys.stdin.readline().split(' ')))\\n    g[a].append([b, w])\\n    g[b].append([a,w])\\n    edges.append([[a, b], w])\\n\\ndfs(1);\\norder.reverse()\\nfor st in order:\\n    dp[st] = 1;\\n    for i in g[st]:\\n        dp[st] += dp[i[0]];\\ntot = 0;\\ncurr = 1;\\ndiv = n * (n-1) / 2;\\nfor i in edges:\\n    a = i[0][0];\\n    b = i[0][1];\\n    sa = dp[a];\\n    sb = dp[b];\\n    tot += min(sa, sb) * (n - min(sa, sb)) * i[1];\\n    prod[curr] = min(sa, sb) * (n - min(sa, sb));\\n    curr += 1;\\n\\nq = int(input())\\nfor i in range(q):\\n    q1, q2 = list(map(int, sys.stdin.readline().split(' ')))\\n    tot -= prod[q1] * edges[q1-1][1];\\n    edges[q1-1][1] = q2;\\n    tot += prod[q1] * edges[q1-1][1];\\n    sys.stdout.write(str(tot*3/div)+\\\"\\\\n\\\")\\n\"]",
        "difficulty": "interview",
        "input": "3\n2 3 5\n1 3 3\n5\n1 4\n2 2\n1 2\n2 1\n1 1\n",
        "output": "14.0000000000\n12.0000000000\n8.0000000000\n6.0000000000\n4.0000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/500/D"
    },
    {
        "id": 686,
        "task_id": 2263,
        "test_case_id": 2,
        "question": "New Year is coming in Tree World! In this world, as the name implies, there are n cities connected by n - 1 roads, and for any two distinct cities there always exists a path between them. The cities are numbered by integers from 1 to n, and the roads are numbered by integers from 1 to n - 1. Let's define d(u, v) as total length of roads on the path between city u and city v.\n\nAs an annual event, people in Tree World repairs exactly one road per year. As a result, the length of one road decreases. It is already known that in the i-th year, the length of the r_{i}-th road is going to become w_{i}, which is shorter than its length before. Assume that the current year is year 1.\n\nThree Santas are planning to give presents annually to all the children in Tree World. In order to do that, they need some preparation, so they are going to choose three distinct cities c_1, c_2, c_3 and make exactly one warehouse in each city. The k-th (1 ≤ k ≤ 3) Santa will take charge of the warehouse in city c_{k}.\n\nIt is really boring for the three Santas to keep a warehouse alone. So, they decided to build an only-for-Santa network! The cost needed to build this network equals to d(c_1, c_2) + d(c_2, c_3) + d(c_3, c_1) dollars. Santas are too busy to find the best place, so they decided to choose c_1, c_2, c_3 randomly uniformly over all triples of distinct numbers from 1 to n. Santas would like to know the expected value of the cost needed to build the network.\n\nHowever, as mentioned, each year, the length of exactly one road decreases. So, the Santas want to calculate the expected after each length change. Help them to calculate the value.\n\n\n-----Input-----\n\nThe first line contains an integer n (3 ≤ n ≤ 10^5) — the number of cities in Tree World.\n\nNext n - 1 lines describe the roads. The i-th line of them (1 ≤ i ≤ n - 1) contains three space-separated integers a_{i}, b_{i}, l_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}, 1 ≤ l_{i} ≤ 10^3), denoting that the i-th road connects cities a_{i} and b_{i}, and the length of i-th road is l_{i}.\n\nThe next line contains an integer q (1 ≤ q ≤ 10^5) — the number of road length changes.\n\nNext q lines describe the length changes. The j-th line of them (1 ≤ j ≤ q) contains two space-separated integers r_{j}, w_{j} (1 ≤ r_{j} ≤ n - 1, 1 ≤ w_{j} ≤ 10^3). It means that in the j-th repair, the length of the r_{j}-th road becomes w_{j}. It is guaranteed that w_{j} is smaller than the current length of the r_{j}-th road. The same road can be repaired several times.\n\n\n-----Output-----\n\nOutput q numbers. For each given change, print a line containing the expected cost needed to build the network in Tree World. The answer will be considered correct if its absolute and relative error doesn't exceed 10^{ - 6}.\n\n\n-----Examples-----\nInput\n3\n2 3 5\n1 3 3\n5\n1 4\n2 2\n1 2\n2 1\n1 1\n\nOutput\n14.0000000000\n12.0000000000\n8.0000000000\n6.0000000000\n4.0000000000\n\nInput\n6\n1 5 3\n5 3 2\n6 1 7\n1 4 4\n5 2 3\n5\n1 2\n2 1\n3 5\n4 1\n5 2\n\nOutput\n19.6000000000\n18.6000000000\n16.6000000000\n13.6000000000\n12.6000000000\n\n\n\n-----Note-----\n\nConsider the first sample. There are 6 triples: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). Because n = 3, the cost needed to build the network is always d(1, 2) + d(2, 3) + d(3, 1) for all the triples. So, the expected cost equals to d(1, 2) + d(2, 3) + d(3, 1).",
        "solutions": "[\"\\nfrom queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return list(map(int, input().split(' ')))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n\\tu, v, c = readarray()\\n\\tu, v = u - 1, v - 1\\n\\tcost.append(c)\\n\\tgraph[u].append((v, i))\\n\\tgraph[v].append((u, i))\\n\\t\\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n\\tv = q[qt]\\n\\tqt += 1\\n\\t\\n\\torder.append(v)\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif used[to]:\\n\\t\\t\\tcontinue\\n\\t\\tused[to] = 1\\n\\t\\tq[qh] = to\\n\\t\\tqh += 1\\n\\t\\t\\n\\n\\t\\t\\norder.reverse()\\n\\t\\t\\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n\\tsz[v] = 1\\n\\tfor (to, e) in graph[v]:\\n\\t\\tsz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n\\tsz[v] = 1\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif to != p:\\n\\t\\t\\tdfs(to, v)\\n\\t\\t\\tsz[v] += sz[to]\\n\\t\\t\\t\\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n\\tfor (to, e) in graph[v]:\\n\\t\\tx = min(sz[v], sz[to])\\n\\t\\tedgeMult[e] = x\\n\\t\\tdistanceSum += 1.0 * cost[e] * x * (n - x)\\n\\t\\t\\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n\\tx, y = readarray()\\n\\tx -= 1\\n\\t\\n\\tdistanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\tcost[x] = y\\n\\tdistanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\t\\n\\tans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\\n\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n#coo\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"import sys\\nsys.setrecursionlimit(1500)\\n\\nMAX = 100005;\\ng = [[] for _ in range(MAX)]\\nvis = [False] * MAX\\ndp = [0] * MAX\\nprod = [0] * MAX\\n\\nedges = []\\norder = []\\ndef dfs(st):\\n    stack = []\\n    stack.append((st, -1))\\n    vis[st] = True\\n    while stack:\\n        st, parent = stack.pop()\\n        order.append(st)\\n        vis[st] = True\\n        if (st == parent): continue;\\n        for i in g[st]:\\n            if (vis[i[0]]): continue;\\n            stack.append((i[0], st))\\n\\nn = int(input())\\nfor i in range(n-1):\\n    a, b, w = list(map(int, sys.stdin.readline().split(' ')))\\n    g[a].append([b, w])\\n    g[b].append([a,w])\\n    edges.append([[a, b], w])\\n\\ndfs(1);\\norder.reverse()\\nfor st in order:\\n    dp[st] = 1;\\n    for i in g[st]:\\n        dp[st] += dp[i[0]];\\ntot = 0;\\ncurr = 1;\\ndiv = n * (n-1) / 2;\\nfor i in edges:\\n    a = i[0][0];\\n    b = i[0][1];\\n    sa = dp[a];\\n    sb = dp[b];\\n    tot += min(sa, sb) * (n - min(sa, sb)) * i[1];\\n    prod[curr] = min(sa, sb) * (n - min(sa, sb));\\n    curr += 1;\\n\\nq = int(input())\\nfor i in range(q):\\n    q1, q2 = list(map(int, sys.stdin.readline().split(' ')))\\n    tot -= prod[q1] * edges[q1-1][1];\\n    edges[q1-1][1] = q2;\\n    tot += prod[q1] * edges[q1-1][1];\\n    sys.stdout.write(str(tot*3/div)+\\\"\\\\n\\\")\\n\"]",
        "difficulty": "interview",
        "input": "6\n1 5 3\n5 3 2\n6 1 7\n1 4 4\n5 2 3\n5\n1 2\n2 1\n3 5\n4 1\n5 2\n",
        "output": "19.6000000000\n18.6000000000\n16.6000000000\n13.6000000000\n12.6000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/500/D"
    },
    {
        "id": 687,
        "task_id": 2263,
        "test_case_id": 3,
        "question": "New Year is coming in Tree World! In this world, as the name implies, there are n cities connected by n - 1 roads, and for any two distinct cities there always exists a path between them. The cities are numbered by integers from 1 to n, and the roads are numbered by integers from 1 to n - 1. Let's define d(u, v) as total length of roads on the path between city u and city v.\n\nAs an annual event, people in Tree World repairs exactly one road per year. As a result, the length of one road decreases. It is already known that in the i-th year, the length of the r_{i}-th road is going to become w_{i}, which is shorter than its length before. Assume that the current year is year 1.\n\nThree Santas are planning to give presents annually to all the children in Tree World. In order to do that, they need some preparation, so they are going to choose three distinct cities c_1, c_2, c_3 and make exactly one warehouse in each city. The k-th (1 ≤ k ≤ 3) Santa will take charge of the warehouse in city c_{k}.\n\nIt is really boring for the three Santas to keep a warehouse alone. So, they decided to build an only-for-Santa network! The cost needed to build this network equals to d(c_1, c_2) + d(c_2, c_3) + d(c_3, c_1) dollars. Santas are too busy to find the best place, so they decided to choose c_1, c_2, c_3 randomly uniformly over all triples of distinct numbers from 1 to n. Santas would like to know the expected value of the cost needed to build the network.\n\nHowever, as mentioned, each year, the length of exactly one road decreases. So, the Santas want to calculate the expected after each length change. Help them to calculate the value.\n\n\n-----Input-----\n\nThe first line contains an integer n (3 ≤ n ≤ 10^5) — the number of cities in Tree World.\n\nNext n - 1 lines describe the roads. The i-th line of them (1 ≤ i ≤ n - 1) contains three space-separated integers a_{i}, b_{i}, l_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}, 1 ≤ l_{i} ≤ 10^3), denoting that the i-th road connects cities a_{i} and b_{i}, and the length of i-th road is l_{i}.\n\nThe next line contains an integer q (1 ≤ q ≤ 10^5) — the number of road length changes.\n\nNext q lines describe the length changes. The j-th line of them (1 ≤ j ≤ q) contains two space-separated integers r_{j}, w_{j} (1 ≤ r_{j} ≤ n - 1, 1 ≤ w_{j} ≤ 10^3). It means that in the j-th repair, the length of the r_{j}-th road becomes w_{j}. It is guaranteed that w_{j} is smaller than the current length of the r_{j}-th road. The same road can be repaired several times.\n\n\n-----Output-----\n\nOutput q numbers. For each given change, print a line containing the expected cost needed to build the network in Tree World. The answer will be considered correct if its absolute and relative error doesn't exceed 10^{ - 6}.\n\n\n-----Examples-----\nInput\n3\n2 3 5\n1 3 3\n5\n1 4\n2 2\n1 2\n2 1\n1 1\n\nOutput\n14.0000000000\n12.0000000000\n8.0000000000\n6.0000000000\n4.0000000000\n\nInput\n6\n1 5 3\n5 3 2\n6 1 7\n1 4 4\n5 2 3\n5\n1 2\n2 1\n3 5\n4 1\n5 2\n\nOutput\n19.6000000000\n18.6000000000\n16.6000000000\n13.6000000000\n12.6000000000\n\n\n\n-----Note-----\n\nConsider the first sample. There are 6 triples: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). Because n = 3, the cost needed to build the network is always d(1, 2) + d(2, 3) + d(3, 1) for all the triples. So, the expected cost equals to d(1, 2) + d(2, 3) + d(3, 1).",
        "solutions": "[\"\\nfrom queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return list(map(int, input().split(' ')))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n\\tu, v, c = readarray()\\n\\tu, v = u - 1, v - 1\\n\\tcost.append(c)\\n\\tgraph[u].append((v, i))\\n\\tgraph[v].append((u, i))\\n\\t\\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n\\tv = q[qt]\\n\\tqt += 1\\n\\t\\n\\torder.append(v)\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif used[to]:\\n\\t\\t\\tcontinue\\n\\t\\tused[to] = 1\\n\\t\\tq[qh] = to\\n\\t\\tqh += 1\\n\\t\\t\\n\\n\\t\\t\\norder.reverse()\\n\\t\\t\\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n\\tsz[v] = 1\\n\\tfor (to, e) in graph[v]:\\n\\t\\tsz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n\\tsz[v] = 1\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif to != p:\\n\\t\\t\\tdfs(to, v)\\n\\t\\t\\tsz[v] += sz[to]\\n\\t\\t\\t\\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n\\tfor (to, e) in graph[v]:\\n\\t\\tx = min(sz[v], sz[to])\\n\\t\\tedgeMult[e] = x\\n\\t\\tdistanceSum += 1.0 * cost[e] * x * (n - x)\\n\\t\\t\\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n\\tx, y = readarray()\\n\\tx -= 1\\n\\t\\n\\tdistanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\tcost[x] = y\\n\\tdistanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\t\\n\\tans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\\n\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n#coo\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"import sys\\nsys.setrecursionlimit(1500)\\n\\nMAX = 100005;\\ng = [[] for _ in range(MAX)]\\nvis = [False] * MAX\\ndp = [0] * MAX\\nprod = [0] * MAX\\n\\nedges = []\\norder = []\\ndef dfs(st):\\n    stack = []\\n    stack.append((st, -1))\\n    vis[st] = True\\n    while stack:\\n        st, parent = stack.pop()\\n        order.append(st)\\n        vis[st] = True\\n        if (st == parent): continue;\\n        for i in g[st]:\\n            if (vis[i[0]]): continue;\\n            stack.append((i[0], st))\\n\\nn = int(input())\\nfor i in range(n-1):\\n    a, b, w = list(map(int, sys.stdin.readline().split(' ')))\\n    g[a].append([b, w])\\n    g[b].append([a,w])\\n    edges.append([[a, b], w])\\n\\ndfs(1);\\norder.reverse()\\nfor st in order:\\n    dp[st] = 1;\\n    for i in g[st]:\\n        dp[st] += dp[i[0]];\\ntot = 0;\\ncurr = 1;\\ndiv = n * (n-1) / 2;\\nfor i in edges:\\n    a = i[0][0];\\n    b = i[0][1];\\n    sa = dp[a];\\n    sb = dp[b];\\n    tot += min(sa, sb) * (n - min(sa, sb)) * i[1];\\n    prod[curr] = min(sa, sb) * (n - min(sa, sb));\\n    curr += 1;\\n\\nq = int(input())\\nfor i in range(q):\\n    q1, q2 = list(map(int, sys.stdin.readline().split(' ')))\\n    tot -= prod[q1] * edges[q1-1][1];\\n    edges[q1-1][1] = q2;\\n    tot += prod[q1] * edges[q1-1][1];\\n    sys.stdout.write(str(tot*3/div)+\\\"\\\\n\\\")\\n\"]",
        "difficulty": "interview",
        "input": "4\n1 2 66\n1 3 565\n3 4 469\n4\n2 186\n3 226\n2 143\n1 54\n",
        "output": "1174.5000000000\n810.0000000000\n724.0000000000\n706.0000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/500/D"
    },
    {
        "id": 688,
        "task_id": 2263,
        "test_case_id": 4,
        "question": "New Year is coming in Tree World! In this world, as the name implies, there are n cities connected by n - 1 roads, and for any two distinct cities there always exists a path between them. The cities are numbered by integers from 1 to n, and the roads are numbered by integers from 1 to n - 1. Let's define d(u, v) as total length of roads on the path between city u and city v.\n\nAs an annual event, people in Tree World repairs exactly one road per year. As a result, the length of one road decreases. It is already known that in the i-th year, the length of the r_{i}-th road is going to become w_{i}, which is shorter than its length before. Assume that the current year is year 1.\n\nThree Santas are planning to give presents annually to all the children in Tree World. In order to do that, they need some preparation, so they are going to choose three distinct cities c_1, c_2, c_3 and make exactly one warehouse in each city. The k-th (1 ≤ k ≤ 3) Santa will take charge of the warehouse in city c_{k}.\n\nIt is really boring for the three Santas to keep a warehouse alone. So, they decided to build an only-for-Santa network! The cost needed to build this network equals to d(c_1, c_2) + d(c_2, c_3) + d(c_3, c_1) dollars. Santas are too busy to find the best place, so they decided to choose c_1, c_2, c_3 randomly uniformly over all triples of distinct numbers from 1 to n. Santas would like to know the expected value of the cost needed to build the network.\n\nHowever, as mentioned, each year, the length of exactly one road decreases. So, the Santas want to calculate the expected after each length change. Help them to calculate the value.\n\n\n-----Input-----\n\nThe first line contains an integer n (3 ≤ n ≤ 10^5) — the number of cities in Tree World.\n\nNext n - 1 lines describe the roads. The i-th line of them (1 ≤ i ≤ n - 1) contains three space-separated integers a_{i}, b_{i}, l_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}, 1 ≤ l_{i} ≤ 10^3), denoting that the i-th road connects cities a_{i} and b_{i}, and the length of i-th road is l_{i}.\n\nThe next line contains an integer q (1 ≤ q ≤ 10^5) — the number of road length changes.\n\nNext q lines describe the length changes. The j-th line of them (1 ≤ j ≤ q) contains two space-separated integers r_{j}, w_{j} (1 ≤ r_{j} ≤ n - 1, 1 ≤ w_{j} ≤ 10^3). It means that in the j-th repair, the length of the r_{j}-th road becomes w_{j}. It is guaranteed that w_{j} is smaller than the current length of the r_{j}-th road. The same road can be repaired several times.\n\n\n-----Output-----\n\nOutput q numbers. For each given change, print a line containing the expected cost needed to build the network in Tree World. The answer will be considered correct if its absolute and relative error doesn't exceed 10^{ - 6}.\n\n\n-----Examples-----\nInput\n3\n2 3 5\n1 3 3\n5\n1 4\n2 2\n1 2\n2 1\n1 1\n\nOutput\n14.0000000000\n12.0000000000\n8.0000000000\n6.0000000000\n4.0000000000\n\nInput\n6\n1 5 3\n5 3 2\n6 1 7\n1 4 4\n5 2 3\n5\n1 2\n2 1\n3 5\n4 1\n5 2\n\nOutput\n19.6000000000\n18.6000000000\n16.6000000000\n13.6000000000\n12.6000000000\n\n\n\n-----Note-----\n\nConsider the first sample. There are 6 triples: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). Because n = 3, the cost needed to build the network is always d(1, 2) + d(2, 3) + d(3, 1) for all the triples. So, the expected cost equals to d(1, 2) + d(2, 3) + d(3, 1).",
        "solutions": "[\"\\nfrom queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return list(map(int, input().split(' ')))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n\\tu, v, c = readarray()\\n\\tu, v = u - 1, v - 1\\n\\tcost.append(c)\\n\\tgraph[u].append((v, i))\\n\\tgraph[v].append((u, i))\\n\\t\\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n\\tv = q[qt]\\n\\tqt += 1\\n\\t\\n\\torder.append(v)\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif used[to]:\\n\\t\\t\\tcontinue\\n\\t\\tused[to] = 1\\n\\t\\tq[qh] = to\\n\\t\\tqh += 1\\n\\t\\t\\n\\n\\t\\t\\norder.reverse()\\n\\t\\t\\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n\\tsz[v] = 1\\n\\tfor (to, e) in graph[v]:\\n\\t\\tsz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n\\tsz[v] = 1\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif to != p:\\n\\t\\t\\tdfs(to, v)\\n\\t\\t\\tsz[v] += sz[to]\\n\\t\\t\\t\\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n\\tfor (to, e) in graph[v]:\\n\\t\\tx = min(sz[v], sz[to])\\n\\t\\tedgeMult[e] = x\\n\\t\\tdistanceSum += 1.0 * cost[e] * x * (n - x)\\n\\t\\t\\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n\\tx, y = readarray()\\n\\tx -= 1\\n\\t\\n\\tdistanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\tcost[x] = y\\n\\tdistanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\t\\n\\tans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\\n\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n#coo\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"import sys\\nsys.setrecursionlimit(1500)\\n\\nMAX = 100005;\\ng = [[] for _ in range(MAX)]\\nvis = [False] * MAX\\ndp = [0] * MAX\\nprod = [0] * MAX\\n\\nedges = []\\norder = []\\ndef dfs(st):\\n    stack = []\\n    stack.append((st, -1))\\n    vis[st] = True\\n    while stack:\\n        st, parent = stack.pop()\\n        order.append(st)\\n        vis[st] = True\\n        if (st == parent): continue;\\n        for i in g[st]:\\n            if (vis[i[0]]): continue;\\n            stack.append((i[0], st))\\n\\nn = int(input())\\nfor i in range(n-1):\\n    a, b, w = list(map(int, sys.stdin.readline().split(' ')))\\n    g[a].append([b, w])\\n    g[b].append([a,w])\\n    edges.append([[a, b], w])\\n\\ndfs(1);\\norder.reverse()\\nfor st in order:\\n    dp[st] = 1;\\n    for i in g[st]:\\n        dp[st] += dp[i[0]];\\ntot = 0;\\ncurr = 1;\\ndiv = n * (n-1) / 2;\\nfor i in edges:\\n    a = i[0][0];\\n    b = i[0][1];\\n    sa = dp[a];\\n    sb = dp[b];\\n    tot += min(sa, sb) * (n - min(sa, sb)) * i[1];\\n    prod[curr] = min(sa, sb) * (n - min(sa, sb));\\n    curr += 1;\\n\\nq = int(input())\\nfor i in range(q):\\n    q1, q2 = list(map(int, sys.stdin.readline().split(' ')))\\n    tot -= prod[q1] * edges[q1-1][1];\\n    edges[q1-1][1] = q2;\\n    tot += prod[q1] * edges[q1-1][1];\\n    sys.stdout.write(str(tot*3/div)+\\\"\\\\n\\\")\\n\"]",
        "difficulty": "interview",
        "input": "10\n1 2 938\n1 6 23\n1 10 727\n1 9 814\n9 5 644\n8 5 39\n2 7 999\n1 3 666\n5 4 577\n10\n9 556\n5 64\n9 318\n3 373\n4 548\n7 219\n5 43\n5 27\n3 155\n8 359\n",
        "output": "5010.5333333333\n4198.5333333333\n4055.7333333333\n3843.3333333333\n3417.7333333333\n2949.7333333333\n2920.3333333333\n2897.9333333333\n2767.1333333333\n2582.9333333333\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/500/D"
    },
    {
        "id": 689,
        "task_id": 2263,
        "test_case_id": 5,
        "question": "New Year is coming in Tree World! In this world, as the name implies, there are n cities connected by n - 1 roads, and for any two distinct cities there always exists a path between them. The cities are numbered by integers from 1 to n, and the roads are numbered by integers from 1 to n - 1. Let's define d(u, v) as total length of roads on the path between city u and city v.\n\nAs an annual event, people in Tree World repairs exactly one road per year. As a result, the length of one road decreases. It is already known that in the i-th year, the length of the r_{i}-th road is going to become w_{i}, which is shorter than its length before. Assume that the current year is year 1.\n\nThree Santas are planning to give presents annually to all the children in Tree World. In order to do that, they need some preparation, so they are going to choose three distinct cities c_1, c_2, c_3 and make exactly one warehouse in each city. The k-th (1 ≤ k ≤ 3) Santa will take charge of the warehouse in city c_{k}.\n\nIt is really boring for the three Santas to keep a warehouse alone. So, they decided to build an only-for-Santa network! The cost needed to build this network equals to d(c_1, c_2) + d(c_2, c_3) + d(c_3, c_1) dollars. Santas are too busy to find the best place, so they decided to choose c_1, c_2, c_3 randomly uniformly over all triples of distinct numbers from 1 to n. Santas would like to know the expected value of the cost needed to build the network.\n\nHowever, as mentioned, each year, the length of exactly one road decreases. So, the Santas want to calculate the expected after each length change. Help them to calculate the value.\n\n\n-----Input-----\n\nThe first line contains an integer n (3 ≤ n ≤ 10^5) — the number of cities in Tree World.\n\nNext n - 1 lines describe the roads. The i-th line of them (1 ≤ i ≤ n - 1) contains three space-separated integers a_{i}, b_{i}, l_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}, 1 ≤ l_{i} ≤ 10^3), denoting that the i-th road connects cities a_{i} and b_{i}, and the length of i-th road is l_{i}.\n\nThe next line contains an integer q (1 ≤ q ≤ 10^5) — the number of road length changes.\n\nNext q lines describe the length changes. The j-th line of them (1 ≤ j ≤ q) contains two space-separated integers r_{j}, w_{j} (1 ≤ r_{j} ≤ n - 1, 1 ≤ w_{j} ≤ 10^3). It means that in the j-th repair, the length of the r_{j}-th road becomes w_{j}. It is guaranteed that w_{j} is smaller than the current length of the r_{j}-th road. The same road can be repaired several times.\n\n\n-----Output-----\n\nOutput q numbers. For each given change, print a line containing the expected cost needed to build the network in Tree World. The answer will be considered correct if its absolute and relative error doesn't exceed 10^{ - 6}.\n\n\n-----Examples-----\nInput\n3\n2 3 5\n1 3 3\n5\n1 4\n2 2\n1 2\n2 1\n1 1\n\nOutput\n14.0000000000\n12.0000000000\n8.0000000000\n6.0000000000\n4.0000000000\n\nInput\n6\n1 5 3\n5 3 2\n6 1 7\n1 4 4\n5 2 3\n5\n1 2\n2 1\n3 5\n4 1\n5 2\n\nOutput\n19.6000000000\n18.6000000000\n16.6000000000\n13.6000000000\n12.6000000000\n\n\n\n-----Note-----\n\nConsider the first sample. There are 6 triples: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). Because n = 3, the cost needed to build the network is always d(1, 2) + d(2, 3) + d(3, 1) for all the triples. So, the expected cost equals to d(1, 2) + d(2, 3) + d(3, 1).",
        "solutions": "[\"\\nfrom queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return list(map(int, input().split(' ')))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n\\tu, v, c = readarray()\\n\\tu, v = u - 1, v - 1\\n\\tcost.append(c)\\n\\tgraph[u].append((v, i))\\n\\tgraph[v].append((u, i))\\n\\t\\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n\\tv = q[qt]\\n\\tqt += 1\\n\\t\\n\\torder.append(v)\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif used[to]:\\n\\t\\t\\tcontinue\\n\\t\\tused[to] = 1\\n\\t\\tq[qh] = to\\n\\t\\tqh += 1\\n\\t\\t\\n\\n\\t\\t\\norder.reverse()\\n\\t\\t\\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n\\tsz[v] = 1\\n\\tfor (to, e) in graph[v]:\\n\\t\\tsz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n\\tsz[v] = 1\\n\\t\\n\\tfor (to, e) in graph[v]:\\n\\t\\tif to != p:\\n\\t\\t\\tdfs(to, v)\\n\\t\\t\\tsz[v] += sz[to]\\n\\t\\t\\t\\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n\\tfor (to, e) in graph[v]:\\n\\t\\tx = min(sz[v], sz[to])\\n\\t\\tedgeMult[e] = x\\n\\t\\tdistanceSum += 1.0 * cost[e] * x * (n - x)\\n\\t\\t\\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n\\tx, y = readarray()\\n\\tx -= 1\\n\\t\\n\\tdistanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\tcost[x] = y\\n\\tdistanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n\\t\\n\\tans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\\n\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"from queue import Queue\\nimport sys\\n\\ncost = []\\n#coo\\ndef readarray(): return map(int, input().split(' '))\\n\\nn = int(input())\\ngraph = [[] for i in range(n)]\\n\\nfor i in range(n - 1):\\n    u, v, c = readarray()\\n    u, v = u - 1, v - 1\\n    cost.append(c)\\n    graph[u].append((v, i))\\n    graph[v].append((u, i))\\n    \\n\\norder = []\\nused = [0] * n\\nq = [0] * (n + n)\\n\\nqh = qt = 0\\n\\n\\nused[qh] = 1\\nqh += 1\\n\\nwhile qt < qh:\\n    v = q[qt]\\n    qt += 1\\n    \\n    order.append(v)\\n    \\n    for (to, e) in graph[v]:\\n        if used[to]:\\n            continue\\n        used[to] = 1\\n        q[qh] = to\\n        qh += 1\\n        \\n\\n        \\norder.reverse()\\n        \\nsz = [0 for x in range(n)]\\n\\nfor v in order:\\n    sz[v] = 1\\n    for (to, e) in graph[v]:\\n        sz[v] += sz[to]\\n\\\"\\\"\\\"\\n\\nsz = [0] * n\\n\\nsys.setrecursionlimit(100505)\\n\\ndef dfs(v, p):\\n    sz[v] = 1\\n    \\n    for (to, e) in graph[v]:\\n        if to != p:\\n            dfs(to, v)\\n            sz[v] += sz[to]\\n            \\ndfs(0, -1)\\n\\n\\\"\\\"\\\"\\n\\ndistanceSum = 0.0\\nedgeMult = [0] * n\\n\\nfor v in range(n):\\n    for (to, e) in graph[v]:\\n        x = min(sz[v], sz[to])\\n        edgeMult[e] = x\\n        distanceSum += 1.0 * cost[e] * x * (n - x)\\n        \\ndistanceSum /= 2.0\\n\\nqueryCnt = int(input())\\n\\nans = []\\n\\nfor i in range(queryCnt):\\n    x, y = readarray()\\n    x -= 1\\n    \\n    distanceSum -= 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    cost[x] = y\\n    distanceSum += 1.0 * cost[x] * edgeMult[x] * (n - edgeMult[x])\\n    \\n    ans.append('%.10lf' % (distanceSum / n / (n - 1) * 6.0))\\n\\nprint('\\\\n'.join(ans))\", \"import sys\\nsys.setrecursionlimit(1500)\\n\\nMAX = 100005;\\ng = [[] for _ in range(MAX)]\\nvis = [False] * MAX\\ndp = [0] * MAX\\nprod = [0] * MAX\\n\\nedges = []\\norder = []\\ndef dfs(st):\\n    stack = []\\n    stack.append((st, -1))\\n    vis[st] = True\\n    while stack:\\n        st, parent = stack.pop()\\n        order.append(st)\\n        vis[st] = True\\n        if (st == parent): continue;\\n        for i in g[st]:\\n            if (vis[i[0]]): continue;\\n            stack.append((i[0], st))\\n\\nn = int(input())\\nfor i in range(n-1):\\n    a, b, w = list(map(int, sys.stdin.readline().split(' ')))\\n    g[a].append([b, w])\\n    g[b].append([a,w])\\n    edges.append([[a, b], w])\\n\\ndfs(1);\\norder.reverse()\\nfor st in order:\\n    dp[st] = 1;\\n    for i in g[st]:\\n        dp[st] += dp[i[0]];\\ntot = 0;\\ncurr = 1;\\ndiv = n * (n-1) / 2;\\nfor i in edges:\\n    a = i[0][0];\\n    b = i[0][1];\\n    sa = dp[a];\\n    sb = dp[b];\\n    tot += min(sa, sb) * (n - min(sa, sb)) * i[1];\\n    prod[curr] = min(sa, sb) * (n - min(sa, sb));\\n    curr += 1;\\n\\nq = int(input())\\nfor i in range(q):\\n    q1, q2 = list(map(int, sys.stdin.readline().split(' ')))\\n    tot -= prod[q1] * edges[q1-1][1];\\n    edges[q1-1][1] = q2;\\n    tot += prod[q1] * edges[q1-1][1];\\n    sys.stdout.write(str(tot*3/div)+\\\"\\\\n\\\")\\n\"]",
        "difficulty": "interview",
        "input": "3\n1 3 925\n2 1 778\n3\n2 482\n2 206\n1 512\n",
        "output": "2814.0000000000\n2262.0000000000\n1436.0000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/500/D"
    },
    {
        "id": 690,
        "task_id": 2305,
        "test_case_id": 1,
        "question": "We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.\nAdditionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.\nFor each k=1, 2, ..., N, solve the following problem:\n - Find the number of simple paths that visit a vertex painted in the color k one or more times.\nNote: The simple paths from Vertex u to v and from v to u are not distinguished.\n\n-----Constraints-----\n - 1 \\leq N \\leq 2 \\times 10^5\n - 1 \\leq c_i \\leq N\n - 1 \\leq a_i,b_i \\leq N\n - The given graph is a tree.\n - All values in input are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN\nc_1 c_2 ... c_N\na_1 b_1\n:\na_{N-1} b_{N-1}\n\n-----Output-----\nPrint the answers for k = 1, 2, ..., N in order, each in its own line.\n\n-----Sample Input-----\n3\n1 2 1\n1 2\n2 3\n\n-----Sample Output-----\n5\n4\n0\n\nLet P_{i,j} denote the simple path connecting Vertex i and j.\nThere are 5 simple paths that visit a vertex painted in the color 1 one or more times:\nP_{1,1}\\,,\\,P_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,3}\\,,\\,P_{3,3} \nThere are 4 simple paths that visit a vertex painted in the color 2 one or more times:\nP_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,2}\\,,\\,P_{2,3} \nThere are no simple paths that visit a vertex painted in the color 3 one or more times.",
        "solutions": "[\"import sys\\nfrom collections import defaultdict\\n\\n# \\u518d\\u5e30\\u5236\\u9650\\u3092\\u7de9\\u548c\\u3059\\u308b\\u304a\\u307e\\u3058\\u306a\\u3044\\nsys.setrecursionlimit(10**6)\\n\\ndef sum(n):return n*(n+1)//2\\n\\n# \\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3068\\u3001\\u300c\\u8272i\\u3092\\u5c01\\u9396\\u3057\\u305f\\u3068\\u304d\\u306b\\u5230\\u9054\\u3067\\u304d\\u306a\\u3044\\u9802\\u70b9\\u306e\\u500b\\u6570\\u300d\\u3092\\u6301\\u3064\\u8f9e\\u66f8\\u3092\\u8fd4\\u3059\\ndef dfs(v,p):\\n    ret=defaultdict(int)\\n    size=1\\n    for vv in g[v]:\\n        if vv==p:\\n            continue\\n        ss,d=dfs(vv,v)\\n        size+=ss\\n        ans[c[v]]+=sum(ss-d[c[v]])\\n        \\n        # \\u30de\\u30fc\\u30b8\\u30c6\\u30af\\n        if len(ret)<len(d):\\n            ret,d=d,ret\\n        for vvv in d:\\n            ret[vvv]+=d[vvv]\\n    ret[c[v]]=size\\n    return size,ret\\n\\nn=int(input())\\nc=list(map(int,input().split()))\\ng=[[] for _ in range(n+1)]\\nans=[0]*(n+1)\\nfor _ in range(n-1):\\n    s,t=list(map(int, input().split()))\\n    s-=1\\n    t-=1\\n    g[s].append(t)\\n    g[t].append(s)\\n_,ret=dfs(0,-1)\\nfor i in range(1,n+1):\\n    print((sum(n)-ans[i]-sum(n-ret[i])))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nc = list(map(int, input().split()))\\nans = [n*(n+1)//2 for _ in range(n)]\\nfor i in range(n):\\n\\tc[i] -= 1\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n\\ta, b = map(int, input().split())\\n\\tadj[a-1].append(b-1)\\n\\tadj[b-1].append(a-1)\\n\\ns = 0\\ndp = [0 for _ in range(n)]\\n\\ndef dfs(x, p):\\n\\tnonlocal s\\n\\tpres = s + dp[c[x]]\\n\\tfor v in adj[x]:\\n\\t\\tif v == p:\\n\\t\\t\\tcontinue\\n\\t\\tdp[c[x]] = -s\\n\\t\\tdfs(v, x)\\n\\t\\tans[c[x]] -= (s+dp[c[x]]) * (s+dp[c[x]]+1) // 2\\n\\ts += 1\\n\\tdp[c[x]] = pres - s\\n\\treturn\\n\\ndfs(0, -1)\\nfor i in range(n):\\n\\tans[i] -= (s+dp[i]) * (s+dp[i]+1) // 2\\n\\nprint(*ans, sep=\\\"\\\\n\\\")\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append(t)\\n        self.E[t].append(f)\\n\\n\\ndef make_order(g, v):\\n    seen = [False]*g.N\\n    last_order = [-1]*g.N\\n    parent = [-1]*g.N\\n    size = [1]*g.N\\n    counter = {\\\"last\\\": 0}\\n\\n    def recur(v):\\n        seen[v] = True\\n        for to in g.E[v]:\\n            if seen[to] == True:\\n                continue\\n            parent[to] = v\\n            recur(to)\\n            size[v] += size[to]\\n\\n        last_order[counter[\\\"last\\\"]] = v\\n        counter[\\\"last\\\"] += 1\\n\\n    recur(v)\\n    return last_order, parent, size\\n\\n\\ndef merge(d: dict, e: dict):\\n    if len(d) < len(e):\\n        d, e = e, d\\n    for k in e:\\n        d[k] += e[k]\\n    return d\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\nans = [0]*N\\n\\nret = {}\\nlast_order, parent, size = make_order(g, 0)\\nfor curr in last_order:\\n    cn = c[curr]\\n    rrr = defaultdict(int)\\n    for dest in g.E[curr]:\\n        if dest == parent[curr]:\\n            continue\\n        child = ret.pop(dest)\\n\\n        n = size[dest]-child[cn]\\n        ans[cn] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        rrr = merge(rrr, child)\\n\\n    rrr[cn] = size[curr]\\n    ret[curr] = rrr\\n\\n\\ntot = N*(N+1)//2\\nfor color in range(N):\\n    if color != c[0]:\\n        n = N-ret[0][color]\\n        ans[color] += n*(n+1)//2\\n    print((tot-ans[color]))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\n \\nN = int(input())  # \\u9802\\u70b9\\u6570\\nC = [int(x) - 1  for x in input().split()]  # \\u9802\\u70b9\\u306e\\u8272\\u756a\\u53f7 (1~N) -> (0~N-1)\\n \\nE = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    E[a-1].append(b-1)\\n    E[b-1].append(a-1)\\n    \\ndef dfs(u, p = -1):\\n \\n    color = C[u]  # \\u8272\\u756a\\u53f7\\n    VC[color].append(u)  # \\u8272\\u3054\\u3068\\u306b stack \\u3057\\u3066\\u304a\\u304f\\n    \\n    s = 1  # \\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\uff09\\n    for v in E[u]:\\n        if v == p:\\n            continue\\n            \\n        child[u] = 0  # v \\u306e\\u65b9\\u5411\\u306b\\u3042\\u308b\\u5b50\\u5b6b\\u306e\\u3046\\u3061\\u8fbf\\u308c\\u308b\\u9802\\u70b9\\u306e\\u6570 = s - \\u5b50\\u5b6b\\u306b\\u3042\\u308b\\u3001\\u540c\\u8272\\u3092\\u6839\\u3068\\u3059\\u308b\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u306e\\u5408\\u8a08\\n        ret = dfs(v, u)\\n \\n        s += ret  # \\u7d14\\u7c8b\\u306a\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3092\\u8a08\\u7b97\\n        child[u] += ret  # \\u540c\\u8272\\u306e\\u9802\\u70b9\\u304c\\u5207\\u308a\\u96e2\\u3055\\u308c\\u305f\\u3042\\u3068\\u306e\\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\n        \\n        ans[color] -= child[u] * (child[u] + 1) // 2        \\n \\n    VC[color].pop()\\n    if VC[color]: # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u3042\\u308b\\u5834\\u5408\\n        child[VC[color][-1]] -= s  # \\u4e00\\u756a\\u8fd1\\u3044\\u540c\\u8272\\u304b\\u3089\\u3001\\u81ea\\u5206\\u306e\\u90e8\\u5206\\u6728\\u306e\\u5927\\u304d\\u3055\\u3092\\u3072\\u304f\\n    else:  # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u306a\\u3044\\u5834\\u5408\\n        root_size[color] -= s  # \\u4e0a\\u4f4d\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\n        \\n    return s\\n\\nchild = [0] * N  # \\u8fbf\\u3063\\u3066\\u3044\\u308b\\u5b50\\u306e\\u4e2d\\u3067\\u306e\\u306e\\u6b8b\\u308a\\u6570\\u3000\\u3068\\u308a\\u3042\\u3048\\u305a 0\\u3067\\u521d\\u671f\\u5316\\nroot_size = [N] * N  # \\u6839\\u65b9\\u5411\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\u3002\\u64cd\\u4f5c\\u524d\\u306f N \\u5168\\u90e8\\uff09\\nVC = [ [] for _ in range(N)]  # \\u8272\\u756a\\u53f7\\u3054\\u3068\\nans = [N*(N+1)//2] * N  # \\u8272\\u3054\\u3068\\u306e\\uff08\\u6b8b\\u308a\\u306e\\uff09\\u7d44\\u307f\\u5408\\u308f\\u305b\\u6570\\u3000\\u521d\\u671f\\u5024\\u306f \\u5168\\u70b9\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\n \\ndfs(0)\\n \\nfor i in range(N):\\n    print(ans[i] - root_size[i] * (root_size[i] + 1) // 2)\", \"import sys\\n\\ninput=sys.stdin.readline\\nsys.setrecursionlimit(10**7)\\n\\nN=int(input())\\nc=list(map(int,input().split()))\\nfor i in range(N):\\n    c[i]-=1\\nedge=[[] for i in range(N)]\\nfor i in range(N-1):\\n    a,b=map(int,input().split())\\n    edge[a-1].append(b-1)\\n    edge[b-1].append(a-1)\\n\\nsubtree=[1]*N\\ndef size(v,pv):\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            size(nv,v)\\n            subtree[v]+=subtree[nv]\\n\\nsize(0,-1)\\n\\ndata=[[] for i in range(N)]\\nParent=[[0] for i in range(N)]\\nparent=[0]*N\\ndef dfs(v,pv,nop):\\n    pp=parent[nop]\\n    parent[nop]=v\\n    Parent[nop].append(v)\\n    data[c[v]].append((parent[c[v]],v))\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            dfs(nv,v,c[v])\\n    parent[nop]=pp\\n\\ndfs(0,-1,0)\\n\\nfor i in range(N):\\n    dic={v:subtree[v] for v in Parent[i]}\\n    for p,ch in data[i]:\\n        dic[p]-=subtree[ch]\\n\\n    res=N*(N+1)//2\\n    for p in dic:\\n        res-=dic[p]*(dic[p]+1)//2\\n    print(res)\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nhead = [-1] * (N + 1)\\nto = [0] * (N - 1 << 1)\\nnxt = [0] * (N - 1 << 1)\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    x, y = x-1, y-1\\n    nxt[i] = head[x]\\n    to[i] = y\\n    head[x] = i\\n    j = i + N - 1\\n    nxt[j] = head[y]\\n    to[j] = x\\n    head[y] = j\\n\\ndef EulerTour(n, i=0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    ANS = [f(n)] * n\\n    \\n    P = [-1] * n\\n    ct = 0\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    while i >= 0:\\n        e = head[i]\\n        if e < 0:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = to[e]\\n        if P[i] == j:\\n            head[i] = nxt[e]\\n            continue\\n        \\n        P[j] = i\\n        head[i] = nxt[e]\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    done = [0] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            continue\\n        done[i] = 1\\n        if i: ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n        for a in X[i]:\\n            if done[a]: continue\\n            P[a] = i\\n            Q.append(~a)\\n            Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\nn = int(input())\\nc = list(map(lambda x: int(x) - 1, input().split()))\\n\\ngraph = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    graph[a].append(b)\\n    graph[b].append(a)\\n\\nans = [n * (n + 1) // 2] * n\\nn_subtree = [1] * n  # n_st[i] = (\\u9802\\u70b9i\\u306e\\u90e8\\u5206\\u6728\\u306e\\u9802\\u70b9\\u6570)\\ncnt = [0] * n  # cnt[i] = (\\u8a2a\\u554f\\u6e08\\u307f\\u9802\\u70b9\\u306e\\u3046\\u3061\\u3001\\u8272i\\u3092\\u901a\\u904e\\u3057\\u306a\\u3044\\u3068\\u305f\\u3069\\u308a\\u7740\\u3051\\u306a\\u3044\\u9802\\u70b9\\u6570)\\nn_visited = 0\\n\\n\\ndef dfs(v, v_p):\\n    nonlocal n_visited\\n    cnt_v_before = cnt[c[v]]\\n    for v_next in graph[v]:\\n        if v_next == v_p:\\n            continue\\n\\n        m_prev = n_visited - cnt[c[v]]\\n        dfs(v_next, v)\\n        n_subtree[v] += n_subtree[v_next]\\n        m_next = n_visited - cnt[c[v]]\\n\\n        m = m_next - m_prev\\n        ans[c[v]] -= m * (m + 1) // 2\\n\\n    cnt[c[v]] = cnt_v_before + n_subtree[v]\\n    n_visited += 1\\n\\n\\ndfs(0, -1)\\n\\nfor i in range(n):\\n    m = n - cnt[i]\\n    ans[i] -= m * (m + 1) // 2\\n\\nprint(*ans, sep='\\\\n')\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\n# ci = [set() for _ in range(n)]\\n# for i,c in enumerate(C):\\n#     ci[c-1].add(i)\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = list(map(int,input().split()))\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n\\nans = [n*(n+1)//2]*n\\ncparent = [[] for _ in range(n)]\\nroot_size = [n]*n\\n# index:color\\nreached = [False]*n\\nsize = [0]*n\\n# index:vertex\\n\\ndef dfs(p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        if reached[nxt]:continue\\n        reached[nxt] = True\\n        size[p] = 0\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] += ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    # size[p] += s\\n    if cparent[c]:\\n        size[cparent[c][-1]] -= s\\n    else:\\n        root_size[c] -= s\\n    # ans[c] -= size[p] * (size[p] - 1)//2\\n    return s\\n\\nreached = [False]*n\\nreached[0] = True\\n\\ndfs(0)\\nfor i in range(n):\\n    print((ans[i] - root_size[i]*(root_size[i]+1)//2))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if ind == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if ind >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind]\\n        if P[i] == j:\\n            IND[i] += 1\\n            continue\\n        P[j] = i\\n        IND[i] += 1\\n        i = j\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n    \\nans = [n*(n+1)//2]*n\\nsize = [0]*n\\ncparent = [[] for _ in range(n)]\\nreached = [False]*n\\nroot_size = [n]*n\\n\\ndef dfs (p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        #print (nxt)\\n        \\n        if reached[nxt]:\\n            continue\\n        size[p] = 0\\n        reached[nxt] = True\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] +=ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    if cparent[c]:\\n        size[cparent[c][-1]] -=s\\n    else:\\n        root_size[c] -=s    \\n    return s\\n\\nreached[0] = True\\ndfs (0)\\n\\nfor i in range(n):\\n    print (ans[i]-root_size[i]*(root_size[i]+1)//2)\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef main():\\n    n=int(input())\\n    color=list(map(int,input().split()))\\n    nodes=[[]for i in range(n)]\\n    cut=[[0]for i in range(n+1)]\\n    ans=[0]*(n+1)\\n    for i in range(n-1):\\n        a,b=map(int,input().split())\\n        nodes[a-1].append(b-1)\\n        nodes[b-1].append(a-1)\\n    def num(p,parent):\\n        s=0\\n        for child in nodes[p]:\\n            if child==parent:\\n                continue\\n            cut[color[p]].append(0)\\n            nc=num(child,p)\\n            group=nc-cut[color[p]].pop()\\n            ans[color[p]]+=group*(group+1)//2\\n            s+=nc\\n        s+=1\\n        cut[color[p]][-1]+=s\\n        return s\\n    num(0,-1)\\n    for a,c in zip(ans[1:],cut[1:]):\\n        group=n-c[0]\\n        c=group*(group+1)//2\\n        print(n*(n+1)//2-c-a)\\nmain()\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                p = P[i]\\n                k = ET2[i] - ET1[i] + 1 - USED[C[p]] + ORG[i]\\n                ANS[C[p]] -= f(k)\\n                TMP[p] += k\\n            continue\\n        if i >= 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            if ET1[i] == 0: ET1[i] = ct\\n        for a in X[i][::-1]:\\n            if a != P[i]:\\n                P[a] = i\\n                for k in range(len(X[a])):\\n                    if X[a][k] == i:\\n                        del X[a][k]\\n                        break\\n                Q.append(~a)\\n                Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\ndef solve():\\n    N = int(input())\\n    Cs = [A-1 for A in map(int, input().split())]\\n    adjL = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = list(map(int, input().split()))\\n        a, b = a-1, b-1\\n        adjL[a].append(b)\\n        adjL[b].append(a)\\n\\n    sizeSubtrees = [1] * N\\n    numVs = [0] * N\\n    anss = [0] * N\\n\\n    def dfs(vNow, vPar, clrPar):\\n        clrNow = Cs[vNow]\\n        numVNow = numVs[clrNow]\\n        numVPar = numVs[clrPar]\\n        for v2 in adjL[vNow]:\\n            if v2 == vPar: continue\\n            dfs(v2, vNow, clrNow)\\n            sizeSubtrees[vNow] += sizeSubtrees[v2]\\n        numVs[clrNow] = numVNow + sizeSubtrees[vNow]\\n        if clrPar != -1:\\n            k = sizeSubtrees[vNow] - (numVs[clrPar]-numVPar)\\n            anss[clrPar] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for clr in range(N):\\n        if clr == Cs[0]: continue\\n        k = N - numVs[clr]\\n        anss[clr] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    anss = [numAll-ans for ans in anss]\\n\\n    print(('\\\\n'.join(map(str, anss))))\\n\\n\\nsolve()\\n\", \"import sys\\nimport itertools\\n\\nsys.setrecursionlimit(3 * 10**5)\\n\\n\\nclass Node:\\n  def __init__(self, color):\\n    self.color = int(color) - 1\\n    self.children = []\\n\\n  def add_child(self, node):\\n    self.children.append(node)\\n\\n  @staticmethod\\n  def prepare_nodes(n):\\n    nodes = tuple(map(Node, input().split()))\\n    for _ in range(n - 1):\\n      a, b = [int(x) - 1 for x in input().split()]\\n      nodes[a].add_child(nodes[b])\\n      nodes[b].add_child(nodes[a])\\n\\n    return nodes\\n\\n\\ndef main():\\n  n = int(input())\\n  nodes = Node.prepare_nodes(n)\\n\\n  dp = [0 for _ in range(n)]\\n  ans_list = [n*(n + 1) // 2 for _ in range(n)]\\n\\n  def update_ans(s, color):\\n    ans_list[color] -= (s + dp[color]) * (s + dp[color] + 1) // 2\\n\\n  def dfs(s, node, parent):\\n    color = node.color\\n    pre_s = s + dp[color]\\n    for child in node.children:\\n      if child == parent: continue\\n\\n      dp[color] = -s\\n      s = dfs(s, child, node)\\n      update_ans(s, color)\\n\\n    s += 1\\n    dp[color] = pre_s - s\\n\\n    return s\\n\\n  s = dfs(0, nodes[0], Node(-1))\\n\\n  for i in range(n):\\n    update_ans(s, i)\\n    print((ans_list[i]))\\n\\n\\nmain()\\n\", \"def main():\\n    import sys\\n    sys.setrecursionlimit(10**9)\\n    input = sys.stdin.readline\\n\\n    N = int(input())\\n    C = [c-1 for c in map(int, input().split())]\\n    tree = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = map(int, input().split())\\n        tree[a-1].append(b-1)\\n        tree[b-1].append(a-1)\\n\\n    # 1\\u56de\\u4ee5\\u4e0a > \\u5168\\u4f53 - 1\\u56de\\u3082\\u901a\\u3089\\u306a\\u3044\\n    # \\u9078\\u3079\\u308b\\u6570 = n(n+1)//2\\n    size_subtree = [1] * N\\n    num_v = [0] * N\\n    ans = [0] * N\\n\\n    def dfs(v_now, v_parent, color_parent):\\n        color_now = C[v_now]\\n        num_v_now = num_v[color_now]\\n        num_v_parent = num_v[color_parent]\\n        for v2 in tree[v_now]:\\n            if v2 == v_parent: continue\\n            dfs(v2, v_now, color_now)\\n            size_subtree[v_now] += size_subtree[v2]\\n        num_v[color_now] = num_v_now + size_subtree[v_now]\\n        if color_parent != -1:\\n            k = size_subtree[v_now] - (num_v[color_parent]-num_v_parent)\\n            ans[color_parent] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for color in range(N):\\n        if color == C[0]: continue\\n        k = N - num_v[color]\\n        ans[color] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    ans = [numAll-ans for ans in ans]\\n\\n    print(*ans, sep='\\\\n')\\n\\nmain()\\n\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append((t, w))\\n        self.E[t].append((f, w))\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\n# k=1, 2, ..., N\\u306b\\u5bfe\\u3057\\u3066\\n# \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u4ee5\\u4e0a\\u901a\\u308b\\u3088\\u3046\\u306a\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\u3092\\u6c42\\u3081\\u308b\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570 - \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u3082\\u901a\\u3089\\u306a\\u3044\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570\\u306fN*(N+1)/2\\n\\n# \\u30b0\\u30e9\\u30d5\\u3092\\u8272k\\u306e\\u30ce\\u30fc\\u30c9\\u3067\\u5206\\u5272\\u3057\\u3066\\u3001\\u90e8\\u5206\\u30b0\\u30e9\\u30d5\\u5185\\u3067\\u306e\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u7dcf\\u6570\\u3092\\u6c42\\u3081\\u308c\\u3070\\u826f\\u3044\\n# \\u5404\\u30ce\\u30fc\\u30c9\\u306b\\u72b6\\u614b\\u3068\\u3057\\u3066\\u8f9e\\u66f8\\u3092\\u3082\\u305f\\u305b\\u308b\\u3002\\n# x\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n# o\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u4e0d\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n\\n# \\u56de\\u7b54\\u7528\\nans = [0]*N\\n\\n\\ndef f(curr, par=-1):\\n    # \\u518d\\u5e30\\u95a2\\u6570\\n    # curr: \\u73fe\\u5728\\u306e\\u7bc0\\u70b9\\n    # par : \\u89aa\\u7bc0\\u70b9\\u306e\\u756a\\u53f7\\n    ret = defaultdict(int)\\n    size = 1\\n    for dest, w in g.E[curr]:\\n        if dest == par:\\n            continue\\n        sz, child = f(dest, curr)\\n        size += sz\\n\\n        # \\u81ea\\u8eab\\u306e\\u8272\\u3068\\u540c\\u3058\\u5834\\u5408\\u3001\\u5b50\\u306e\\u9802\\u70b9\\u306e\\u6570\\u304b\\u3089\\u52a0\\u7b97\\n        n = sz-child[c[curr]]\\n        ans[c[curr]] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        if len(ret) < len(child):\\n            child, ret = ret, child\\n        for key in child:\\n            ret[key] += child[key]\\n\\n    ret[c[curr]] = size\\n    return size, ret\\n\\n\\nsz, ret = f(0)\\nfor color in range(N):\\n    if color != c[0]:\\n        n = sz-ret[color]\\n        ans[color] += n*(n+1)//2\\n\\ntot = N*(N+1)//2\\nfor a in ans:\\n    print((tot-a))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        if IND[i] == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if IND[i] >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        if P[i] == X[i][IND[i]]:\\n            IND[i] += 1\\n            continue\\n        P[X[i][IND[i]]] = i\\n        IND[i], i = IND[i] + 1, X[i][IND[i]]\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [len(x) for x in X]\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if not ind:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind - 1]\\n        if P[i] == j:\\n            IND[i] -= 1\\n            continue\\n        \\n        P[j] = i\\n        IND[i] -= 1\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\nstdin = sys.stdin\\n\\nns = lambda: stdin.readline().rstrip()\\nni = lambda: int(stdin.readline().rstrip())\\nnm = lambda: map(int, stdin.readline().split())\\nnl = lambda: list(map(int, stdin.readline().split()))\\n\\nn = ni()\\ncolor = nl()\\ng = [list() for _ in range(n)]\\nfor _ in range(n-1):\\n    a, b = nm()\\n    g[a-1].append(b-1)\\n    g[b-1].append(a-1)\\n\\nsize = [1]*n\\npar = [-1]*n\\nh = [list() for _ in range(n+1)]\\nidx = [0]*n\\ndef dfs(v, p):\\n    h[color[v]].append(v)\\n    for x in g[v]:\\n        if x == p: continue\\n        par[x] = v\\n        size[v] += dfs(x, v)\\n        h[color[v]].append(v)\\n    h[color[v]].append(v)\\n    return size[v]\\ndfs(0, -1)\\n# print(size)\\nfor v in range(n):\\n    if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n        idx[v] += 1\\nacc = [0]*(n+10)\\nans = [n*(n+1)//2]*(n+1)\\nfor col in range(1, n+1):\\n    # print(h[col])\\n    acc[-1] = 0\\n    p = [-1]\\n    for v in h[col]:\\n        if p[-1] == v:\\n            p.pop()\\n            if idx[v] >= len(g[v]):\\n                acc[p[-1]] += size[v]\\n                continue\\n            q = size[g[v][idx[v]]] - acc[v]\\n            idx[v] += 1\\n            if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n              idx[v] += 1\\n            ans[col] -= q*(q+1)//2\\n            acc[v] = 0\\n            # print(v, q)\\n            p.append(v)\\n        else:\\n            p.append(v)\\n    q = n - acc[-1]\\n    # print(-1, q)\\n    ans[col] -= q*(q+1)//2\\nprint(*ans[1:], sep='\\\\n')\"]",
        "difficulty": "interview",
        "input": "3\n1 2 1\n1 2\n2 3\n",
        "output": "5\n4\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc163/tasks/abc163_f"
    },
    {
        "id": 691,
        "task_id": 2305,
        "test_case_id": 3,
        "question": "We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.\nAdditionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.\nFor each k=1, 2, ..., N, solve the following problem:\n - Find the number of simple paths that visit a vertex painted in the color k one or more times.\nNote: The simple paths from Vertex u to v and from v to u are not distinguished.\n\n-----Constraints-----\n - 1 \\leq N \\leq 2 \\times 10^5\n - 1 \\leq c_i \\leq N\n - 1 \\leq a_i,b_i \\leq N\n - The given graph is a tree.\n - All values in input are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN\nc_1 c_2 ... c_N\na_1 b_1\n:\na_{N-1} b_{N-1}\n\n-----Output-----\nPrint the answers for k = 1, 2, ..., N in order, each in its own line.\n\n-----Sample Input-----\n3\n1 2 1\n1 2\n2 3\n\n-----Sample Output-----\n5\n4\n0\n\nLet P_{i,j} denote the simple path connecting Vertex i and j.\nThere are 5 simple paths that visit a vertex painted in the color 1 one or more times:\nP_{1,1}\\,,\\,P_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,3}\\,,\\,P_{3,3} \nThere are 4 simple paths that visit a vertex painted in the color 2 one or more times:\nP_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,2}\\,,\\,P_{2,3} \nThere are no simple paths that visit a vertex painted in the color 3 one or more times.",
        "solutions": "[\"import sys\\nfrom collections import defaultdict\\n\\n# \\u518d\\u5e30\\u5236\\u9650\\u3092\\u7de9\\u548c\\u3059\\u308b\\u304a\\u307e\\u3058\\u306a\\u3044\\nsys.setrecursionlimit(10**6)\\n\\ndef sum(n):return n*(n+1)//2\\n\\n# \\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3068\\u3001\\u300c\\u8272i\\u3092\\u5c01\\u9396\\u3057\\u305f\\u3068\\u304d\\u306b\\u5230\\u9054\\u3067\\u304d\\u306a\\u3044\\u9802\\u70b9\\u306e\\u500b\\u6570\\u300d\\u3092\\u6301\\u3064\\u8f9e\\u66f8\\u3092\\u8fd4\\u3059\\ndef dfs(v,p):\\n    ret=defaultdict(int)\\n    size=1\\n    for vv in g[v]:\\n        if vv==p:\\n            continue\\n        ss,d=dfs(vv,v)\\n        size+=ss\\n        ans[c[v]]+=sum(ss-d[c[v]])\\n        \\n        # \\u30de\\u30fc\\u30b8\\u30c6\\u30af\\n        if len(ret)<len(d):\\n            ret,d=d,ret\\n        for vvv in d:\\n            ret[vvv]+=d[vvv]\\n    ret[c[v]]=size\\n    return size,ret\\n\\nn=int(input())\\nc=list(map(int,input().split()))\\ng=[[] for _ in range(n+1)]\\nans=[0]*(n+1)\\nfor _ in range(n-1):\\n    s,t=list(map(int, input().split()))\\n    s-=1\\n    t-=1\\n    g[s].append(t)\\n    g[t].append(s)\\n_,ret=dfs(0,-1)\\nfor i in range(1,n+1):\\n    print((sum(n)-ans[i]-sum(n-ret[i])))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nc = list(map(int, input().split()))\\nans = [n*(n+1)//2 for _ in range(n)]\\nfor i in range(n):\\n\\tc[i] -= 1\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n\\ta, b = map(int, input().split())\\n\\tadj[a-1].append(b-1)\\n\\tadj[b-1].append(a-1)\\n\\ns = 0\\ndp = [0 for _ in range(n)]\\n\\ndef dfs(x, p):\\n\\tnonlocal s\\n\\tpres = s + dp[c[x]]\\n\\tfor v in adj[x]:\\n\\t\\tif v == p:\\n\\t\\t\\tcontinue\\n\\t\\tdp[c[x]] = -s\\n\\t\\tdfs(v, x)\\n\\t\\tans[c[x]] -= (s+dp[c[x]]) * (s+dp[c[x]]+1) // 2\\n\\ts += 1\\n\\tdp[c[x]] = pres - s\\n\\treturn\\n\\ndfs(0, -1)\\nfor i in range(n):\\n\\tans[i] -= (s+dp[i]) * (s+dp[i]+1) // 2\\n\\nprint(*ans, sep=\\\"\\\\n\\\")\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append(t)\\n        self.E[t].append(f)\\n\\n\\ndef make_order(g, v):\\n    seen = [False]*g.N\\n    last_order = [-1]*g.N\\n    parent = [-1]*g.N\\n    size = [1]*g.N\\n    counter = {\\\"last\\\": 0}\\n\\n    def recur(v):\\n        seen[v] = True\\n        for to in g.E[v]:\\n            if seen[to] == True:\\n                continue\\n            parent[to] = v\\n            recur(to)\\n            size[v] += size[to]\\n\\n        last_order[counter[\\\"last\\\"]] = v\\n        counter[\\\"last\\\"] += 1\\n\\n    recur(v)\\n    return last_order, parent, size\\n\\n\\ndef merge(d: dict, e: dict):\\n    if len(d) < len(e):\\n        d, e = e, d\\n    for k in e:\\n        d[k] += e[k]\\n    return d\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\nans = [0]*N\\n\\nret = {}\\nlast_order, parent, size = make_order(g, 0)\\nfor curr in last_order:\\n    cn = c[curr]\\n    rrr = defaultdict(int)\\n    for dest in g.E[curr]:\\n        if dest == parent[curr]:\\n            continue\\n        child = ret.pop(dest)\\n\\n        n = size[dest]-child[cn]\\n        ans[cn] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        rrr = merge(rrr, child)\\n\\n    rrr[cn] = size[curr]\\n    ret[curr] = rrr\\n\\n\\ntot = N*(N+1)//2\\nfor color in range(N):\\n    if color != c[0]:\\n        n = N-ret[0][color]\\n        ans[color] += n*(n+1)//2\\n    print((tot-ans[color]))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\n \\nN = int(input())  # \\u9802\\u70b9\\u6570\\nC = [int(x) - 1  for x in input().split()]  # \\u9802\\u70b9\\u306e\\u8272\\u756a\\u53f7 (1~N) -> (0~N-1)\\n \\nE = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    E[a-1].append(b-1)\\n    E[b-1].append(a-1)\\n    \\ndef dfs(u, p = -1):\\n \\n    color = C[u]  # \\u8272\\u756a\\u53f7\\n    VC[color].append(u)  # \\u8272\\u3054\\u3068\\u306b stack \\u3057\\u3066\\u304a\\u304f\\n    \\n    s = 1  # \\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\uff09\\n    for v in E[u]:\\n        if v == p:\\n            continue\\n            \\n        child[u] = 0  # v \\u306e\\u65b9\\u5411\\u306b\\u3042\\u308b\\u5b50\\u5b6b\\u306e\\u3046\\u3061\\u8fbf\\u308c\\u308b\\u9802\\u70b9\\u306e\\u6570 = s - \\u5b50\\u5b6b\\u306b\\u3042\\u308b\\u3001\\u540c\\u8272\\u3092\\u6839\\u3068\\u3059\\u308b\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u306e\\u5408\\u8a08\\n        ret = dfs(v, u)\\n \\n        s += ret  # \\u7d14\\u7c8b\\u306a\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3092\\u8a08\\u7b97\\n        child[u] += ret  # \\u540c\\u8272\\u306e\\u9802\\u70b9\\u304c\\u5207\\u308a\\u96e2\\u3055\\u308c\\u305f\\u3042\\u3068\\u306e\\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\n        \\n        ans[color] -= child[u] * (child[u] + 1) // 2        \\n \\n    VC[color].pop()\\n    if VC[color]: # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u3042\\u308b\\u5834\\u5408\\n        child[VC[color][-1]] -= s  # \\u4e00\\u756a\\u8fd1\\u3044\\u540c\\u8272\\u304b\\u3089\\u3001\\u81ea\\u5206\\u306e\\u90e8\\u5206\\u6728\\u306e\\u5927\\u304d\\u3055\\u3092\\u3072\\u304f\\n    else:  # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u306a\\u3044\\u5834\\u5408\\n        root_size[color] -= s  # \\u4e0a\\u4f4d\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\n        \\n    return s\\n\\nchild = [0] * N  # \\u8fbf\\u3063\\u3066\\u3044\\u308b\\u5b50\\u306e\\u4e2d\\u3067\\u306e\\u306e\\u6b8b\\u308a\\u6570\\u3000\\u3068\\u308a\\u3042\\u3048\\u305a 0\\u3067\\u521d\\u671f\\u5316\\nroot_size = [N] * N  # \\u6839\\u65b9\\u5411\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\u3002\\u64cd\\u4f5c\\u524d\\u306f N \\u5168\\u90e8\\uff09\\nVC = [ [] for _ in range(N)]  # \\u8272\\u756a\\u53f7\\u3054\\u3068\\nans = [N*(N+1)//2] * N  # \\u8272\\u3054\\u3068\\u306e\\uff08\\u6b8b\\u308a\\u306e\\uff09\\u7d44\\u307f\\u5408\\u308f\\u305b\\u6570\\u3000\\u521d\\u671f\\u5024\\u306f \\u5168\\u70b9\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\n \\ndfs(0)\\n \\nfor i in range(N):\\n    print(ans[i] - root_size[i] * (root_size[i] + 1) // 2)\", \"import sys\\n\\ninput=sys.stdin.readline\\nsys.setrecursionlimit(10**7)\\n\\nN=int(input())\\nc=list(map(int,input().split()))\\nfor i in range(N):\\n    c[i]-=1\\nedge=[[] for i in range(N)]\\nfor i in range(N-1):\\n    a,b=map(int,input().split())\\n    edge[a-1].append(b-1)\\n    edge[b-1].append(a-1)\\n\\nsubtree=[1]*N\\ndef size(v,pv):\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            size(nv,v)\\n            subtree[v]+=subtree[nv]\\n\\nsize(0,-1)\\n\\ndata=[[] for i in range(N)]\\nParent=[[0] for i in range(N)]\\nparent=[0]*N\\ndef dfs(v,pv,nop):\\n    pp=parent[nop]\\n    parent[nop]=v\\n    Parent[nop].append(v)\\n    data[c[v]].append((parent[c[v]],v))\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            dfs(nv,v,c[v])\\n    parent[nop]=pp\\n\\ndfs(0,-1,0)\\n\\nfor i in range(N):\\n    dic={v:subtree[v] for v in Parent[i]}\\n    for p,ch in data[i]:\\n        dic[p]-=subtree[ch]\\n\\n    res=N*(N+1)//2\\n    for p in dic:\\n        res-=dic[p]*(dic[p]+1)//2\\n    print(res)\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nhead = [-1] * (N + 1)\\nto = [0] * (N - 1 << 1)\\nnxt = [0] * (N - 1 << 1)\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    x, y = x-1, y-1\\n    nxt[i] = head[x]\\n    to[i] = y\\n    head[x] = i\\n    j = i + N - 1\\n    nxt[j] = head[y]\\n    to[j] = x\\n    head[y] = j\\n\\ndef EulerTour(n, i=0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    ANS = [f(n)] * n\\n    \\n    P = [-1] * n\\n    ct = 0\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    while i >= 0:\\n        e = head[i]\\n        if e < 0:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = to[e]\\n        if P[i] == j:\\n            head[i] = nxt[e]\\n            continue\\n        \\n        P[j] = i\\n        head[i] = nxt[e]\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    done = [0] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            continue\\n        done[i] = 1\\n        if i: ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n        for a in X[i]:\\n            if done[a]: continue\\n            P[a] = i\\n            Q.append(~a)\\n            Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\nn = int(input())\\nc = list(map(lambda x: int(x) - 1, input().split()))\\n\\ngraph = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    graph[a].append(b)\\n    graph[b].append(a)\\n\\nans = [n * (n + 1) // 2] * n\\nn_subtree = [1] * n  # n_st[i] = (\\u9802\\u70b9i\\u306e\\u90e8\\u5206\\u6728\\u306e\\u9802\\u70b9\\u6570)\\ncnt = [0] * n  # cnt[i] = (\\u8a2a\\u554f\\u6e08\\u307f\\u9802\\u70b9\\u306e\\u3046\\u3061\\u3001\\u8272i\\u3092\\u901a\\u904e\\u3057\\u306a\\u3044\\u3068\\u305f\\u3069\\u308a\\u7740\\u3051\\u306a\\u3044\\u9802\\u70b9\\u6570)\\nn_visited = 0\\n\\n\\ndef dfs(v, v_p):\\n    nonlocal n_visited\\n    cnt_v_before = cnt[c[v]]\\n    for v_next in graph[v]:\\n        if v_next == v_p:\\n            continue\\n\\n        m_prev = n_visited - cnt[c[v]]\\n        dfs(v_next, v)\\n        n_subtree[v] += n_subtree[v_next]\\n        m_next = n_visited - cnt[c[v]]\\n\\n        m = m_next - m_prev\\n        ans[c[v]] -= m * (m + 1) // 2\\n\\n    cnt[c[v]] = cnt_v_before + n_subtree[v]\\n    n_visited += 1\\n\\n\\ndfs(0, -1)\\n\\nfor i in range(n):\\n    m = n - cnt[i]\\n    ans[i] -= m * (m + 1) // 2\\n\\nprint(*ans, sep='\\\\n')\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\n# ci = [set() for _ in range(n)]\\n# for i,c in enumerate(C):\\n#     ci[c-1].add(i)\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = list(map(int,input().split()))\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n\\nans = [n*(n+1)//2]*n\\ncparent = [[] for _ in range(n)]\\nroot_size = [n]*n\\n# index:color\\nreached = [False]*n\\nsize = [0]*n\\n# index:vertex\\n\\ndef dfs(p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        if reached[nxt]:continue\\n        reached[nxt] = True\\n        size[p] = 0\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] += ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    # size[p] += s\\n    if cparent[c]:\\n        size[cparent[c][-1]] -= s\\n    else:\\n        root_size[c] -= s\\n    # ans[c] -= size[p] * (size[p] - 1)//2\\n    return s\\n\\nreached = [False]*n\\nreached[0] = True\\n\\ndfs(0)\\nfor i in range(n):\\n    print((ans[i] - root_size[i]*(root_size[i]+1)//2))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if ind == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if ind >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind]\\n        if P[i] == j:\\n            IND[i] += 1\\n            continue\\n        P[j] = i\\n        IND[i] += 1\\n        i = j\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n    \\nans = [n*(n+1)//2]*n\\nsize = [0]*n\\ncparent = [[] for _ in range(n)]\\nreached = [False]*n\\nroot_size = [n]*n\\n\\ndef dfs (p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        #print (nxt)\\n        \\n        if reached[nxt]:\\n            continue\\n        size[p] = 0\\n        reached[nxt] = True\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] +=ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    if cparent[c]:\\n        size[cparent[c][-1]] -=s\\n    else:\\n        root_size[c] -=s    \\n    return s\\n\\nreached[0] = True\\ndfs (0)\\n\\nfor i in range(n):\\n    print (ans[i]-root_size[i]*(root_size[i]+1)//2)\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef main():\\n    n=int(input())\\n    color=list(map(int,input().split()))\\n    nodes=[[]for i in range(n)]\\n    cut=[[0]for i in range(n+1)]\\n    ans=[0]*(n+1)\\n    for i in range(n-1):\\n        a,b=map(int,input().split())\\n        nodes[a-1].append(b-1)\\n        nodes[b-1].append(a-1)\\n    def num(p,parent):\\n        s=0\\n        for child in nodes[p]:\\n            if child==parent:\\n                continue\\n            cut[color[p]].append(0)\\n            nc=num(child,p)\\n            group=nc-cut[color[p]].pop()\\n            ans[color[p]]+=group*(group+1)//2\\n            s+=nc\\n        s+=1\\n        cut[color[p]][-1]+=s\\n        return s\\n    num(0,-1)\\n    for a,c in zip(ans[1:],cut[1:]):\\n        group=n-c[0]\\n        c=group*(group+1)//2\\n        print(n*(n+1)//2-c-a)\\nmain()\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                p = P[i]\\n                k = ET2[i] - ET1[i] + 1 - USED[C[p]] + ORG[i]\\n                ANS[C[p]] -= f(k)\\n                TMP[p] += k\\n            continue\\n        if i >= 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            if ET1[i] == 0: ET1[i] = ct\\n        for a in X[i][::-1]:\\n            if a != P[i]:\\n                P[a] = i\\n                for k in range(len(X[a])):\\n                    if X[a][k] == i:\\n                        del X[a][k]\\n                        break\\n                Q.append(~a)\\n                Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\ndef solve():\\n    N = int(input())\\n    Cs = [A-1 for A in map(int, input().split())]\\n    adjL = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = list(map(int, input().split()))\\n        a, b = a-1, b-1\\n        adjL[a].append(b)\\n        adjL[b].append(a)\\n\\n    sizeSubtrees = [1] * N\\n    numVs = [0] * N\\n    anss = [0] * N\\n\\n    def dfs(vNow, vPar, clrPar):\\n        clrNow = Cs[vNow]\\n        numVNow = numVs[clrNow]\\n        numVPar = numVs[clrPar]\\n        for v2 in adjL[vNow]:\\n            if v2 == vPar: continue\\n            dfs(v2, vNow, clrNow)\\n            sizeSubtrees[vNow] += sizeSubtrees[v2]\\n        numVs[clrNow] = numVNow + sizeSubtrees[vNow]\\n        if clrPar != -1:\\n            k = sizeSubtrees[vNow] - (numVs[clrPar]-numVPar)\\n            anss[clrPar] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for clr in range(N):\\n        if clr == Cs[0]: continue\\n        k = N - numVs[clr]\\n        anss[clr] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    anss = [numAll-ans for ans in anss]\\n\\n    print(('\\\\n'.join(map(str, anss))))\\n\\n\\nsolve()\\n\", \"import sys\\nimport itertools\\n\\nsys.setrecursionlimit(3 * 10**5)\\n\\n\\nclass Node:\\n  def __init__(self, color):\\n    self.color = int(color) - 1\\n    self.children = []\\n\\n  def add_child(self, node):\\n    self.children.append(node)\\n\\n  @staticmethod\\n  def prepare_nodes(n):\\n    nodes = tuple(map(Node, input().split()))\\n    for _ in range(n - 1):\\n      a, b = [int(x) - 1 for x in input().split()]\\n      nodes[a].add_child(nodes[b])\\n      nodes[b].add_child(nodes[a])\\n\\n    return nodes\\n\\n\\ndef main():\\n  n = int(input())\\n  nodes = Node.prepare_nodes(n)\\n\\n  dp = [0 for _ in range(n)]\\n  ans_list = [n*(n + 1) // 2 for _ in range(n)]\\n\\n  def update_ans(s, color):\\n    ans_list[color] -= (s + dp[color]) * (s + dp[color] + 1) // 2\\n\\n  def dfs(s, node, parent):\\n    color = node.color\\n    pre_s = s + dp[color]\\n    for child in node.children:\\n      if child == parent: continue\\n\\n      dp[color] = -s\\n      s = dfs(s, child, node)\\n      update_ans(s, color)\\n\\n    s += 1\\n    dp[color] = pre_s - s\\n\\n    return s\\n\\n  s = dfs(0, nodes[0], Node(-1))\\n\\n  for i in range(n):\\n    update_ans(s, i)\\n    print((ans_list[i]))\\n\\n\\nmain()\\n\", \"def main():\\n    import sys\\n    sys.setrecursionlimit(10**9)\\n    input = sys.stdin.readline\\n\\n    N = int(input())\\n    C = [c-1 for c in map(int, input().split())]\\n    tree = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = map(int, input().split())\\n        tree[a-1].append(b-1)\\n        tree[b-1].append(a-1)\\n\\n    # 1\\u56de\\u4ee5\\u4e0a > \\u5168\\u4f53 - 1\\u56de\\u3082\\u901a\\u3089\\u306a\\u3044\\n    # \\u9078\\u3079\\u308b\\u6570 = n(n+1)//2\\n    size_subtree = [1] * N\\n    num_v = [0] * N\\n    ans = [0] * N\\n\\n    def dfs(v_now, v_parent, color_parent):\\n        color_now = C[v_now]\\n        num_v_now = num_v[color_now]\\n        num_v_parent = num_v[color_parent]\\n        for v2 in tree[v_now]:\\n            if v2 == v_parent: continue\\n            dfs(v2, v_now, color_now)\\n            size_subtree[v_now] += size_subtree[v2]\\n        num_v[color_now] = num_v_now + size_subtree[v_now]\\n        if color_parent != -1:\\n            k = size_subtree[v_now] - (num_v[color_parent]-num_v_parent)\\n            ans[color_parent] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for color in range(N):\\n        if color == C[0]: continue\\n        k = N - num_v[color]\\n        ans[color] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    ans = [numAll-ans for ans in ans]\\n\\n    print(*ans, sep='\\\\n')\\n\\nmain()\\n\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append((t, w))\\n        self.E[t].append((f, w))\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\n# k=1, 2, ..., N\\u306b\\u5bfe\\u3057\\u3066\\n# \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u4ee5\\u4e0a\\u901a\\u308b\\u3088\\u3046\\u306a\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\u3092\\u6c42\\u3081\\u308b\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570 - \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u3082\\u901a\\u3089\\u306a\\u3044\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570\\u306fN*(N+1)/2\\n\\n# \\u30b0\\u30e9\\u30d5\\u3092\\u8272k\\u306e\\u30ce\\u30fc\\u30c9\\u3067\\u5206\\u5272\\u3057\\u3066\\u3001\\u90e8\\u5206\\u30b0\\u30e9\\u30d5\\u5185\\u3067\\u306e\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u7dcf\\u6570\\u3092\\u6c42\\u3081\\u308c\\u3070\\u826f\\u3044\\n# \\u5404\\u30ce\\u30fc\\u30c9\\u306b\\u72b6\\u614b\\u3068\\u3057\\u3066\\u8f9e\\u66f8\\u3092\\u3082\\u305f\\u305b\\u308b\\u3002\\n# x\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n# o\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u4e0d\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n\\n# \\u56de\\u7b54\\u7528\\nans = [0]*N\\n\\n\\ndef f(curr, par=-1):\\n    # \\u518d\\u5e30\\u95a2\\u6570\\n    # curr: \\u73fe\\u5728\\u306e\\u7bc0\\u70b9\\n    # par : \\u89aa\\u7bc0\\u70b9\\u306e\\u756a\\u53f7\\n    ret = defaultdict(int)\\n    size = 1\\n    for dest, w in g.E[curr]:\\n        if dest == par:\\n            continue\\n        sz, child = f(dest, curr)\\n        size += sz\\n\\n        # \\u81ea\\u8eab\\u306e\\u8272\\u3068\\u540c\\u3058\\u5834\\u5408\\u3001\\u5b50\\u306e\\u9802\\u70b9\\u306e\\u6570\\u304b\\u3089\\u52a0\\u7b97\\n        n = sz-child[c[curr]]\\n        ans[c[curr]] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        if len(ret) < len(child):\\n            child, ret = ret, child\\n        for key in child:\\n            ret[key] += child[key]\\n\\n    ret[c[curr]] = size\\n    return size, ret\\n\\n\\nsz, ret = f(0)\\nfor color in range(N):\\n    if color != c[0]:\\n        n = sz-ret[color]\\n        ans[color] += n*(n+1)//2\\n\\ntot = N*(N+1)//2\\nfor a in ans:\\n    print((tot-a))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        if IND[i] == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if IND[i] >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        if P[i] == X[i][IND[i]]:\\n            IND[i] += 1\\n            continue\\n        P[X[i][IND[i]]] = i\\n        IND[i], i = IND[i] + 1, X[i][IND[i]]\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [len(x) for x in X]\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if not ind:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind - 1]\\n        if P[i] == j:\\n            IND[i] -= 1\\n            continue\\n        \\n        P[j] = i\\n        IND[i] -= 1\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\nstdin = sys.stdin\\n\\nns = lambda: stdin.readline().rstrip()\\nni = lambda: int(stdin.readline().rstrip())\\nnm = lambda: map(int, stdin.readline().split())\\nnl = lambda: list(map(int, stdin.readline().split()))\\n\\nn = ni()\\ncolor = nl()\\ng = [list() for _ in range(n)]\\nfor _ in range(n-1):\\n    a, b = nm()\\n    g[a-1].append(b-1)\\n    g[b-1].append(a-1)\\n\\nsize = [1]*n\\npar = [-1]*n\\nh = [list() for _ in range(n+1)]\\nidx = [0]*n\\ndef dfs(v, p):\\n    h[color[v]].append(v)\\n    for x in g[v]:\\n        if x == p: continue\\n        par[x] = v\\n        size[v] += dfs(x, v)\\n        h[color[v]].append(v)\\n    h[color[v]].append(v)\\n    return size[v]\\ndfs(0, -1)\\n# print(size)\\nfor v in range(n):\\n    if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n        idx[v] += 1\\nacc = [0]*(n+10)\\nans = [n*(n+1)//2]*(n+1)\\nfor col in range(1, n+1):\\n    # print(h[col])\\n    acc[-1] = 0\\n    p = [-1]\\n    for v in h[col]:\\n        if p[-1] == v:\\n            p.pop()\\n            if idx[v] >= len(g[v]):\\n                acc[p[-1]] += size[v]\\n                continue\\n            q = size[g[v][idx[v]]] - acc[v]\\n            idx[v] += 1\\n            if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n              idx[v] += 1\\n            ans[col] -= q*(q+1)//2\\n            acc[v] = 0\\n            # print(v, q)\\n            p.append(v)\\n        else:\\n            p.append(v)\\n    q = n - acc[-1]\\n    # print(-1, q)\\n    ans[col] -= q*(q+1)//2\\nprint(*ans[1:], sep='\\\\n')\"]",
        "difficulty": "interview",
        "input": "2\n1 2\n1 2\n",
        "output": "2\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc163/tasks/abc163_f"
    },
    {
        "id": 692,
        "task_id": 2305,
        "test_case_id": 4,
        "question": "We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.\nAdditionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.\nFor each k=1, 2, ..., N, solve the following problem:\n - Find the number of simple paths that visit a vertex painted in the color k one or more times.\nNote: The simple paths from Vertex u to v and from v to u are not distinguished.\n\n-----Constraints-----\n - 1 \\leq N \\leq 2 \\times 10^5\n - 1 \\leq c_i \\leq N\n - 1 \\leq a_i,b_i \\leq N\n - The given graph is a tree.\n - All values in input are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN\nc_1 c_2 ... c_N\na_1 b_1\n:\na_{N-1} b_{N-1}\n\n-----Output-----\nPrint the answers for k = 1, 2, ..., N in order, each in its own line.\n\n-----Sample Input-----\n3\n1 2 1\n1 2\n2 3\n\n-----Sample Output-----\n5\n4\n0\n\nLet P_{i,j} denote the simple path connecting Vertex i and j.\nThere are 5 simple paths that visit a vertex painted in the color 1 one or more times:\nP_{1,1}\\,,\\,P_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,3}\\,,\\,P_{3,3} \nThere are 4 simple paths that visit a vertex painted in the color 2 one or more times:\nP_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,2}\\,,\\,P_{2,3} \nThere are no simple paths that visit a vertex painted in the color 3 one or more times.",
        "solutions": "[\"import sys\\nfrom collections import defaultdict\\n\\n# \\u518d\\u5e30\\u5236\\u9650\\u3092\\u7de9\\u548c\\u3059\\u308b\\u304a\\u307e\\u3058\\u306a\\u3044\\nsys.setrecursionlimit(10**6)\\n\\ndef sum(n):return n*(n+1)//2\\n\\n# \\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3068\\u3001\\u300c\\u8272i\\u3092\\u5c01\\u9396\\u3057\\u305f\\u3068\\u304d\\u306b\\u5230\\u9054\\u3067\\u304d\\u306a\\u3044\\u9802\\u70b9\\u306e\\u500b\\u6570\\u300d\\u3092\\u6301\\u3064\\u8f9e\\u66f8\\u3092\\u8fd4\\u3059\\ndef dfs(v,p):\\n    ret=defaultdict(int)\\n    size=1\\n    for vv in g[v]:\\n        if vv==p:\\n            continue\\n        ss,d=dfs(vv,v)\\n        size+=ss\\n        ans[c[v]]+=sum(ss-d[c[v]])\\n        \\n        # \\u30de\\u30fc\\u30b8\\u30c6\\u30af\\n        if len(ret)<len(d):\\n            ret,d=d,ret\\n        for vvv in d:\\n            ret[vvv]+=d[vvv]\\n    ret[c[v]]=size\\n    return size,ret\\n\\nn=int(input())\\nc=list(map(int,input().split()))\\ng=[[] for _ in range(n+1)]\\nans=[0]*(n+1)\\nfor _ in range(n-1):\\n    s,t=list(map(int, input().split()))\\n    s-=1\\n    t-=1\\n    g[s].append(t)\\n    g[t].append(s)\\n_,ret=dfs(0,-1)\\nfor i in range(1,n+1):\\n    print((sum(n)-ans[i]-sum(n-ret[i])))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nc = list(map(int, input().split()))\\nans = [n*(n+1)//2 for _ in range(n)]\\nfor i in range(n):\\n\\tc[i] -= 1\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n\\ta, b = map(int, input().split())\\n\\tadj[a-1].append(b-1)\\n\\tadj[b-1].append(a-1)\\n\\ns = 0\\ndp = [0 for _ in range(n)]\\n\\ndef dfs(x, p):\\n\\tnonlocal s\\n\\tpres = s + dp[c[x]]\\n\\tfor v in adj[x]:\\n\\t\\tif v == p:\\n\\t\\t\\tcontinue\\n\\t\\tdp[c[x]] = -s\\n\\t\\tdfs(v, x)\\n\\t\\tans[c[x]] -= (s+dp[c[x]]) * (s+dp[c[x]]+1) // 2\\n\\ts += 1\\n\\tdp[c[x]] = pres - s\\n\\treturn\\n\\ndfs(0, -1)\\nfor i in range(n):\\n\\tans[i] -= (s+dp[i]) * (s+dp[i]+1) // 2\\n\\nprint(*ans, sep=\\\"\\\\n\\\")\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append(t)\\n        self.E[t].append(f)\\n\\n\\ndef make_order(g, v):\\n    seen = [False]*g.N\\n    last_order = [-1]*g.N\\n    parent = [-1]*g.N\\n    size = [1]*g.N\\n    counter = {\\\"last\\\": 0}\\n\\n    def recur(v):\\n        seen[v] = True\\n        for to in g.E[v]:\\n            if seen[to] == True:\\n                continue\\n            parent[to] = v\\n            recur(to)\\n            size[v] += size[to]\\n\\n        last_order[counter[\\\"last\\\"]] = v\\n        counter[\\\"last\\\"] += 1\\n\\n    recur(v)\\n    return last_order, parent, size\\n\\n\\ndef merge(d: dict, e: dict):\\n    if len(d) < len(e):\\n        d, e = e, d\\n    for k in e:\\n        d[k] += e[k]\\n    return d\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\nans = [0]*N\\n\\nret = {}\\nlast_order, parent, size = make_order(g, 0)\\nfor curr in last_order:\\n    cn = c[curr]\\n    rrr = defaultdict(int)\\n    for dest in g.E[curr]:\\n        if dest == parent[curr]:\\n            continue\\n        child = ret.pop(dest)\\n\\n        n = size[dest]-child[cn]\\n        ans[cn] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        rrr = merge(rrr, child)\\n\\n    rrr[cn] = size[curr]\\n    ret[curr] = rrr\\n\\n\\ntot = N*(N+1)//2\\nfor color in range(N):\\n    if color != c[0]:\\n        n = N-ret[0][color]\\n        ans[color] += n*(n+1)//2\\n    print((tot-ans[color]))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\n \\nN = int(input())  # \\u9802\\u70b9\\u6570\\nC = [int(x) - 1  for x in input().split()]  # \\u9802\\u70b9\\u306e\\u8272\\u756a\\u53f7 (1~N) -> (0~N-1)\\n \\nE = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    E[a-1].append(b-1)\\n    E[b-1].append(a-1)\\n    \\ndef dfs(u, p = -1):\\n \\n    color = C[u]  # \\u8272\\u756a\\u53f7\\n    VC[color].append(u)  # \\u8272\\u3054\\u3068\\u306b stack \\u3057\\u3066\\u304a\\u304f\\n    \\n    s = 1  # \\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\uff09\\n    for v in E[u]:\\n        if v == p:\\n            continue\\n            \\n        child[u] = 0  # v \\u306e\\u65b9\\u5411\\u306b\\u3042\\u308b\\u5b50\\u5b6b\\u306e\\u3046\\u3061\\u8fbf\\u308c\\u308b\\u9802\\u70b9\\u306e\\u6570 = s - \\u5b50\\u5b6b\\u306b\\u3042\\u308b\\u3001\\u540c\\u8272\\u3092\\u6839\\u3068\\u3059\\u308b\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u306e\\u5408\\u8a08\\n        ret = dfs(v, u)\\n \\n        s += ret  # \\u7d14\\u7c8b\\u306a\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3092\\u8a08\\u7b97\\n        child[u] += ret  # \\u540c\\u8272\\u306e\\u9802\\u70b9\\u304c\\u5207\\u308a\\u96e2\\u3055\\u308c\\u305f\\u3042\\u3068\\u306e\\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\n        \\n        ans[color] -= child[u] * (child[u] + 1) // 2        \\n \\n    VC[color].pop()\\n    if VC[color]: # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u3042\\u308b\\u5834\\u5408\\n        child[VC[color][-1]] -= s  # \\u4e00\\u756a\\u8fd1\\u3044\\u540c\\u8272\\u304b\\u3089\\u3001\\u81ea\\u5206\\u306e\\u90e8\\u5206\\u6728\\u306e\\u5927\\u304d\\u3055\\u3092\\u3072\\u304f\\n    else:  # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u306a\\u3044\\u5834\\u5408\\n        root_size[color] -= s  # \\u4e0a\\u4f4d\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\n        \\n    return s\\n\\nchild = [0] * N  # \\u8fbf\\u3063\\u3066\\u3044\\u308b\\u5b50\\u306e\\u4e2d\\u3067\\u306e\\u306e\\u6b8b\\u308a\\u6570\\u3000\\u3068\\u308a\\u3042\\u3048\\u305a 0\\u3067\\u521d\\u671f\\u5316\\nroot_size = [N] * N  # \\u6839\\u65b9\\u5411\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\u3002\\u64cd\\u4f5c\\u524d\\u306f N \\u5168\\u90e8\\uff09\\nVC = [ [] for _ in range(N)]  # \\u8272\\u756a\\u53f7\\u3054\\u3068\\nans = [N*(N+1)//2] * N  # \\u8272\\u3054\\u3068\\u306e\\uff08\\u6b8b\\u308a\\u306e\\uff09\\u7d44\\u307f\\u5408\\u308f\\u305b\\u6570\\u3000\\u521d\\u671f\\u5024\\u306f \\u5168\\u70b9\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\n \\ndfs(0)\\n \\nfor i in range(N):\\n    print(ans[i] - root_size[i] * (root_size[i] + 1) // 2)\", \"import sys\\n\\ninput=sys.stdin.readline\\nsys.setrecursionlimit(10**7)\\n\\nN=int(input())\\nc=list(map(int,input().split()))\\nfor i in range(N):\\n    c[i]-=1\\nedge=[[] for i in range(N)]\\nfor i in range(N-1):\\n    a,b=map(int,input().split())\\n    edge[a-1].append(b-1)\\n    edge[b-1].append(a-1)\\n\\nsubtree=[1]*N\\ndef size(v,pv):\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            size(nv,v)\\n            subtree[v]+=subtree[nv]\\n\\nsize(0,-1)\\n\\ndata=[[] for i in range(N)]\\nParent=[[0] for i in range(N)]\\nparent=[0]*N\\ndef dfs(v,pv,nop):\\n    pp=parent[nop]\\n    parent[nop]=v\\n    Parent[nop].append(v)\\n    data[c[v]].append((parent[c[v]],v))\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            dfs(nv,v,c[v])\\n    parent[nop]=pp\\n\\ndfs(0,-1,0)\\n\\nfor i in range(N):\\n    dic={v:subtree[v] for v in Parent[i]}\\n    for p,ch in data[i]:\\n        dic[p]-=subtree[ch]\\n\\n    res=N*(N+1)//2\\n    for p in dic:\\n        res-=dic[p]*(dic[p]+1)//2\\n    print(res)\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nhead = [-1] * (N + 1)\\nto = [0] * (N - 1 << 1)\\nnxt = [0] * (N - 1 << 1)\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    x, y = x-1, y-1\\n    nxt[i] = head[x]\\n    to[i] = y\\n    head[x] = i\\n    j = i + N - 1\\n    nxt[j] = head[y]\\n    to[j] = x\\n    head[y] = j\\n\\ndef EulerTour(n, i=0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    ANS = [f(n)] * n\\n    \\n    P = [-1] * n\\n    ct = 0\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    while i >= 0:\\n        e = head[i]\\n        if e < 0:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = to[e]\\n        if P[i] == j:\\n            head[i] = nxt[e]\\n            continue\\n        \\n        P[j] = i\\n        head[i] = nxt[e]\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    done = [0] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            continue\\n        done[i] = 1\\n        if i: ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n        for a in X[i]:\\n            if done[a]: continue\\n            P[a] = i\\n            Q.append(~a)\\n            Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\nn = int(input())\\nc = list(map(lambda x: int(x) - 1, input().split()))\\n\\ngraph = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    graph[a].append(b)\\n    graph[b].append(a)\\n\\nans = [n * (n + 1) // 2] * n\\nn_subtree = [1] * n  # n_st[i] = (\\u9802\\u70b9i\\u306e\\u90e8\\u5206\\u6728\\u306e\\u9802\\u70b9\\u6570)\\ncnt = [0] * n  # cnt[i] = (\\u8a2a\\u554f\\u6e08\\u307f\\u9802\\u70b9\\u306e\\u3046\\u3061\\u3001\\u8272i\\u3092\\u901a\\u904e\\u3057\\u306a\\u3044\\u3068\\u305f\\u3069\\u308a\\u7740\\u3051\\u306a\\u3044\\u9802\\u70b9\\u6570)\\nn_visited = 0\\n\\n\\ndef dfs(v, v_p):\\n    nonlocal n_visited\\n    cnt_v_before = cnt[c[v]]\\n    for v_next in graph[v]:\\n        if v_next == v_p:\\n            continue\\n\\n        m_prev = n_visited - cnt[c[v]]\\n        dfs(v_next, v)\\n        n_subtree[v] += n_subtree[v_next]\\n        m_next = n_visited - cnt[c[v]]\\n\\n        m = m_next - m_prev\\n        ans[c[v]] -= m * (m + 1) // 2\\n\\n    cnt[c[v]] = cnt_v_before + n_subtree[v]\\n    n_visited += 1\\n\\n\\ndfs(0, -1)\\n\\nfor i in range(n):\\n    m = n - cnt[i]\\n    ans[i] -= m * (m + 1) // 2\\n\\nprint(*ans, sep='\\\\n')\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\n# ci = [set() for _ in range(n)]\\n# for i,c in enumerate(C):\\n#     ci[c-1].add(i)\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = list(map(int,input().split()))\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n\\nans = [n*(n+1)//2]*n\\ncparent = [[] for _ in range(n)]\\nroot_size = [n]*n\\n# index:color\\nreached = [False]*n\\nsize = [0]*n\\n# index:vertex\\n\\ndef dfs(p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        if reached[nxt]:continue\\n        reached[nxt] = True\\n        size[p] = 0\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] += ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    # size[p] += s\\n    if cparent[c]:\\n        size[cparent[c][-1]] -= s\\n    else:\\n        root_size[c] -= s\\n    # ans[c] -= size[p] * (size[p] - 1)//2\\n    return s\\n\\nreached = [False]*n\\nreached[0] = True\\n\\ndfs(0)\\nfor i in range(n):\\n    print((ans[i] - root_size[i]*(root_size[i]+1)//2))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if ind == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if ind >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind]\\n        if P[i] == j:\\n            IND[i] += 1\\n            continue\\n        P[j] = i\\n        IND[i] += 1\\n        i = j\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n    \\nans = [n*(n+1)//2]*n\\nsize = [0]*n\\ncparent = [[] for _ in range(n)]\\nreached = [False]*n\\nroot_size = [n]*n\\n\\ndef dfs (p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        #print (nxt)\\n        \\n        if reached[nxt]:\\n            continue\\n        size[p] = 0\\n        reached[nxt] = True\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] +=ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    if cparent[c]:\\n        size[cparent[c][-1]] -=s\\n    else:\\n        root_size[c] -=s    \\n    return s\\n\\nreached[0] = True\\ndfs (0)\\n\\nfor i in range(n):\\n    print (ans[i]-root_size[i]*(root_size[i]+1)//2)\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef main():\\n    n=int(input())\\n    color=list(map(int,input().split()))\\n    nodes=[[]for i in range(n)]\\n    cut=[[0]for i in range(n+1)]\\n    ans=[0]*(n+1)\\n    for i in range(n-1):\\n        a,b=map(int,input().split())\\n        nodes[a-1].append(b-1)\\n        nodes[b-1].append(a-1)\\n    def num(p,parent):\\n        s=0\\n        for child in nodes[p]:\\n            if child==parent:\\n                continue\\n            cut[color[p]].append(0)\\n            nc=num(child,p)\\n            group=nc-cut[color[p]].pop()\\n            ans[color[p]]+=group*(group+1)//2\\n            s+=nc\\n        s+=1\\n        cut[color[p]][-1]+=s\\n        return s\\n    num(0,-1)\\n    for a,c in zip(ans[1:],cut[1:]):\\n        group=n-c[0]\\n        c=group*(group+1)//2\\n        print(n*(n+1)//2-c-a)\\nmain()\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                p = P[i]\\n                k = ET2[i] - ET1[i] + 1 - USED[C[p]] + ORG[i]\\n                ANS[C[p]] -= f(k)\\n                TMP[p] += k\\n            continue\\n        if i >= 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            if ET1[i] == 0: ET1[i] = ct\\n        for a in X[i][::-1]:\\n            if a != P[i]:\\n                P[a] = i\\n                for k in range(len(X[a])):\\n                    if X[a][k] == i:\\n                        del X[a][k]\\n                        break\\n                Q.append(~a)\\n                Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\ndef solve():\\n    N = int(input())\\n    Cs = [A-1 for A in map(int, input().split())]\\n    adjL = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = list(map(int, input().split()))\\n        a, b = a-1, b-1\\n        adjL[a].append(b)\\n        adjL[b].append(a)\\n\\n    sizeSubtrees = [1] * N\\n    numVs = [0] * N\\n    anss = [0] * N\\n\\n    def dfs(vNow, vPar, clrPar):\\n        clrNow = Cs[vNow]\\n        numVNow = numVs[clrNow]\\n        numVPar = numVs[clrPar]\\n        for v2 in adjL[vNow]:\\n            if v2 == vPar: continue\\n            dfs(v2, vNow, clrNow)\\n            sizeSubtrees[vNow] += sizeSubtrees[v2]\\n        numVs[clrNow] = numVNow + sizeSubtrees[vNow]\\n        if clrPar != -1:\\n            k = sizeSubtrees[vNow] - (numVs[clrPar]-numVPar)\\n            anss[clrPar] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for clr in range(N):\\n        if clr == Cs[0]: continue\\n        k = N - numVs[clr]\\n        anss[clr] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    anss = [numAll-ans for ans in anss]\\n\\n    print(('\\\\n'.join(map(str, anss))))\\n\\n\\nsolve()\\n\", \"import sys\\nimport itertools\\n\\nsys.setrecursionlimit(3 * 10**5)\\n\\n\\nclass Node:\\n  def __init__(self, color):\\n    self.color = int(color) - 1\\n    self.children = []\\n\\n  def add_child(self, node):\\n    self.children.append(node)\\n\\n  @staticmethod\\n  def prepare_nodes(n):\\n    nodes = tuple(map(Node, input().split()))\\n    for _ in range(n - 1):\\n      a, b = [int(x) - 1 for x in input().split()]\\n      nodes[a].add_child(nodes[b])\\n      nodes[b].add_child(nodes[a])\\n\\n    return nodes\\n\\n\\ndef main():\\n  n = int(input())\\n  nodes = Node.prepare_nodes(n)\\n\\n  dp = [0 for _ in range(n)]\\n  ans_list = [n*(n + 1) // 2 for _ in range(n)]\\n\\n  def update_ans(s, color):\\n    ans_list[color] -= (s + dp[color]) * (s + dp[color] + 1) // 2\\n\\n  def dfs(s, node, parent):\\n    color = node.color\\n    pre_s = s + dp[color]\\n    for child in node.children:\\n      if child == parent: continue\\n\\n      dp[color] = -s\\n      s = dfs(s, child, node)\\n      update_ans(s, color)\\n\\n    s += 1\\n    dp[color] = pre_s - s\\n\\n    return s\\n\\n  s = dfs(0, nodes[0], Node(-1))\\n\\n  for i in range(n):\\n    update_ans(s, i)\\n    print((ans_list[i]))\\n\\n\\nmain()\\n\", \"def main():\\n    import sys\\n    sys.setrecursionlimit(10**9)\\n    input = sys.stdin.readline\\n\\n    N = int(input())\\n    C = [c-1 for c in map(int, input().split())]\\n    tree = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = map(int, input().split())\\n        tree[a-1].append(b-1)\\n        tree[b-1].append(a-1)\\n\\n    # 1\\u56de\\u4ee5\\u4e0a > \\u5168\\u4f53 - 1\\u56de\\u3082\\u901a\\u3089\\u306a\\u3044\\n    # \\u9078\\u3079\\u308b\\u6570 = n(n+1)//2\\n    size_subtree = [1] * N\\n    num_v = [0] * N\\n    ans = [0] * N\\n\\n    def dfs(v_now, v_parent, color_parent):\\n        color_now = C[v_now]\\n        num_v_now = num_v[color_now]\\n        num_v_parent = num_v[color_parent]\\n        for v2 in tree[v_now]:\\n            if v2 == v_parent: continue\\n            dfs(v2, v_now, color_now)\\n            size_subtree[v_now] += size_subtree[v2]\\n        num_v[color_now] = num_v_now + size_subtree[v_now]\\n        if color_parent != -1:\\n            k = size_subtree[v_now] - (num_v[color_parent]-num_v_parent)\\n            ans[color_parent] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for color in range(N):\\n        if color == C[0]: continue\\n        k = N - num_v[color]\\n        ans[color] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    ans = [numAll-ans for ans in ans]\\n\\n    print(*ans, sep='\\\\n')\\n\\nmain()\\n\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append((t, w))\\n        self.E[t].append((f, w))\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\n# k=1, 2, ..., N\\u306b\\u5bfe\\u3057\\u3066\\n# \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u4ee5\\u4e0a\\u901a\\u308b\\u3088\\u3046\\u306a\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\u3092\\u6c42\\u3081\\u308b\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570 - \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u3082\\u901a\\u3089\\u306a\\u3044\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570\\u306fN*(N+1)/2\\n\\n# \\u30b0\\u30e9\\u30d5\\u3092\\u8272k\\u306e\\u30ce\\u30fc\\u30c9\\u3067\\u5206\\u5272\\u3057\\u3066\\u3001\\u90e8\\u5206\\u30b0\\u30e9\\u30d5\\u5185\\u3067\\u306e\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u7dcf\\u6570\\u3092\\u6c42\\u3081\\u308c\\u3070\\u826f\\u3044\\n# \\u5404\\u30ce\\u30fc\\u30c9\\u306b\\u72b6\\u614b\\u3068\\u3057\\u3066\\u8f9e\\u66f8\\u3092\\u3082\\u305f\\u305b\\u308b\\u3002\\n# x\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n# o\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u4e0d\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n\\n# \\u56de\\u7b54\\u7528\\nans = [0]*N\\n\\n\\ndef f(curr, par=-1):\\n    # \\u518d\\u5e30\\u95a2\\u6570\\n    # curr: \\u73fe\\u5728\\u306e\\u7bc0\\u70b9\\n    # par : \\u89aa\\u7bc0\\u70b9\\u306e\\u756a\\u53f7\\n    ret = defaultdict(int)\\n    size = 1\\n    for dest, w in g.E[curr]:\\n        if dest == par:\\n            continue\\n        sz, child = f(dest, curr)\\n        size += sz\\n\\n        # \\u81ea\\u8eab\\u306e\\u8272\\u3068\\u540c\\u3058\\u5834\\u5408\\u3001\\u5b50\\u306e\\u9802\\u70b9\\u306e\\u6570\\u304b\\u3089\\u52a0\\u7b97\\n        n = sz-child[c[curr]]\\n        ans[c[curr]] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        if len(ret) < len(child):\\n            child, ret = ret, child\\n        for key in child:\\n            ret[key] += child[key]\\n\\n    ret[c[curr]] = size\\n    return size, ret\\n\\n\\nsz, ret = f(0)\\nfor color in range(N):\\n    if color != c[0]:\\n        n = sz-ret[color]\\n        ans[color] += n*(n+1)//2\\n\\ntot = N*(N+1)//2\\nfor a in ans:\\n    print((tot-a))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        if IND[i] == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if IND[i] >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        if P[i] == X[i][IND[i]]:\\n            IND[i] += 1\\n            continue\\n        P[X[i][IND[i]]] = i\\n        IND[i], i = IND[i] + 1, X[i][IND[i]]\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [len(x) for x in X]\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if not ind:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind - 1]\\n        if P[i] == j:\\n            IND[i] -= 1\\n            continue\\n        \\n        P[j] = i\\n        IND[i] -= 1\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\nstdin = sys.stdin\\n\\nns = lambda: stdin.readline().rstrip()\\nni = lambda: int(stdin.readline().rstrip())\\nnm = lambda: map(int, stdin.readline().split())\\nnl = lambda: list(map(int, stdin.readline().split()))\\n\\nn = ni()\\ncolor = nl()\\ng = [list() for _ in range(n)]\\nfor _ in range(n-1):\\n    a, b = nm()\\n    g[a-1].append(b-1)\\n    g[b-1].append(a-1)\\n\\nsize = [1]*n\\npar = [-1]*n\\nh = [list() for _ in range(n+1)]\\nidx = [0]*n\\ndef dfs(v, p):\\n    h[color[v]].append(v)\\n    for x in g[v]:\\n        if x == p: continue\\n        par[x] = v\\n        size[v] += dfs(x, v)\\n        h[color[v]].append(v)\\n    h[color[v]].append(v)\\n    return size[v]\\ndfs(0, -1)\\n# print(size)\\nfor v in range(n):\\n    if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n        idx[v] += 1\\nacc = [0]*(n+10)\\nans = [n*(n+1)//2]*(n+1)\\nfor col in range(1, n+1):\\n    # print(h[col])\\n    acc[-1] = 0\\n    p = [-1]\\n    for v in h[col]:\\n        if p[-1] == v:\\n            p.pop()\\n            if idx[v] >= len(g[v]):\\n                acc[p[-1]] += size[v]\\n                continue\\n            q = size[g[v][idx[v]]] - acc[v]\\n            idx[v] += 1\\n            if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n              idx[v] += 1\\n            ans[col] -= q*(q+1)//2\\n            acc[v] = 0\\n            # print(v, q)\\n            p.append(v)\\n        else:\\n            p.append(v)\\n    q = n - acc[-1]\\n    # print(-1, q)\\n    ans[col] -= q*(q+1)//2\\nprint(*ans[1:], sep='\\\\n')\"]",
        "difficulty": "interview",
        "input": "5\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n",
        "output": "5\n8\n10\n5\n5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc163/tasks/abc163_f"
    },
    {
        "id": 693,
        "task_id": 2305,
        "test_case_id": 5,
        "question": "We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.\nAdditionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.\nFor each k=1, 2, ..., N, solve the following problem:\n - Find the number of simple paths that visit a vertex painted in the color k one or more times.\nNote: The simple paths from Vertex u to v and from v to u are not distinguished.\n\n-----Constraints-----\n - 1 \\leq N \\leq 2 \\times 10^5\n - 1 \\leq c_i \\leq N\n - 1 \\leq a_i,b_i \\leq N\n - The given graph is a tree.\n - All values in input are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN\nc_1 c_2 ... c_N\na_1 b_1\n:\na_{N-1} b_{N-1}\n\n-----Output-----\nPrint the answers for k = 1, 2, ..., N in order, each in its own line.\n\n-----Sample Input-----\n3\n1 2 1\n1 2\n2 3\n\n-----Sample Output-----\n5\n4\n0\n\nLet P_{i,j} denote the simple path connecting Vertex i and j.\nThere are 5 simple paths that visit a vertex painted in the color 1 one or more times:\nP_{1,1}\\,,\\,P_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,3}\\,,\\,P_{3,3} \nThere are 4 simple paths that visit a vertex painted in the color 2 one or more times:\nP_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,2}\\,,\\,P_{2,3} \nThere are no simple paths that visit a vertex painted in the color 3 one or more times.",
        "solutions": "[\"import sys\\nfrom collections import defaultdict\\n\\n# \\u518d\\u5e30\\u5236\\u9650\\u3092\\u7de9\\u548c\\u3059\\u308b\\u304a\\u307e\\u3058\\u306a\\u3044\\nsys.setrecursionlimit(10**6)\\n\\ndef sum(n):return n*(n+1)//2\\n\\n# \\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3068\\u3001\\u300c\\u8272i\\u3092\\u5c01\\u9396\\u3057\\u305f\\u3068\\u304d\\u306b\\u5230\\u9054\\u3067\\u304d\\u306a\\u3044\\u9802\\u70b9\\u306e\\u500b\\u6570\\u300d\\u3092\\u6301\\u3064\\u8f9e\\u66f8\\u3092\\u8fd4\\u3059\\ndef dfs(v,p):\\n    ret=defaultdict(int)\\n    size=1\\n    for vv in g[v]:\\n        if vv==p:\\n            continue\\n        ss,d=dfs(vv,v)\\n        size+=ss\\n        ans[c[v]]+=sum(ss-d[c[v]])\\n        \\n        # \\u30de\\u30fc\\u30b8\\u30c6\\u30af\\n        if len(ret)<len(d):\\n            ret,d=d,ret\\n        for vvv in d:\\n            ret[vvv]+=d[vvv]\\n    ret[c[v]]=size\\n    return size,ret\\n\\nn=int(input())\\nc=list(map(int,input().split()))\\ng=[[] for _ in range(n+1)]\\nans=[0]*(n+1)\\nfor _ in range(n-1):\\n    s,t=list(map(int, input().split()))\\n    s-=1\\n    t-=1\\n    g[s].append(t)\\n    g[t].append(s)\\n_,ret=dfs(0,-1)\\nfor i in range(1,n+1):\\n    print((sum(n)-ans[i]-sum(n-ret[i])))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nc = list(map(int, input().split()))\\nans = [n*(n+1)//2 for _ in range(n)]\\nfor i in range(n):\\n\\tc[i] -= 1\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n\\ta, b = map(int, input().split())\\n\\tadj[a-1].append(b-1)\\n\\tadj[b-1].append(a-1)\\n\\ns = 0\\ndp = [0 for _ in range(n)]\\n\\ndef dfs(x, p):\\n\\tnonlocal s\\n\\tpres = s + dp[c[x]]\\n\\tfor v in adj[x]:\\n\\t\\tif v == p:\\n\\t\\t\\tcontinue\\n\\t\\tdp[c[x]] = -s\\n\\t\\tdfs(v, x)\\n\\t\\tans[c[x]] -= (s+dp[c[x]]) * (s+dp[c[x]]+1) // 2\\n\\ts += 1\\n\\tdp[c[x]] = pres - s\\n\\treturn\\n\\ndfs(0, -1)\\nfor i in range(n):\\n\\tans[i] -= (s+dp[i]) * (s+dp[i]+1) // 2\\n\\nprint(*ans, sep=\\\"\\\\n\\\")\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append(t)\\n        self.E[t].append(f)\\n\\n\\ndef make_order(g, v):\\n    seen = [False]*g.N\\n    last_order = [-1]*g.N\\n    parent = [-1]*g.N\\n    size = [1]*g.N\\n    counter = {\\\"last\\\": 0}\\n\\n    def recur(v):\\n        seen[v] = True\\n        for to in g.E[v]:\\n            if seen[to] == True:\\n                continue\\n            parent[to] = v\\n            recur(to)\\n            size[v] += size[to]\\n\\n        last_order[counter[\\\"last\\\"]] = v\\n        counter[\\\"last\\\"] += 1\\n\\n    recur(v)\\n    return last_order, parent, size\\n\\n\\ndef merge(d: dict, e: dict):\\n    if len(d) < len(e):\\n        d, e = e, d\\n    for k in e:\\n        d[k] += e[k]\\n    return d\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\nans = [0]*N\\n\\nret = {}\\nlast_order, parent, size = make_order(g, 0)\\nfor curr in last_order:\\n    cn = c[curr]\\n    rrr = defaultdict(int)\\n    for dest in g.E[curr]:\\n        if dest == parent[curr]:\\n            continue\\n        child = ret.pop(dest)\\n\\n        n = size[dest]-child[cn]\\n        ans[cn] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        rrr = merge(rrr, child)\\n\\n    rrr[cn] = size[curr]\\n    ret[curr] = rrr\\n\\n\\ntot = N*(N+1)//2\\nfor color in range(N):\\n    if color != c[0]:\\n        n = N-ret[0][color]\\n        ans[color] += n*(n+1)//2\\n    print((tot-ans[color]))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\n \\nN = int(input())  # \\u9802\\u70b9\\u6570\\nC = [int(x) - 1  for x in input().split()]  # \\u9802\\u70b9\\u306e\\u8272\\u756a\\u53f7 (1~N) -> (0~N-1)\\n \\nE = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    E[a-1].append(b-1)\\n    E[b-1].append(a-1)\\n    \\ndef dfs(u, p = -1):\\n \\n    color = C[u]  # \\u8272\\u756a\\u53f7\\n    VC[color].append(u)  # \\u8272\\u3054\\u3068\\u306b stack \\u3057\\u3066\\u304a\\u304f\\n    \\n    s = 1  # \\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\uff09\\n    for v in E[u]:\\n        if v == p:\\n            continue\\n            \\n        child[u] = 0  # v \\u306e\\u65b9\\u5411\\u306b\\u3042\\u308b\\u5b50\\u5b6b\\u306e\\u3046\\u3061\\u8fbf\\u308c\\u308b\\u9802\\u70b9\\u306e\\u6570 = s - \\u5b50\\u5b6b\\u306b\\u3042\\u308b\\u3001\\u540c\\u8272\\u3092\\u6839\\u3068\\u3059\\u308b\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u306e\\u5408\\u8a08\\n        ret = dfs(v, u)\\n \\n        s += ret  # \\u7d14\\u7c8b\\u306a\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3092\\u8a08\\u7b97\\n        child[u] += ret  # \\u540c\\u8272\\u306e\\u9802\\u70b9\\u304c\\u5207\\u308a\\u96e2\\u3055\\u308c\\u305f\\u3042\\u3068\\u306e\\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\n        \\n        ans[color] -= child[u] * (child[u] + 1) // 2        \\n \\n    VC[color].pop()\\n    if VC[color]: # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u3042\\u308b\\u5834\\u5408\\n        child[VC[color][-1]] -= s  # \\u4e00\\u756a\\u8fd1\\u3044\\u540c\\u8272\\u304b\\u3089\\u3001\\u81ea\\u5206\\u306e\\u90e8\\u5206\\u6728\\u306e\\u5927\\u304d\\u3055\\u3092\\u3072\\u304f\\n    else:  # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u306a\\u3044\\u5834\\u5408\\n        root_size[color] -= s  # \\u4e0a\\u4f4d\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\n        \\n    return s\\n\\nchild = [0] * N  # \\u8fbf\\u3063\\u3066\\u3044\\u308b\\u5b50\\u306e\\u4e2d\\u3067\\u306e\\u306e\\u6b8b\\u308a\\u6570\\u3000\\u3068\\u308a\\u3042\\u3048\\u305a 0\\u3067\\u521d\\u671f\\u5316\\nroot_size = [N] * N  # \\u6839\\u65b9\\u5411\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\u3002\\u64cd\\u4f5c\\u524d\\u306f N \\u5168\\u90e8\\uff09\\nVC = [ [] for _ in range(N)]  # \\u8272\\u756a\\u53f7\\u3054\\u3068\\nans = [N*(N+1)//2] * N  # \\u8272\\u3054\\u3068\\u306e\\uff08\\u6b8b\\u308a\\u306e\\uff09\\u7d44\\u307f\\u5408\\u308f\\u305b\\u6570\\u3000\\u521d\\u671f\\u5024\\u306f \\u5168\\u70b9\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\n \\ndfs(0)\\n \\nfor i in range(N):\\n    print(ans[i] - root_size[i] * (root_size[i] + 1) // 2)\", \"import sys\\n\\ninput=sys.stdin.readline\\nsys.setrecursionlimit(10**7)\\n\\nN=int(input())\\nc=list(map(int,input().split()))\\nfor i in range(N):\\n    c[i]-=1\\nedge=[[] for i in range(N)]\\nfor i in range(N-1):\\n    a,b=map(int,input().split())\\n    edge[a-1].append(b-1)\\n    edge[b-1].append(a-1)\\n\\nsubtree=[1]*N\\ndef size(v,pv):\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            size(nv,v)\\n            subtree[v]+=subtree[nv]\\n\\nsize(0,-1)\\n\\ndata=[[] for i in range(N)]\\nParent=[[0] for i in range(N)]\\nparent=[0]*N\\ndef dfs(v,pv,nop):\\n    pp=parent[nop]\\n    parent[nop]=v\\n    Parent[nop].append(v)\\n    data[c[v]].append((parent[c[v]],v))\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            dfs(nv,v,c[v])\\n    parent[nop]=pp\\n\\ndfs(0,-1,0)\\n\\nfor i in range(N):\\n    dic={v:subtree[v] for v in Parent[i]}\\n    for p,ch in data[i]:\\n        dic[p]-=subtree[ch]\\n\\n    res=N*(N+1)//2\\n    for p in dic:\\n        res-=dic[p]*(dic[p]+1)//2\\n    print(res)\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nhead = [-1] * (N + 1)\\nto = [0] * (N - 1 << 1)\\nnxt = [0] * (N - 1 << 1)\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    x, y = x-1, y-1\\n    nxt[i] = head[x]\\n    to[i] = y\\n    head[x] = i\\n    j = i + N - 1\\n    nxt[j] = head[y]\\n    to[j] = x\\n    head[y] = j\\n\\ndef EulerTour(n, i=0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    ANS = [f(n)] * n\\n    \\n    P = [-1] * n\\n    ct = 0\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    while i >= 0:\\n        e = head[i]\\n        if e < 0:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = to[e]\\n        if P[i] == j:\\n            head[i] = nxt[e]\\n            continue\\n        \\n        P[j] = i\\n        head[i] = nxt[e]\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    done = [0] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            continue\\n        done[i] = 1\\n        if i: ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n        for a in X[i]:\\n            if done[a]: continue\\n            P[a] = i\\n            Q.append(~a)\\n            Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\nn = int(input())\\nc = list(map(lambda x: int(x) - 1, input().split()))\\n\\ngraph = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    graph[a].append(b)\\n    graph[b].append(a)\\n\\nans = [n * (n + 1) // 2] * n\\nn_subtree = [1] * n  # n_st[i] = (\\u9802\\u70b9i\\u306e\\u90e8\\u5206\\u6728\\u306e\\u9802\\u70b9\\u6570)\\ncnt = [0] * n  # cnt[i] = (\\u8a2a\\u554f\\u6e08\\u307f\\u9802\\u70b9\\u306e\\u3046\\u3061\\u3001\\u8272i\\u3092\\u901a\\u904e\\u3057\\u306a\\u3044\\u3068\\u305f\\u3069\\u308a\\u7740\\u3051\\u306a\\u3044\\u9802\\u70b9\\u6570)\\nn_visited = 0\\n\\n\\ndef dfs(v, v_p):\\n    nonlocal n_visited\\n    cnt_v_before = cnt[c[v]]\\n    for v_next in graph[v]:\\n        if v_next == v_p:\\n            continue\\n\\n        m_prev = n_visited - cnt[c[v]]\\n        dfs(v_next, v)\\n        n_subtree[v] += n_subtree[v_next]\\n        m_next = n_visited - cnt[c[v]]\\n\\n        m = m_next - m_prev\\n        ans[c[v]] -= m * (m + 1) // 2\\n\\n    cnt[c[v]] = cnt_v_before + n_subtree[v]\\n    n_visited += 1\\n\\n\\ndfs(0, -1)\\n\\nfor i in range(n):\\n    m = n - cnt[i]\\n    ans[i] -= m * (m + 1) // 2\\n\\nprint(*ans, sep='\\\\n')\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\n# ci = [set() for _ in range(n)]\\n# for i,c in enumerate(C):\\n#     ci[c-1].add(i)\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = list(map(int,input().split()))\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n\\nans = [n*(n+1)//2]*n\\ncparent = [[] for _ in range(n)]\\nroot_size = [n]*n\\n# index:color\\nreached = [False]*n\\nsize = [0]*n\\n# index:vertex\\n\\ndef dfs(p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        if reached[nxt]:continue\\n        reached[nxt] = True\\n        size[p] = 0\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] += ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    # size[p] += s\\n    if cparent[c]:\\n        size[cparent[c][-1]] -= s\\n    else:\\n        root_size[c] -= s\\n    # ans[c] -= size[p] * (size[p] - 1)//2\\n    return s\\n\\nreached = [False]*n\\nreached[0] = True\\n\\ndfs(0)\\nfor i in range(n):\\n    print((ans[i] - root_size[i]*(root_size[i]+1)//2))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if ind == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if ind >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind]\\n        if P[i] == j:\\n            IND[i] += 1\\n            continue\\n        P[j] = i\\n        IND[i] += 1\\n        i = j\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n    \\nans = [n*(n+1)//2]*n\\nsize = [0]*n\\ncparent = [[] for _ in range(n)]\\nreached = [False]*n\\nroot_size = [n]*n\\n\\ndef dfs (p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        #print (nxt)\\n        \\n        if reached[nxt]:\\n            continue\\n        size[p] = 0\\n        reached[nxt] = True\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] +=ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    if cparent[c]:\\n        size[cparent[c][-1]] -=s\\n    else:\\n        root_size[c] -=s    \\n    return s\\n\\nreached[0] = True\\ndfs (0)\\n\\nfor i in range(n):\\n    print (ans[i]-root_size[i]*(root_size[i]+1)//2)\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef main():\\n    n=int(input())\\n    color=list(map(int,input().split()))\\n    nodes=[[]for i in range(n)]\\n    cut=[[0]for i in range(n+1)]\\n    ans=[0]*(n+1)\\n    for i in range(n-1):\\n        a,b=map(int,input().split())\\n        nodes[a-1].append(b-1)\\n        nodes[b-1].append(a-1)\\n    def num(p,parent):\\n        s=0\\n        for child in nodes[p]:\\n            if child==parent:\\n                continue\\n            cut[color[p]].append(0)\\n            nc=num(child,p)\\n            group=nc-cut[color[p]].pop()\\n            ans[color[p]]+=group*(group+1)//2\\n            s+=nc\\n        s+=1\\n        cut[color[p]][-1]+=s\\n        return s\\n    num(0,-1)\\n    for a,c in zip(ans[1:],cut[1:]):\\n        group=n-c[0]\\n        c=group*(group+1)//2\\n        print(n*(n+1)//2-c-a)\\nmain()\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                p = P[i]\\n                k = ET2[i] - ET1[i] + 1 - USED[C[p]] + ORG[i]\\n                ANS[C[p]] -= f(k)\\n                TMP[p] += k\\n            continue\\n        if i >= 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            if ET1[i] == 0: ET1[i] = ct\\n        for a in X[i][::-1]:\\n            if a != P[i]:\\n                P[a] = i\\n                for k in range(len(X[a])):\\n                    if X[a][k] == i:\\n                        del X[a][k]\\n                        break\\n                Q.append(~a)\\n                Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\ndef solve():\\n    N = int(input())\\n    Cs = [A-1 for A in map(int, input().split())]\\n    adjL = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = list(map(int, input().split()))\\n        a, b = a-1, b-1\\n        adjL[a].append(b)\\n        adjL[b].append(a)\\n\\n    sizeSubtrees = [1] * N\\n    numVs = [0] * N\\n    anss = [0] * N\\n\\n    def dfs(vNow, vPar, clrPar):\\n        clrNow = Cs[vNow]\\n        numVNow = numVs[clrNow]\\n        numVPar = numVs[clrPar]\\n        for v2 in adjL[vNow]:\\n            if v2 == vPar: continue\\n            dfs(v2, vNow, clrNow)\\n            sizeSubtrees[vNow] += sizeSubtrees[v2]\\n        numVs[clrNow] = numVNow + sizeSubtrees[vNow]\\n        if clrPar != -1:\\n            k = sizeSubtrees[vNow] - (numVs[clrPar]-numVPar)\\n            anss[clrPar] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for clr in range(N):\\n        if clr == Cs[0]: continue\\n        k = N - numVs[clr]\\n        anss[clr] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    anss = [numAll-ans for ans in anss]\\n\\n    print(('\\\\n'.join(map(str, anss))))\\n\\n\\nsolve()\\n\", \"import sys\\nimport itertools\\n\\nsys.setrecursionlimit(3 * 10**5)\\n\\n\\nclass Node:\\n  def __init__(self, color):\\n    self.color = int(color) - 1\\n    self.children = []\\n\\n  def add_child(self, node):\\n    self.children.append(node)\\n\\n  @staticmethod\\n  def prepare_nodes(n):\\n    nodes = tuple(map(Node, input().split()))\\n    for _ in range(n - 1):\\n      a, b = [int(x) - 1 for x in input().split()]\\n      nodes[a].add_child(nodes[b])\\n      nodes[b].add_child(nodes[a])\\n\\n    return nodes\\n\\n\\ndef main():\\n  n = int(input())\\n  nodes = Node.prepare_nodes(n)\\n\\n  dp = [0 for _ in range(n)]\\n  ans_list = [n*(n + 1) // 2 for _ in range(n)]\\n\\n  def update_ans(s, color):\\n    ans_list[color] -= (s + dp[color]) * (s + dp[color] + 1) // 2\\n\\n  def dfs(s, node, parent):\\n    color = node.color\\n    pre_s = s + dp[color]\\n    for child in node.children:\\n      if child == parent: continue\\n\\n      dp[color] = -s\\n      s = dfs(s, child, node)\\n      update_ans(s, color)\\n\\n    s += 1\\n    dp[color] = pre_s - s\\n\\n    return s\\n\\n  s = dfs(0, nodes[0], Node(-1))\\n\\n  for i in range(n):\\n    update_ans(s, i)\\n    print((ans_list[i]))\\n\\n\\nmain()\\n\", \"def main():\\n    import sys\\n    sys.setrecursionlimit(10**9)\\n    input = sys.stdin.readline\\n\\n    N = int(input())\\n    C = [c-1 for c in map(int, input().split())]\\n    tree = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = map(int, input().split())\\n        tree[a-1].append(b-1)\\n        tree[b-1].append(a-1)\\n\\n    # 1\\u56de\\u4ee5\\u4e0a > \\u5168\\u4f53 - 1\\u56de\\u3082\\u901a\\u3089\\u306a\\u3044\\n    # \\u9078\\u3079\\u308b\\u6570 = n(n+1)//2\\n    size_subtree = [1] * N\\n    num_v = [0] * N\\n    ans = [0] * N\\n\\n    def dfs(v_now, v_parent, color_parent):\\n        color_now = C[v_now]\\n        num_v_now = num_v[color_now]\\n        num_v_parent = num_v[color_parent]\\n        for v2 in tree[v_now]:\\n            if v2 == v_parent: continue\\n            dfs(v2, v_now, color_now)\\n            size_subtree[v_now] += size_subtree[v2]\\n        num_v[color_now] = num_v_now + size_subtree[v_now]\\n        if color_parent != -1:\\n            k = size_subtree[v_now] - (num_v[color_parent]-num_v_parent)\\n            ans[color_parent] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for color in range(N):\\n        if color == C[0]: continue\\n        k = N - num_v[color]\\n        ans[color] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    ans = [numAll-ans for ans in ans]\\n\\n    print(*ans, sep='\\\\n')\\n\\nmain()\\n\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append((t, w))\\n        self.E[t].append((f, w))\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\n# k=1, 2, ..., N\\u306b\\u5bfe\\u3057\\u3066\\n# \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u4ee5\\u4e0a\\u901a\\u308b\\u3088\\u3046\\u306a\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\u3092\\u6c42\\u3081\\u308b\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570 - \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u3082\\u901a\\u3089\\u306a\\u3044\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570\\u306fN*(N+1)/2\\n\\n# \\u30b0\\u30e9\\u30d5\\u3092\\u8272k\\u306e\\u30ce\\u30fc\\u30c9\\u3067\\u5206\\u5272\\u3057\\u3066\\u3001\\u90e8\\u5206\\u30b0\\u30e9\\u30d5\\u5185\\u3067\\u306e\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u7dcf\\u6570\\u3092\\u6c42\\u3081\\u308c\\u3070\\u826f\\u3044\\n# \\u5404\\u30ce\\u30fc\\u30c9\\u306b\\u72b6\\u614b\\u3068\\u3057\\u3066\\u8f9e\\u66f8\\u3092\\u3082\\u305f\\u305b\\u308b\\u3002\\n# x\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n# o\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u4e0d\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n\\n# \\u56de\\u7b54\\u7528\\nans = [0]*N\\n\\n\\ndef f(curr, par=-1):\\n    # \\u518d\\u5e30\\u95a2\\u6570\\n    # curr: \\u73fe\\u5728\\u306e\\u7bc0\\u70b9\\n    # par : \\u89aa\\u7bc0\\u70b9\\u306e\\u756a\\u53f7\\n    ret = defaultdict(int)\\n    size = 1\\n    for dest, w in g.E[curr]:\\n        if dest == par:\\n            continue\\n        sz, child = f(dest, curr)\\n        size += sz\\n\\n        # \\u81ea\\u8eab\\u306e\\u8272\\u3068\\u540c\\u3058\\u5834\\u5408\\u3001\\u5b50\\u306e\\u9802\\u70b9\\u306e\\u6570\\u304b\\u3089\\u52a0\\u7b97\\n        n = sz-child[c[curr]]\\n        ans[c[curr]] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        if len(ret) < len(child):\\n            child, ret = ret, child\\n        for key in child:\\n            ret[key] += child[key]\\n\\n    ret[c[curr]] = size\\n    return size, ret\\n\\n\\nsz, ret = f(0)\\nfor color in range(N):\\n    if color != c[0]:\\n        n = sz-ret[color]\\n        ans[color] += n*(n+1)//2\\n\\ntot = N*(N+1)//2\\nfor a in ans:\\n    print((tot-a))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        if IND[i] == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if IND[i] >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        if P[i] == X[i][IND[i]]:\\n            IND[i] += 1\\n            continue\\n        P[X[i][IND[i]]] = i\\n        IND[i], i = IND[i] + 1, X[i][IND[i]]\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [len(x) for x in X]\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if not ind:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind - 1]\\n        if P[i] == j:\\n            IND[i] -= 1\\n            continue\\n        \\n        P[j] = i\\n        IND[i] -= 1\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\nstdin = sys.stdin\\n\\nns = lambda: stdin.readline().rstrip()\\nni = lambda: int(stdin.readline().rstrip())\\nnm = lambda: map(int, stdin.readline().split())\\nnl = lambda: list(map(int, stdin.readline().split()))\\n\\nn = ni()\\ncolor = nl()\\ng = [list() for _ in range(n)]\\nfor _ in range(n-1):\\n    a, b = nm()\\n    g[a-1].append(b-1)\\n    g[b-1].append(a-1)\\n\\nsize = [1]*n\\npar = [-1]*n\\nh = [list() for _ in range(n+1)]\\nidx = [0]*n\\ndef dfs(v, p):\\n    h[color[v]].append(v)\\n    for x in g[v]:\\n        if x == p: continue\\n        par[x] = v\\n        size[v] += dfs(x, v)\\n        h[color[v]].append(v)\\n    h[color[v]].append(v)\\n    return size[v]\\ndfs(0, -1)\\n# print(size)\\nfor v in range(n):\\n    if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n        idx[v] += 1\\nacc = [0]*(n+10)\\nans = [n*(n+1)//2]*(n+1)\\nfor col in range(1, n+1):\\n    # print(h[col])\\n    acc[-1] = 0\\n    p = [-1]\\n    for v in h[col]:\\n        if p[-1] == v:\\n            p.pop()\\n            if idx[v] >= len(g[v]):\\n                acc[p[-1]] += size[v]\\n                continue\\n            q = size[g[v][idx[v]]] - acc[v]\\n            idx[v] += 1\\n            if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n              idx[v] += 1\\n            ans[col] -= q*(q+1)//2\\n            acc[v] = 0\\n            # print(v, q)\\n            p.append(v)\\n        else:\\n            p.append(v)\\n    q = n - acc[-1]\\n    # print(-1, q)\\n    ans[col] -= q*(q+1)//2\\nprint(*ans[1:], sep='\\\\n')\"]",
        "difficulty": "interview",
        "input": "8\n2 7 2 5 4 1 7 5\n3 1\n1 2\n2 7\n4 5\n5 6\n6 8\n7 8\n",
        "output": "18\n15\n0\n14\n23\n0\n23\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc163/tasks/abc163_f"
    },
    {
        "id": 694,
        "task_id": 3168,
        "test_case_id": 1,
        "question": "A binary search tree is a tree in which every node has at most two children nodes (a left and a right child). Each node has an integer written inside it. If the number $X$ is written inside a node, then the numbers in its left subtree are less than $X$ and the numbers in its right subtree are greater than X. You will be given a sequence of integers between 1 and $N$ (inclusive) such that each number appears in the sequence exactly once. You are to create a binary search tree from the sequence, putting the first number in the root node and inserting every other number in order.\n\nWhen inserting a new number $Y$ in a tree, you first traverse the tree as if you were searching for $Y$. When you arrive at a node $Z$ such that you can’t continue searching for $Y$, you put $Y$ as a left or right son of $Z$ depending on if $Z>Y$ or $Z<Y$, so that after the insertion the tree will still be a binary search tree. After the insertion you add the depth of $Y$ to a counter $C$ and print the value of $C$. The counter $C$ is set to $0$ in the beginning.\n\n-----Input-----\nThe first line contains the integer $N$ $(1 \\leq N \\leq 300000)$, the length of the sequence.\n\nThe remaining $N$ lines contain the numbers in the sequence, integers in the interval $[1, N]$. The numbers will be distinct.\n\n-----Output-----\nOutput $N$ integers each on its own line, the values of the counter $C$ after each number is inserted into the tree.\n\n-----Examples-----\nSample Input 1:\n4\n1\n2\n3\n4\nSample Output 1:\n0\n1\n3\n6\n\nSample Input 2:\n5\n3\n2\n4\n1\n5\nSample Output 2:\n0\n1\n2\n4\n6",
        "solutions": "",
        "difficulty": "competition",
        "input": "4\n1\n2\n3\n4\n",
        "output": "0\n1\n3\n6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/bst"
    },
    {
        "id": 695,
        "task_id": 3168,
        "test_case_id": 2,
        "question": "A binary search tree is a tree in which every node has at most two children nodes (a left and a right child). Each node has an integer written inside it. If the number $X$ is written inside a node, then the numbers in its left subtree are less than $X$ and the numbers in its right subtree are greater than X. You will be given a sequence of integers between 1 and $N$ (inclusive) such that each number appears in the sequence exactly once. You are to create a binary search tree from the sequence, putting the first number in the root node and inserting every other number in order.\n\nWhen inserting a new number $Y$ in a tree, you first traverse the tree as if you were searching for $Y$. When you arrive at a node $Z$ such that you can’t continue searching for $Y$, you put $Y$ as a left or right son of $Z$ depending on if $Z>Y$ or $Z<Y$, so that after the insertion the tree will still be a binary search tree. After the insertion you add the depth of $Y$ to a counter $C$ and print the value of $C$. The counter $C$ is set to $0$ in the beginning.\n\n-----Input-----\nThe first line contains the integer $N$ $(1 \\leq N \\leq 300000)$, the length of the sequence.\n\nThe remaining $N$ lines contain the numbers in the sequence, integers in the interval $[1, N]$. The numbers will be distinct.\n\n-----Output-----\nOutput $N$ integers each on its own line, the values of the counter $C$ after each number is inserted into the tree.\n\n-----Examples-----\nSample Input 1:\n4\n1\n2\n3\n4\nSample Output 1:\n0\n1\n3\n6\n\nSample Input 2:\n5\n3\n2\n4\n1\n5\nSample Output 2:\n0\n1\n2\n4\n6",
        "solutions": "",
        "difficulty": "competition",
        "input": "5\n3\n2\n4\n1\n5\n",
        "output": "0\n1\n2\n4\n6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/bst"
    },
    {
        "id": 696,
        "task_id": 3168,
        "test_case_id": 3,
        "question": "A binary search tree is a tree in which every node has at most two children nodes (a left and a right child). Each node has an integer written inside it. If the number $X$ is written inside a node, then the numbers in its left subtree are less than $X$ and the numbers in its right subtree are greater than X. You will be given a sequence of integers between 1 and $N$ (inclusive) such that each number appears in the sequence exactly once. You are to create a binary search tree from the sequence, putting the first number in the root node and inserting every other number in order.\n\nWhen inserting a new number $Y$ in a tree, you first traverse the tree as if you were searching for $Y$. When you arrive at a node $Z$ such that you can’t continue searching for $Y$, you put $Y$ as a left or right son of $Z$ depending on if $Z>Y$ or $Z<Y$, so that after the insertion the tree will still be a binary search tree. After the insertion you add the depth of $Y$ to a counter $C$ and print the value of $C$. The counter $C$ is set to $0$ in the beginning.\n\n-----Input-----\nThe first line contains the integer $N$ $(1 \\leq N \\leq 300000)$, the length of the sequence.\n\nThe remaining $N$ lines contain the numbers in the sequence, integers in the interval $[1, N]$. The numbers will be distinct.\n\n-----Output-----\nOutput $N$ integers each on its own line, the values of the counter $C$ after each number is inserted into the tree.\n\n-----Examples-----\nSample Input 1:\n4\n1\n2\n3\n4\nSample Output 1:\n0\n1\n3\n6\n\nSample Input 2:\n5\n3\n2\n4\n1\n5\nSample Output 2:\n0\n1\n2\n4\n6",
        "solutions": "",
        "difficulty": "competition",
        "input": "8\n3\n5\n1\n6\n8\n7\n2\n4\n",
        "output": "0\n1\n2\n4\n7\n11\n13\n15\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/bst"
    },
    {
        "id": 697,
        "task_id": 4000,
        "test_case_id": 1,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n",
        "output": "5\n1 8 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 698,
        "task_id": 4000,
        "test_case_id": 2,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n3 2\n3 5\n2 4\n1 3\n",
        "output": "4\n5 1 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 699,
        "task_id": 4000,
        "test_case_id": 3,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 2\n2 3\n3 4\n",
        "output": "3 \n1 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 700,
        "task_id": 4000,
        "test_case_id": 4,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 2\n1 3\n2 4\n2 5\n3 6\n",
        "output": "5\n5 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 701,
        "task_id": 4000,
        "test_case_id": 5,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 3\n3 1\n1 2\n",
        "output": "3 \n2 1 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 702,
        "task_id": 4000,
        "test_case_id": 6,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n1 3\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 703,
        "task_id": 4000,
        "test_case_id": 7,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n1 3\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 704,
        "task_id": 4000,
        "test_case_id": 8,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 1\n1 2\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 705,
        "task_id": 4000,
        "test_case_id": 9,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 3\n1 2\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 706,
        "task_id": 4000,
        "test_case_id": 10,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n1 3\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 707,
        "task_id": 4000,
        "test_case_id": 11,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n1 3\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 708,
        "task_id": 4000,
        "test_case_id": 12,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 1\n3 2\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 709,
        "task_id": 4000,
        "test_case_id": 13,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 3\n2 1\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 710,
        "task_id": 4000,
        "test_case_id": 14,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 3\n1 2\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 711,
        "task_id": 4000,
        "test_case_id": 15,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 1\n2 1\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 712,
        "task_id": 4000,
        "test_case_id": 16,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n1 3\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 713,
        "task_id": 4000,
        "test_case_id": 17,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 1\n3 1\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 714,
        "task_id": 4000,
        "test_case_id": 18,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n3 1\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 715,
        "task_id": 4000,
        "test_case_id": 19,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 3\n3 2\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 716,
        "task_id": 4000,
        "test_case_id": 20,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 3\n1 2\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 717,
        "task_id": 4000,
        "test_case_id": 21,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 2\n1 3\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 718,
        "task_id": 4000,
        "test_case_id": 22,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n3 1\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 719,
        "task_id": 4000,
        "test_case_id": 23,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n1 3\n",
        "output": "2 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 720,
        "task_id": 4000,
        "test_case_id": 24,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 2\n2 1\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 721,
        "task_id": 4000,
        "test_case_id": 25,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 2\n3 1\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 722,
        "task_id": 4000,
        "test_case_id": 26,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n2 3\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 723,
        "task_id": 4000,
        "test_case_id": 27,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 1\n3 2\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 724,
        "task_id": 4000,
        "test_case_id": 28,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n1 3\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 725,
        "task_id": 4000,
        "test_case_id": 29,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n3 2\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 726,
        "task_id": 4000,
        "test_case_id": 30,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 2\n3 1\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 727,
        "task_id": 4000,
        "test_case_id": 31,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 2\n1 2\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 728,
        "task_id": 4000,
        "test_case_id": 32,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n2 1\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 729,
        "task_id": 4000,
        "test_case_id": 33,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 2\n1 3\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 730,
        "task_id": 4000,
        "test_case_id": 34,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n3 1\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 731,
        "task_id": 4000,
        "test_case_id": 35,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n2 1\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 732,
        "task_id": 4000,
        "test_case_id": 36,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n1 2\n2 3\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 733,
        "task_id": 4000,
        "test_case_id": 37,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n3 1\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 734,
        "task_id": 4000,
        "test_case_id": 38,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n3 2\n3 1\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 735,
        "task_id": 4000,
        "test_case_id": 39,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n2 1\n",
        "output": "2 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 736,
        "task_id": 4000,
        "test_case_id": 40,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "3\n2 3\n3 1\n",
        "output": "2 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 737,
        "task_id": 4000,
        "test_case_id": 41,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 3\n1 2\n1 4\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 738,
        "task_id": 4000,
        "test_case_id": 42,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 3\n3 4\n2 3\n",
        "output": "3\n2 1 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 739,
        "task_id": 4000,
        "test_case_id": 43,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 2\n1 3\n1 4\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 740,
        "task_id": 4000,
        "test_case_id": 44,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 4\n1 3\n2 1\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 741,
        "task_id": 4000,
        "test_case_id": 45,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n1 2\n3 1\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 742,
        "task_id": 4000,
        "test_case_id": 46,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n3 1\n4 1\n2 1\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 743,
        "task_id": 4000,
        "test_case_id": 47,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n1 2\n1 3\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 744,
        "task_id": 4000,
        "test_case_id": 48,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 4\n3 1\n1 2\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 745,
        "task_id": 4000,
        "test_case_id": 49,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 3\n4 1\n3 2\n",
        "output": "3 \n4 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 746,
        "task_id": 4000,
        "test_case_id": 50,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 4\n3 4\n1 2\n",
        "output": "3 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 747,
        "task_id": 4000,
        "test_case_id": 51,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 1\n1 4\n2 3\n",
        "output": "3 \n4 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 748,
        "task_id": 4000,
        "test_case_id": 52,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n3 1\n1 2\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 749,
        "task_id": 4000,
        "test_case_id": 53,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 1\n3 1\n4 1\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 750,
        "task_id": 4000,
        "test_case_id": 54,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 3\n1 4\n3 2\n",
        "output": "3 \n4 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 751,
        "task_id": 4000,
        "test_case_id": 55,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n1 2\n3 1\n",
        "output": "3\n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 752,
        "task_id": 4000,
        "test_case_id": 56,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n4 2\n3 1\n",
        "output": "3 \n3 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 753,
        "task_id": 4000,
        "test_case_id": 57,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 4\n3 2\n3 4\n",
        "output": "3 \n1 4 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 754,
        "task_id": 4000,
        "test_case_id": 58,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n3 2\n1 2\n4 1\n",
        "output": "3 \n4 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 755,
        "task_id": 4000,
        "test_case_id": 59,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 1\n1 4\n2 3\n",
        "output": "3 \n4 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 756,
        "task_id": 4000,
        "test_case_id": 60,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n3 4\n2 4\n3 1\n",
        "output": "3 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 757,
        "task_id": 4000,
        "test_case_id": 61,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n1 4\n3 4\n2 4\n",
        "output": "3\n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 758,
        "task_id": 4000,
        "test_case_id": 62,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n3 1\n2 3\n2 4\n",
        "output": "3 \n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 759,
        "task_id": 4000,
        "test_case_id": 63,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 3\n1 2\n1 4\n",
        "output": "3 \n4 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 760,
        "task_id": 4000,
        "test_case_id": 64,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n2 3\n1 3\n",
        "output": "3 \n4 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 761,
        "task_id": 4000,
        "test_case_id": 65,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 1\n2 3\n4 3\n",
        "output": "3 \n1 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 762,
        "task_id": 4000,
        "test_case_id": 66,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n2 4\n1 3\n",
        "output": "3 \n3 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 763,
        "task_id": 4000,
        "test_case_id": 67,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 2\n1 3\n3 4\n",
        "output": "3 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 764,
        "task_id": 4000,
        "test_case_id": 68,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 3\n4 2\n1 3\n",
        "output": "3 \n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 765,
        "task_id": 4000,
        "test_case_id": 69,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 3\n1 4\n4 2\n",
        "output": "3 \n1 4 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 766,
        "task_id": 4000,
        "test_case_id": 70,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n3 4\n2 4\n1 4\n",
        "output": "3\n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 767,
        "task_id": 4000,
        "test_case_id": 71,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 3\n3 1\n2 4\n",
        "output": "3 \n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 768,
        "task_id": 4000,
        "test_case_id": 72,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 3\n4 1\n4 3\n",
        "output": "3 \n1 4 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 769,
        "task_id": 4000,
        "test_case_id": 73,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 3\n3 1\n4 2\n",
        "output": "3 \n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 770,
        "task_id": 4000,
        "test_case_id": 74,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n2 1\n4 3\n3 2\n",
        "output": "3 \n1 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 771,
        "task_id": 4000,
        "test_case_id": 75,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "4\n4 1\n4 2\n3 4\n",
        "output": "3\n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 772,
        "task_id": 4000,
        "test_case_id": 76,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 3\n3 2\n5 1\n1 4\n",
        "output": "4\n5 4 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 773,
        "task_id": 4000,
        "test_case_id": 77,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 3\n1 2\n4 1\n1 5\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 774,
        "task_id": 4000,
        "test_case_id": 78,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 5\n4 1\n1 2\n1 3\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 775,
        "task_id": 4000,
        "test_case_id": 79,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 2\n1 4\n5 1\n1 3\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 776,
        "task_id": 4000,
        "test_case_id": 80,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n2 1\n3 1\n4 1\n5 1\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 777,
        "task_id": 4000,
        "test_case_id": 81,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 1\n4 2\n4 1\n1 3\n",
        "output": "4\n5 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 778,
        "task_id": 4000,
        "test_case_id": 82,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n3 1\n2 1\n1 5\n4 1\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 779,
        "task_id": 4000,
        "test_case_id": 83,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 5\n3 1\n1 4\n2 1\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 780,
        "task_id": 4000,
        "test_case_id": 84,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n2 3\n1 5\n4 1\n3 1\n",
        "output": "4\n5 4 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 781,
        "task_id": 4000,
        "test_case_id": 85,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 2\n4 1\n3 1\n1 5\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 782,
        "task_id": 4000,
        "test_case_id": 86,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 4\n1 4\n1 2\n3 4\n",
        "output": "4\n2 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 783,
        "task_id": 4000,
        "test_case_id": 87,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 1\n5 4\n4 3\n1 2\n",
        "output": "4 \n2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 784,
        "task_id": 4000,
        "test_case_id": 88,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n2 4\n4 1\n1 3\n1 5\n",
        "output": "4\n5 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 785,
        "task_id": 4000,
        "test_case_id": 89,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n3 1\n2 1\n1 5\n1 4\n",
        "output": "3\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 786,
        "task_id": 4000,
        "test_case_id": 90,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 4\n2 1\n5 1\n5 3\n",
        "output": "4\n4 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 787,
        "task_id": 4000,
        "test_case_id": 91,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 5\n3 4\n2 3\n1 3\n",
        "output": "4\n5 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 788,
        "task_id": 4000,
        "test_case_id": 92,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n2 5\n5 3\n1 3\n2 4\n",
        "output": "4 \n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 789,
        "task_id": 4000,
        "test_case_id": 93,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n3 1\n5 1\n2 1\n4 2\n",
        "output": "4\n5 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 790,
        "task_id": 4000,
        "test_case_id": 94,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 3\n5 3\n2 1\n1 4\n",
        "output": "4\n4 2 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 791,
        "task_id": 4000,
        "test_case_id": 95,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n3 5\n3 1\n2 1\n4 3\n",
        "output": "4\n2 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 792,
        "task_id": 4000,
        "test_case_id": 96,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n2 3\n2 1\n1 5\n4 2\n",
        "output": "4\n5 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 793,
        "task_id": 4000,
        "test_case_id": 97,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 2\n4 5\n1 2\n3 5\n",
        "output": "4\n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 794,
        "task_id": 4000,
        "test_case_id": 98,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 3\n2 4\n1 2\n4 3\n",
        "output": "4 \n1 2 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 795,
        "task_id": 4000,
        "test_case_id": 99,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n4 1\n1 5\n3 5\n2 5\n",
        "output": "4\n4 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 796,
        "task_id": 4000,
        "test_case_id": 100,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 3\n4 2\n1 2\n2 5\n",
        "output": "4\n4 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 797,
        "task_id": 4000,
        "test_case_id": 101,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 1\n2 3\n5 4\n3 5\n",
        "output": "4\n4 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 798,
        "task_id": 4000,
        "test_case_id": 102,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n4 5\n1 5\n2 5\n2 3\n",
        "output": "4\n4 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 799,
        "task_id": 4000,
        "test_case_id": 103,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 1\n4 2\n5 4\n3 5\n",
        "output": "4\n3 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 800,
        "task_id": 4000,
        "test_case_id": 104,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n1 5\n2 3\n2 5\n3 4\n",
        "output": "4 \n1 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 801,
        "task_id": 4000,
        "test_case_id": 105,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n2 3\n4 1\n5 4\n5 2\n",
        "output": "4 \n1 4 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 802,
        "task_id": 4000,
        "test_case_id": 106,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 4\n2 3\n2 5\n1 3\n",
        "output": "4 \n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 803,
        "task_id": 4000,
        "test_case_id": 107,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 4\n5 2\n1 3\n4 3\n",
        "output": "4 \n1 3 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 804,
        "task_id": 4000,
        "test_case_id": 108,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n2 5\n2 1\n3 2\n4 3\n",
        "output": "4\n5 1 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 805,
        "task_id": 4000,
        "test_case_id": 109,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n4 5\n2 1\n3 2\n5 3\n",
        "output": "4 \n1 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 806,
        "task_id": 4000,
        "test_case_id": 110,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "5\n5 2\n4 3\n1 2\n2 4\n",
        "output": "4\n5 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 807,
        "task_id": 4000,
        "test_case_id": 111,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n4 2\n5 1\n6 1\n2 1\n1 3\n",
        "output": "4\n6 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 808,
        "task_id": 4000,
        "test_case_id": 112,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 1\n1 4\n1 6\n1 3\n5 1\n",
        "output": "3\n5 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 809,
        "task_id": 4000,
        "test_case_id": 113,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 1\n1 4\n2 3\n1 6\n1 3\n",
        "output": "4\n6 5 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 810,
        "task_id": 4000,
        "test_case_id": 114,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 5\n1 3\n6 1\n4 1\n1 2\n",
        "output": "3\n5 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 811,
        "task_id": 4000,
        "test_case_id": 115,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 2\n5 3\n3 4\n1 3\n6 3\n",
        "output": "4\n2 5 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 812,
        "task_id": 4000,
        "test_case_id": 116,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 4\n4 1\n4 3\n5 4\n1 6\n",
        "output": "4\n6 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 813,
        "task_id": 4000,
        "test_case_id": 117,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 1\n5 1\n1 4\n6 1\n1 3\n",
        "output": "3\n5 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 814,
        "task_id": 4000,
        "test_case_id": 118,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 3\n5 1\n2 4\n1 2\n1 6\n",
        "output": "4\n6 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 815,
        "task_id": 4000,
        "test_case_id": 119,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 5\n1 6\n4 1\n1 2\n3 1\n",
        "output": "3\n5 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 816,
        "task_id": 4000,
        "test_case_id": 120,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 6\n4 1\n3 1\n6 1\n2 6\n",
        "output": "4\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 817,
        "task_id": 4000,
        "test_case_id": 121,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n6 1\n5 4\n1 3\n1 5\n5 2\n",
        "output": "4\n6 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 818,
        "task_id": 4000,
        "test_case_id": 122,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 6\n6 3\n4 3\n5 6\n2 1\n",
        "output": "5\n2 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 819,
        "task_id": 4000,
        "test_case_id": 123,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 3\n1 6\n1 3\n1 4\n5 6\n",
        "output": "5\n2 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 820,
        "task_id": 4000,
        "test_case_id": 124,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 1\n3 1\n4 1\n1 6\n2 6\n",
        "output": "4\n5 4 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 821,
        "task_id": 4000,
        "test_case_id": 125,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 1\n1 6\n3 1\n4 2\n2 5\n",
        "output": "4\n6 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 822,
        "task_id": 4000,
        "test_case_id": 126,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 3\n1 6\n6 4\n5 1\n2 5\n",
        "output": "5\n2 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 823,
        "task_id": 4000,
        "test_case_id": 127,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 5\n4 1\n2 1\n1 6\n3 4\n",
        "output": "5\n3 6 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 824,
        "task_id": 4000,
        "test_case_id": 128,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 4\n6 1\n5 1\n6 2\n3 6\n",
        "output": "5\n5 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 825,
        "task_id": 4000,
        "test_case_id": 129,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 5\n1 6\n6 2\n1 4\n4 3\n",
        "output": "5\n2 5 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 826,
        "task_id": 4000,
        "test_case_id": 130,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 4\n5 3\n3 1\n2 5\n1 6\n",
        "output": "5\n6 4 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 827,
        "task_id": 4000,
        "test_case_id": 131,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 3\n5 4\n2 1\n2 4\n6 4\n",
        "output": "5\n1 6 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 828,
        "task_id": 4000,
        "test_case_id": 132,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n4 3\n5 6\n6 1\n5 2\n6 3\n",
        "output": "5\n2 1 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 829,
        "task_id": 4000,
        "test_case_id": 133,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 3\n3 1\n5 6\n4 5\n2 3\n",
        "output": "4\n2 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 830,
        "task_id": 4000,
        "test_case_id": 134,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n4 3\n4 2\n2 1\n2 6\n5 4\n",
        "output": "4\n6 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 831,
        "task_id": 4000,
        "test_case_id": 135,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n3 6\n6 2\n6 5\n2 1\n6 4\n",
        "output": "4\n1 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 832,
        "task_id": 4000,
        "test_case_id": 136,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 4\n5 3\n2 4\n5 6\n2 1\n",
        "output": "5\n1 3 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 833,
        "task_id": 4000,
        "test_case_id": 137,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n6 3\n4 2\n6 5\n4 1\n2 5\n",
        "output": "5 \n1 4 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 834,
        "task_id": 4000,
        "test_case_id": 138,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 3\n4 6\n6 2\n1 2\n2 3\n",
        "output": "5\n4 1 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 835,
        "task_id": 4000,
        "test_case_id": 139,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 4\n6 2\n2 5\n3 5\n2 4\n",
        "output": "5\n1 6 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 836,
        "task_id": 4000,
        "test_case_id": 140,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n2 4\n3 6\n5 1\n3 5\n6 2\n",
        "output": "5 \n1 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 837,
        "task_id": 4000,
        "test_case_id": 141,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n5 4\n6 2\n1 5\n4 6\n5 3\n",
        "output": "5\n3 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 838,
        "task_id": 4000,
        "test_case_id": 142,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n6 2\n3 5\n4 2\n5 4\n1 2\n",
        "output": "5\n6 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 839,
        "task_id": 4000,
        "test_case_id": 143,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n1 4\n2 3\n5 6\n5 3\n2 4\n",
        "output": "5 \n1 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 840,
        "task_id": 4000,
        "test_case_id": 144,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n6 4\n4 1\n5 6\n3 2\n4 2\n",
        "output": "5\n3 1 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 841,
        "task_id": 4000,
        "test_case_id": 145,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "6\n6 1\n5 6\n5 3\n2 5\n2 4\n",
        "output": "5\n1 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 842,
        "task_id": 4000,
        "test_case_id": 146,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 7\n6 1\n5 1\n2 1\n3 2\n2 4\n",
        "output": "4\n7 6 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 843,
        "task_id": 4000,
        "test_case_id": 147,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n3 1\n6 1\n4 1\n1 5\n5 2\n1 7\n",
        "output": "4\n7 6 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 844,
        "task_id": 4000,
        "test_case_id": 148,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n7 1\n5 1\n1 3\n1 2\n6 1\n4 7\n",
        "output": "4\n6 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 845,
        "task_id": 4000,
        "test_case_id": 149,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 7\n2 1\n1 5\n4 1\n3 1\n5 6\n",
        "output": "4\n7 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 846,
        "task_id": 4000,
        "test_case_id": 150,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n3 1\n1 5\n1 7\n1 4\n6 1\n2 1\n",
        "output": "3\n6 5 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 847,
        "task_id": 4000,
        "test_case_id": 151,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 4\n7 4\n5 1\n3 2\n6 4\n3 1\n",
        "output": "5\n2 6 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 848,
        "task_id": 4000,
        "test_case_id": 152,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 3\n6 1\n1 7\n1 4\n5 4\n1 2\n",
        "output": "4\n7 6 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 849,
        "task_id": 4000,
        "test_case_id": 153,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n5 1\n6 1\n2 1\n1 3\n1 7\n1 4\n",
        "output": "3\n6 5 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 850,
        "task_id": 4000,
        "test_case_id": 154,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n5 1\n5 7\n1 2\n5 6\n3 1\n4 5\n",
        "output": "4\n3 6 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 851,
        "task_id": 4000,
        "test_case_id": 155,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 4\n6 1\n2 1\n7 5\n1 7\n1 3\n",
        "output": "4\n6 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 852,
        "task_id": 4000,
        "test_case_id": 156,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 2\n7 3\n1 6\n5 1\n2 7\n4 6\n",
        "output": "6\n4 5 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 853,
        "task_id": 4000,
        "test_case_id": 157,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n2 6\n5 6\n4 1\n1 7\n1 6\n3 5\n",
        "output": "5\n7 4 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 854,
        "task_id": 4000,
        "test_case_id": 158,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n3 2\n1 4\n3 1\n1 6\n7 1\n5 4\n",
        "output": "5\n2 7 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 855,
        "task_id": 4000,
        "test_case_id": 159,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 5\n4 1\n2 1\n2 3\n1 7\n6 2\n",
        "output": "4\n7 5 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 856,
        "task_id": 4000,
        "test_case_id": 160,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n7 2\n3 4\n4 2\n7 1\n6 7\n5 7\n",
        "output": "5\n6 5 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 857,
        "task_id": 4000,
        "test_case_id": 161,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 6\n3 1\n5 6\n1 4\n1 2\n1 7\n",
        "output": "4\n7 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 858,
        "task_id": 4000,
        "test_case_id": 162,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n2 7\n4 6\n7 1\n5 1\n3 1\n7 6\n",
        "output": "5\n5 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 859,
        "task_id": 4000,
        "test_case_id": 163,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n3 2\n5 1\n3 5\n2 7\n3 6\n4 2\n",
        "output": "5\n1 6 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 860,
        "task_id": 4000,
        "test_case_id": 164,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n4 1\n4 3\n7 5\n1 6\n7 4\n2 6\n",
        "output": "6\n2 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 861,
        "task_id": 4000,
        "test_case_id": 165,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n5 3\n2 4\n6 1\n4 7\n3 1\n2 3\n",
        "output": "6\n6 5 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 862,
        "task_id": 4000,
        "test_case_id": 166,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 7\n6 5\n4 3\n7 4\n2 5\n5 4\n",
        "output": "5\n1 3 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 863,
        "task_id": 4000,
        "test_case_id": 167,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 4\n3 6\n6 7\n3 5\n4 2\n2 7\n",
        "output": "6 \n1 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 864,
        "task_id": 4000,
        "test_case_id": 168,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n2 7\n2 4\n1 3\n5 6\n5 3\n3 2\n",
        "output": "5\n6 4 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 865,
        "task_id": 4000,
        "test_case_id": 169,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n4 2\n6 3\n5 1\n6 1\n7 4\n6 4\n",
        "output": "5\n5 3 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 866,
        "task_id": 4000,
        "test_case_id": 170,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n3 2\n1 7\n7 2\n6 5\n6 1\n4 5\n",
        "output": "6 \n3 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 867,
        "task_id": 4000,
        "test_case_id": 171,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n3 2\n3 5\n4 7\n3 6\n4 5\n5 1\n",
        "output": "5\n6 2 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 868,
        "task_id": 4000,
        "test_case_id": 172,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n7 6\n7 1\n4 2\n4 5\n7 3\n5 3\n",
        "output": "6\n6 1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 869,
        "task_id": 4000,
        "test_case_id": 173,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 2\n3 6\n6 5\n4 3\n7 5\n2 6\n",
        "output": "6\n4 1 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 870,
        "task_id": 4000,
        "test_case_id": 174,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n2 7\n6 7\n5 4\n1 2\n5 3\n3 6\n",
        "output": "6 \n1 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 871,
        "task_id": 4000,
        "test_case_id": 175,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 5\n4 2\n5 3\n6 5\n7 2\n1 2\n",
        "output": "5\n6 4 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 872,
        "task_id": 4000,
        "test_case_id": 176,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n5 2\n3 6\n7 1\n3 7\n3 4\n2 6\n",
        "output": "6\n1 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 873,
        "task_id": 4000,
        "test_case_id": 177,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n1 5\n3 5\n7 2\n7 6\n3 7\n4 5\n",
        "output": "5\n4 2 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 874,
        "task_id": 4000,
        "test_case_id": 178,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n7 6\n2 1\n6 5\n3 2\n3 6\n7 4\n",
        "output": "6\n1 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 875,
        "task_id": 4000,
        "test_case_id": 179,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n7 6\n2 3\n3 6\n5 4\n4 2\n1 5\n",
        "output": "6 \n1 5 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 876,
        "task_id": 4000,
        "test_case_id": 180,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "7\n2 4\n7 1\n6 5\n3 6\n2 7\n7 6\n",
        "output": "5\n4 3 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 877,
        "task_id": 4000,
        "test_case_id": 181,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 5\n1 8\n7 1\n1 6\n1 2\n4 2\n1 3\n",
        "output": "4\n8 7 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 878,
        "task_id": 4000,
        "test_case_id": 182,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n6 1\n2 1\n5 4\n8 1\n7 3\n7 1\n4 7\n",
        "output": "5\n8 6 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 879,
        "task_id": 4000,
        "test_case_id": 183,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n6 8\n1 8\n8 2\n1 7\n5 7\n1 3\n4 1\n",
        "output": "5\n5 4 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 880,
        "task_id": 4000,
        "test_case_id": 184,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n2 1\n6 1\n4 1\n7 1\n1 3\n1 5\n1 8\n",
        "output": "3\n7 6 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 881,
        "task_id": 4000,
        "test_case_id": 185,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 8\n2 1\n1 7\n1 5\n1 3\n4 8\n1 6\n",
        "output": "4\n7 6 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 882,
        "task_id": 4000,
        "test_case_id": 186,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n2 1\n2 8\n7 8\n5 1\n2 6\n3 1\n4 6\n",
        "output": "6\n5 4 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 883,
        "task_id": 4000,
        "test_case_id": 187,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n6 4\n7 6\n1 5\n1 3\n1 6\n8 1\n2 7\n",
        "output": "5\n8 5 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 884,
        "task_id": 4000,
        "test_case_id": 188,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n8 5\n6 4\n2 4\n1 5\n1 7\n1 4\n4 3\n",
        "output": "5\n6 7 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 885,
        "task_id": 4000,
        "test_case_id": 189,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n6 5\n7 4\n1 3\n8 7\n1 7\n2 1\n6 1\n",
        "output": "5\n5 4 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 886,
        "task_id": 4000,
        "test_case_id": 190,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n7 3\n2 8\n4 1\n1 3\n2 1\n6 3\n5 1\n",
        "output": "5\n7 6 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 887,
        "task_id": 4000,
        "test_case_id": 191,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 4\n6 5\n1 6\n7 1\n2 1\n3 6\n5 8\n",
        "output": "5\n7 4 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 888,
        "task_id": 4000,
        "test_case_id": 192,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n4 8\n4 6\n1 3\n7 4\n7 5\n1 4\n1 2\n",
        "output": "5\n3 8 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 889,
        "task_id": 4000,
        "test_case_id": 193,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n7 8\n5 4\n8 1\n8 6\n1 2\n4 3\n1 4\n",
        "output": "5\n5 6 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 890,
        "task_id": 4000,
        "test_case_id": 194,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n2 3\n6 5\n1 7\n2 1\n4 6\n8 4\n1 6\n",
        "output": "6\n3 7 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 891,
        "task_id": 4000,
        "test_case_id": 195,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n5 3\n4 2\n6 5\n8 1\n5 1\n7 8\n1 2\n",
        "output": "6\n6 4 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 892,
        "task_id": 4000,
        "test_case_id": 196,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n7 3\n5 6\n6 1\n7 4\n6 2\n2 8\n6 4\n",
        "output": "6\n8 5 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 893,
        "task_id": 4000,
        "test_case_id": 197,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n8 3\n4 2\n4 1\n1 7\n1 3\n6 2\n8 5\n",
        "output": "7\n5 7 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 894,
        "task_id": 4000,
        "test_case_id": 198,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n5 1\n1 6\n6 3\n4 3\n5 2\n1 7\n3 8\n",
        "output": "6\n2 7 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 895,
        "task_id": 4000,
        "test_case_id": 199,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n5 1\n7 2\n4 3\n8 5\n7 1\n4 6\n5 4\n",
        "output": "6\n2 8 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 896,
        "task_id": 4000,
        "test_case_id": 200,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n8 7\n3 2\n1 3\n6 4\n1 8\n1 4\n8 5\n",
        "output": "6\n6 2 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 897,
        "task_id": 4000,
        "test_case_id": 201,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n7 2\n5 4\n2 6\n1 3\n3 2\n4 2\n4 8\n",
        "output": "5\n1 7 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 898,
        "task_id": 4000,
        "test_case_id": 202,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n3 5\n3 6\n8 7\n2 7\n2 1\n2 6\n1 4\n",
        "output": "7\n8 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 899,
        "task_id": 4000,
        "test_case_id": 203,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 8\n5 4\n2 5\n6 3\n1 5\n1 7\n8 6\n",
        "output": "6\n4 7 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 900,
        "task_id": 4000,
        "test_case_id": 204,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 7\n6 7\n2 3\n8 4\n5 7\n2 1\n6 8\n",
        "output": "7\n3 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 901,
        "task_id": 4000,
        "test_case_id": 205,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n8 1\n3 4\n6 4\n3 5\n2 4\n8 4\n7 3\n",
        "output": "5\n1 6 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 902,
        "task_id": 4000,
        "test_case_id": 206,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n2 3\n4 7\n3 7\n2 6\n5 6\n4 1\n3 8\n",
        "output": "7\n1 8 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 903,
        "task_id": 4000,
        "test_case_id": 207,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n3 6\n8 2\n3 1\n8 4\n8 7\n6 4\n1 5\n",
        "output": "7\n5 2 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 904,
        "task_id": 4000,
        "test_case_id": 208,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 6\n2 7\n4 5\n6 5\n4 8\n2 5\n3 7\n",
        "output": "7\n8 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 905,
        "task_id": 4000,
        "test_case_id": 209,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n2 3\n6 5\n1 8\n4 5\n4 8\n7 6\n2 6\n",
        "output": "7\n1 7 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 906,
        "task_id": 4000,
        "test_case_id": 210,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 2\n5 8\n6 8\n4 5\n6 2\n3 7\n7 4\n",
        "output": "7 \n1 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 907,
        "task_id": 4000,
        "test_case_id": 211,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n1 7\n2 8\n7 8\n4 5\n3 4\n1 5\n6 3\n",
        "output": "7 \n2 8 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 908,
        "task_id": 4000,
        "test_case_id": 212,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n2 8\n8 1\n5 7\n6 4\n4 7\n7 2\n7 3\n",
        "output": "6\n1 5 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 909,
        "task_id": 4000,
        "test_case_id": 213,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n8 5\n6 3\n8 3\n7 2\n1 2\n5 4\n6 7\n",
        "output": "7 \n1 2 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 910,
        "task_id": 4000,
        "test_case_id": 214,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n6 7\n5 8\n4 1\n3 5\n3 6\n7 2\n4 2\n",
        "output": "7 \n1 4 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 911,
        "task_id": 4000,
        "test_case_id": 215,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "8\n6 7\n6 8\n1 3\n2 3\n5 6\n8 4\n7 3\n",
        "output": "6\n2 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 912,
        "task_id": 4000,
        "test_case_id": 216,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n3 1\n7 4\n1 4\n1 8\n2 1\n2 6\n9 1\n1 5\n",
        "output": "5\n6 9 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 913,
        "task_id": 4000,
        "test_case_id": 217,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n8 9\n6 2\n1 6\n1 4\n3 1\n9 1\n1 5\n1 7\n",
        "output": "5\n2 7 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 914,
        "task_id": 4000,
        "test_case_id": 218,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n3 9\n5 1\n4 1\n7 6\n3 1\n3 2\n8 1\n7 1\n",
        "output": "5\n6 8 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 915,
        "task_id": 4000,
        "test_case_id": 219,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n1 3\n6 4\n4 1\n5 1\n7 5\n1 9\n8 5\n1 2\n",
        "output": "5\n6 9 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 916,
        "task_id": 4000,
        "test_case_id": 220,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n4 1\n8 2\n6 1\n1 5\n3 1\n6 7\n9 5\n1 2\n",
        "output": "6\n8 7 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 917,
        "task_id": 4000,
        "test_case_id": 221,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n1 9\n4 9\n7 1\n3 2\n1 2\n1 6\n1 8\n2 5\n",
        "output": "5\n4 8 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 918,
        "task_id": 4000,
        "test_case_id": 222,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n4 1\n2 9\n1 2\n8 1\n9 5\n3 2\n7 6\n7 1\n",
        "output": "6\n6 8 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 919,
        "task_id": 4000,
        "test_case_id": 223,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n8 4\n5 8\n3 1\n2 8\n1 7\n9 8\n1 6\n1 8\n",
        "output": "4\n7 6 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 920,
        "task_id": 4000,
        "test_case_id": 224,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n8 3\n6 8\n9 4\n1 8\n8 5\n9 3\n2 1\n1 7\n",
        "output": "6\n7 6 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 921,
        "task_id": 4000,
        "test_case_id": 225,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n1 8\n5 1\n4 3\n9 1\n2 1\n1 4\n7 1\n6 1\n",
        "output": "4\n9 8 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 922,
        "task_id": 4000,
        "test_case_id": 226,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n7 2\n2 4\n5 8\n8 3\n2 1\n1 6\n6 9\n1 8\n",
        "output": "6\n7 5 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 923,
        "task_id": 4000,
        "test_case_id": 227,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n5 1\n8 1\n2 1\n4 1\n3 4\n1 7\n7 6\n5 9\n",
        "output": "6\n6 3 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 924,
        "task_id": 4000,
        "test_case_id": 228,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n8 1\n9 6\n6 1\n1 3\n2 1\n7 2\n1 4\n4 5\n",
        "output": "6\n7 5 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 925,
        "task_id": 4000,
        "test_case_id": 229,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n1 2\n9 8\n5 8\n7 5\n6 2\n5 3\n1 4\n1 5\n",
        "output": "6\n6 7 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 926,
        "task_id": 4000,
        "test_case_id": 230,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n7 1\n6 5\n1 8\n4 9\n2 8\n4 1\n3 8\n8 6\n",
        "output": "6\n9 7 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 927,
        "task_id": 4000,
        "test_case_id": 231,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n3 7\n3 9\n1 5\n6 1\n1 2\n4 3\n8 2\n3 2\n",
        "output": "5\n6 8 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 928,
        "task_id": 4000,
        "test_case_id": 232,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n6 2\n3 7\n5 1\n6 5\n3 1\n9 4\n6 8\n4 3\n",
        "output": "7\n8 7 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 929,
        "task_id": 4000,
        "test_case_id": 233,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n4 3\n6 9\n1 9\n1 3\n5 1\n7 1\n8 7\n2 5\n",
        "output": "6\n6 4 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 930,
        "task_id": 4000,
        "test_case_id": 234,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n3 5\n9 1\n4 1\n7 4\n3 8\n2 6\n9 2\n3 2\n",
        "output": "7\n7 6 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 931,
        "task_id": 4000,
        "test_case_id": 235,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n4 2\n9 5\n1 4\n1 7\n4 9\n5 3\n3 6\n9 8\n",
        "output": "7\n7 8 6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 932,
        "task_id": 4000,
        "test_case_id": 236,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n5 2\n3 9\n2 4\n7 8\n5 6\n9 8\n1 2\n9 4\n",
        "output": "7\n6 3 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 933,
        "task_id": 4000,
        "test_case_id": 237,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n5 8\n8 9\n2 6\n2 7\n3 6\n2 1\n8 1\n4 9\n",
        "output": "7\n3 7 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 934,
        "task_id": 4000,
        "test_case_id": 238,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n4 2\n2 6\n4 1\n8 5\n8 7\n6 5\n3 4\n9 2\n",
        "output": "7\n3 9 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 935,
        "task_id": 4000,
        "test_case_id": 239,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n1 4\n5 8\n8 7\n3 4\n6 3\n5 3\n2 4\n2 9\n",
        "output": "7\n9 6 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 936,
        "task_id": 4000,
        "test_case_id": 240,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n3 6\n9 7\n1 6\n2 8\n7 4\n9 5\n8 6\n3 9\n",
        "output": "7\n2 5 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 937,
        "task_id": 4000,
        "test_case_id": 241,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n7 9\n6 7\n4 6\n3 9\n9 8\n2 5\n2 8\n1 2\n",
        "output": "7\n5 3 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 938,
        "task_id": 4000,
        "test_case_id": 242,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n9 3\n5 8\n2 7\n2 3\n9 6\n1 7\n4 5\n4 2\n",
        "output": "8\n6 1 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 939,
        "task_id": 4000,
        "test_case_id": 243,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n2 7\n3 7\n8 2\n6 7\n1 3\n2 9\n5 2\n4 6\n",
        "output": "6\n4 1 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 940,
        "task_id": 4000,
        "test_case_id": 244,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n9 2\n6 8\n4 1\n2 5\n1 9\n8 7\n3 6\n5 7\n",
        "output": "8 \n4 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 941,
        "task_id": 4000,
        "test_case_id": 245,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n4 3\n6 9\n4 8\n6 5\n7 5\n1 6\n8 5\n7 2\n",
        "output": "7\n9 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 942,
        "task_id": 4000,
        "test_case_id": 246,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n6 7\n8 6\n1 4\n3 8\n4 7\n5 6\n9 3\n2 7\n",
        "output": "7\n1 5 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 943,
        "task_id": 4000,
        "test_case_id": 247,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n6 4\n9 5\n7 8\n1 8\n9 3\n4 2\n7 5\n7 4\n",
        "output": "7\n6 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 944,
        "task_id": 4000,
        "test_case_id": 248,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n9 2\n3 8\n4 6\n7 9\n2 5\n5 3\n1 6\n9 1\n",
        "output": "8\n4 7 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 945,
        "task_id": 4000,
        "test_case_id": 249,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n6 9\n1 3\n6 7\n2 8\n4 6\n2 6\n5 2\n3 7\n",
        "output": "6\n1 9 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 946,
        "task_id": 4000,
        "test_case_id": 250,
        "question": "You are given an unweighted tree with $n$ vertices. Recall that a tree is a connected undirected graph without cycles.\n\nYour task is to choose three distinct vertices $a, b, c$ on this tree such that the number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$ is the maximum possible. See the notes section for a better understanding.\n\nThe simple path is the path that visits each vertex at most once.\n\n\n-----Input-----\n\nThe first line contains one integer number $n$ ($3 \\le n \\le 2 \\cdot 10^5$) — the number of vertices in the tree. \n\nNext $n - 1$ lines describe the edges of the tree in form $a_i, b_i$ ($1 \\le a_i$, $b_i \\le n$, $a_i \\ne b_i$). It is guaranteed that given graph is a tree.\n\n\n-----Output-----\n\nIn the first line print one integer $res$ — the maximum number of edges which belong to at least one of the simple paths between $a$ and $b$, $b$ and $c$, or $a$ and $c$.\n\nIn the second line print three integers $a, b, c$ such that $1 \\le a, b, c \\le n$ and $a \\ne, b \\ne c, a \\ne c$.\n\nIf there are several answers, you can print any.\n\n\n-----Example-----\nInput\n8\n1 2\n2 3\n3 4\n4 5\n4 6\n3 7\n3 8\n\nOutput\n5\n1 8 6\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example (and another one correct answer):\n\n[Image]\n\nIf you choose vertices $1, 5, 6$ then the path between $1$ and $5$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 5)$, the path between $1$ and $6$ consists of edges $(1, 2), (2, 3), (3, 4), (4, 6)$ and the path between $5$ and $6$ consists of edges $(4, 5), (4, 6)$. The union of these paths is $(1, 2), (2, 3), (3, 4), (4, 5), (4, 6)$ so the answer is $5$. It can be shown that there is no better answer.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\nNEGINF = -1000000\\n\\nn = int(input())\\nadj = [[] for i in range(n)]\\nparent = [-1] * n\\n\\nvisited = [False] * n\\nfor _ in range(n - 1):\\n    a, b = list(map(int, input().split()))\\n    adj[a - 1].append(b - 1)\\n    adj[b - 1].append(a - 1)\\n\\ntup = tuple()\\nouts = [tup] * n\\nq = [(0, 0)]\\n\\nwhile q:\\n    node, type = q.pop()\\n    if type == 0:\\n        visited[node] = True\\n        q.append((node, 1))\\n        for v in adj[node]:\\n            if not visited[v]:\\n                parent[v] = node\\n                q.append((v, 0))\\n    else:\\n        ones = [(0, node)]\\n        twos = []\\n        threes = []\\n        for v in adj[node]:\\n            if v != parent[node]:\\n                a, b, c = outs[v]\\n                ones.append((a[0] + 1, a[1], v))\\n                twos.append((b[0] + 1, b[1], v))\\n                threes.append(c)\\n        ones.sort(reverse = True)\\n        twos.sort(reverse = True)\\n        threes.sort(reverse = True)\\n\\n        bestOne = (ones[0][0], ones[0][1])\\n        \\n        bestsTwo = [(NEGINF, (0, 0))]\\n        if len(twos) > 0:\\n            bestsTwo.append((twos[0][0], twos[0][1]))\\n        if len(ones) > 1:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            bestsTwo.append((o1[0] + o2[0], (o1[1], o2[1])))\\n\\n        bestsThree = [(NEGINF, (0, 0, 0))]\\n        if len(threes) > 0:\\n            bestsThree.append(threes[0])\\n        if len(ones) > 2:\\n            o1 = ones[0]\\n            o2 = ones[1]\\n            o3 = ones[2]\\n            bestsThree.append((o1[0] + o2[0] + o3[0], (o1[1], o2[1], o3[1])))\\n        if len(twos) > 0:\\n            o1 = ones[0]\\n            t1 = twos[0]\\n            if o1[2] != t1[2]:\\n                bestsThree.append((o1[0] + t1[0], (o1[1], t1[1][0], t1[1][1])))\\n            else:\\n                if len(twos) > 1:\\n                    t2 = twos[1]\\n                    bestsThree.append((o1[0] + t2[0], (o1[1], t2[1][0], t2[1][1])))\\n                if len(ones) > 1:\\n                    o2 = ones[1]\\n                    bestsThree.append((o2[0] + t1[0], (o2[1], t1[1][0], t1[1][1])))\\n                    \\n\\n        outs[node] = (bestOne, max(bestsTwo), max(bestsThree))\\n\\nfinal = outs[0][2]\\nprint(final[0])\\nprint(' '.join([str(x + 1) for x in final[1]]))\\n\", \"import sys\\nfrom collections import deque\\n\\n# inf = open('input.txt', 'r')\\n# reader = (line.rstrip() for line in inf)\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\ng = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    v, to = list(map(int, input().split()))\\n    v -= 1\\n    to -= 1\\n    g[v].append(to)\\n    g[to].append(v)\\n\\n# (vertex, distance)\\nqueue = deque([(0, 0)])\\nvisited = [False] * n\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\na = v\\n\\n# (vertex, distance)\\nqueue = deque([(a, 0)])\\nprev = [-1] * n\\nfor i in range(n):\\n    visited[i] = False\\nwhile queue:\\n    v, d = queue.popleft()\\n    visited[v] = True\\n    for to in g[v]:\\n        if not visited[to]:\\n            queue.append((to, d + 1))\\n            prev[to] = v\\nb, ctr = v, d\\n\\nfor i in range(n):\\n    visited[i] = False\\ncurr = prev[b]\\nnxt = b\\nprv = prev[curr]\\nadd = 0\\nif a != 0 and b != 0:\\n    c = 0\\nelif a != 1 and b != 1:\\n    c = 1\\nelse:\\n    c = 2\\nwhile curr != a:\\n    visited[curr] = True\\n    for to in g[curr]:\\n        if to == nxt or to == prv:\\n            continue\\n        queue = deque([(to, 1)])\\n        while queue:\\n            v, d = queue.popleft()\\n            visited[v] = True\\n            for to in g[v]:\\n                if not visited[to]:\\n                    queue.append((to, d + 1))\\n        if add < d:\\n            c, add = v, d    \\n    nxt = curr\\n    curr = prev[curr]\\n    prv = prev[curr]\\n    \\nprint(ctr + add)\\nprint(a + 1, b + 1, c + 1)\\n\\n# inf.close()\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nE=[[] for i in range(n+1)]\\n\\nfor i in range(n-1):\\n    x,y=list(map(int,input().split()))\\n    E[x].append(y)\\n    E[y].append(x)\\n\\ndef dfs(t):\\n    L=[-1]*(n+1)\\n    L[t]=0\\n\\n    Q=[t]\\n\\n    while Q:\\n        x=Q.pop()\\n        for to in E[x]:\\n            if L[to]==-1:\\n                L[to]=L[x]+1\\n                Q.append(to)\\n\\n    return L.index(max(L))\\n\\nA=dfs(1)\\nB=dfs(A)\\n\\nDEPTH=[-1]*(n+1)\\nDEPTH[1]=0\\n\\nfrom collections import deque\\nQUE = deque([1])\\nQUE2 = deque()\\nEULER=[]\\n\\nUSED=[0]*(n+1)\\nwhile QUE:\\n    x=QUE.pop()\\n    EULER.append((DEPTH[x],x))\\n    if USED[x]==1:\\n        continue\\n    for to in E[x]:\\n        \\n        if USED[to]==0:\\n            DEPTH[to]=DEPTH[x]+1\\n            QUE2.append(to)\\n        else:\\n            QUE.append(to)\\n    QUE.extend(QUE2)\\n    QUE2=deque()\\n \\n    USED[x]=1\\n\\nMINP=[1<<30]*(n+1)\\nMAXP=[-1]*(n+1)\\n\\nfor ind,(depth,p) in enumerate(EULER):\\n    MINP[p]=min(MINP[p],ind)\\n    MAXP[p]=max(MAXP[p],ind)\\n\\nLEN=len(EULER)\\n\\nseg_el=1<<(LEN.bit_length())\\nSEG=[(1<<30,0)]*(2*seg_el)\\n\\nfor i in range(LEN):\\n    SEG[i+seg_el]=EULER[i]\\n\\nfor i in range(seg_el-1,0,-1):\\n    SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n\\ndef update(n,x,seg_el):\\n    i=n+seg_el\\n    SEG[i]=x\\n    i>>=1\\n    \\n    while i!=0:\\n        SEG[i]=min(SEG[i*2],SEG[i*2+1])\\n        i>>=1\\n        \\ndef getvalues(l,r):\\n    L=l+seg_el\\n    R=r+seg_el\\n    ANS=(1<<30,0)\\n\\n    while L<R:\\n        if L & 1:\\n            ANS=min(ANS , SEG[L])\\n            L+=1\\n\\n        if R & 1:\\n            R-=1\\n            ANS=min(ANS , SEG[R])\\n        L>>=1\\n        R>>=1\\n\\n    return ANS\\n\\ndef LCA(l,r):\\n    return getvalues(min(MINP[l],MINP[r]),max(MAXP[l],MAXP[r])+1)\\n\\nA2=DEPTH[A]*2+DEPTH[B]*2-LCA(A,B)[0]*2\\nANS=0\\n\\nfor i in range(1,n+1):\\n    if i==A or i==B:\\n        continue\\n\\n    if ANS<A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2:\\n        ANS=A2+DEPTH[i]*2-LCA(i,A)[0]*2-LCA(i,B)[0]*2\\n        Aind=i\\n\\nprint(ANS//2)\\nprint(A,B,Aind)\\n    \\n    \\n\", \"import sys\\nfrom collections import deque\\nn = int(input())\\nadj = [[] for _ in range(n)]\\nfor u, v in (list(map(int, l.split())) for l in sys.stdin):\\n    adj[u-1].append(v-1)\\n    adj[v-1].append(u-1)\\n\\ninf = 10**9\\n\\n\\ndef rec(s):\\n    prev = [-1]*n\\n    prev[s] = inf\\n    dq = deque([s])\\n    last = s\\n\\n    while dq:\\n        v = dq.popleft()\\n        last = v\\n        for dest in adj[v]:\\n            if prev[dest] > -1:\\n                continue\\n            prev[dest] = v\\n            dq.append(dest)\\n\\n    return last, prev\\n\\n\\nv1, _ = rec(0)\\nv2, prev = rec(v1)\\nv = prev[v2]\\nvisited = [0]*n\\nvisited[v] = visited[v1] = visited[v2] = 1\\ndia = 0\\nmax_e, max_e_i = 0, v\\n\\nwhile v != inf:\\n    dia += 1\\n    if prev[v] != inf:\\n        visited[prev[v]] = 1\\n\\n    stack = [(v, 0)]\\n    while stack:\\n        cv, e = stack.pop()\\n        if max_e < e:\\n            max_e, max_e_i = e, cv\\n        e += 1\\n\\n        for dest in adj[cv]:\\n            if visited[dest]:\\n                continue\\n            visited[dest] = 1\\n            stack.append((dest, e))\\n\\n    v = prev[v]\\n\\nprint(dia + max_e)\\nprint(v1+1, v2+1, max_e_i+1)\\n\"]",
        "difficulty": "introductory",
        "input": "9\n1 8\n6 9\n6 7\n4 3\n3 5\n8 7\n2 6\n9 3\n",
        "output": "7\n1 4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1294/F"
    },
    {
        "id": 947,
        "task_id": 4161,
        "test_case_id": 1,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "2\n",
        "output": "9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 948,
        "task_id": 4161,
        "test_case_id": 2,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "200\n",
        "output": "10813692\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 949,
        "task_id": 4161,
        "test_case_id": 3,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "1\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 950,
        "task_id": 4161,
        "test_case_id": 4,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "199\n",
        "output": "10611772\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 951,
        "task_id": 4161,
        "test_case_id": 5,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "198\n",
        "output": "10493367\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 952,
        "task_id": 4161,
        "test_case_id": 6,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "197\n",
        "output": "10290813\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 953,
        "task_id": 4161,
        "test_case_id": 7,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "196\n",
        "output": "10174780\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 954,
        "task_id": 4161,
        "test_case_id": 8,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "195\n",
        "output": "9997134\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 955,
        "task_id": 4161,
        "test_case_id": 9,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "53\n",
        "output": "194157\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 956,
        "task_id": 4161,
        "test_case_id": 10,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "157\n",
        "output": "5205481\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 957,
        "task_id": 4161,
        "test_case_id": 11,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "120\n",
        "output": "2332104\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 958,
        "task_id": 4161,
        "test_case_id": 12,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "161\n",
        "output": "5610705\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 959,
        "task_id": 4161,
        "test_case_id": 13,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "98\n",
        "output": "1255101\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 960,
        "task_id": 4161,
        "test_case_id": 14,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "3\n",
        "output": "30\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 961,
        "task_id": 4161,
        "test_case_id": 15,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "4\n",
        "output": "76\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 962,
        "task_id": 4161,
        "test_case_id": 16,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "5\n",
        "output": "141\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 963,
        "task_id": 4161,
        "test_case_id": 17,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "6\n",
        "output": "267\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 964,
        "task_id": 4161,
        "test_case_id": 18,
        "question": "Find \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\nHere \\gcd(a,b,c) denotes the greatest common divisor of a, b, and c.\n\n-----Constraints-----\n - 1 \\leq K \\leq 200\n - K is an integer.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nK\n\n-----Output-----\nPrint the value of \\displaystyle{\\sum_{a=1}^{K}\\sum_{b=1}^{K}\\sum_{c=1}^{K} \\gcd(a,b,c)}.\n\n-----Sample Input-----\n2\n\n-----Sample Output-----\n9\n\n\\gcd(1,1,1)+\\gcd(1,1,2)+\\gcd(1,2,1)+\\gcd(1,2,2)+\\gcd(2,1,1)+\\gcd(2,1,2)+\\gcd(2,2,1)+\\gcd(2,2,2)=1+1+1+1+1+1+1+2=9\nThus, the answer is 9.",
        "solutions": "[\"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"from math import gcd\\nn = int(input())\\nans = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    for k in range(1,n+1):\\n      ans += gcd(gcd(i,j),k)\\nprint(ans)\", \"def gcd(a, b):\\n  if b == 0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nk = int(input())\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(a,k+1):\\n    for c in range(b,k+1):\\n      d = gcd(a, b)\\n      if len({a,b,c}) == 1:\\n        ans += gcd(c, d)\\n      elif len({a,b,c}) == 2:\\n        ans += 3*gcd(c, d)\\n      else:\\n        ans += 6*gcd(c, d)\\nprint(ans)\", \"K = int(input())\\nans = 0\\n\\ndef gcd(x, y):\\n  if x % y == 0:\\n    return y\\n  else:\\n    return gcd(y, x % y)\\n\\nans = 0\\nfor a in range(1, K+1):\\n  for b in range(1, K+1):\\n    d = gcd(a, b)\\n    for c in range(1, K+1):\\n      ans += gcd(c, d)\\n\\nprint(ans)\", \"import itertools as itt\\nimport math\\n\\nk = int(input())\\n\\nans = 0\\nfor i in itt.combinations_with_replacement(range(1, k+1), 3):\\n    if i[0] == i[1] and i[0] != i[2]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[1] == i[2] and i[1] != i[0]:\\n        ans += 3 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n    elif i[0] == i[1] and i[0] == i[2]:\\n        ans += math.gcd(math.gcd(i[0], i[1]), i[2])\\n    else:\\n        ans += 6 * math.gcd(math.gcd(i[0], i[1]), i[2])\\n\\nprint(ans)\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(1,1+k):\\n        for j in range(1,1+k):\\n            for l in range(1,1+k):\\n                ans += gcd(i,gcd(j,l))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input());print(sum(gcd(a+1,gcd(b+1,c+1))for a in range(K)for b in range(K)for c in range(K)))\", \"import math \\n\\nk = int(input())\\n\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ab = math.gcd(i,j)\\n        for x in range(1,k + 1):\\n            ans += math.gcd(ab,x)\\n    \\n\\nprint(ans)\", \"import math\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(a, K+1):\\n        for c in range(b, K+1):\\n            s = math.gcd(a, b)\\n            t = math.gcd(s, c)\\n            if a == c:\\n                ans += t\\n            elif (a == b or b == c) and a != c:\\n                ans += 3*t\\n            else:\\n                ans += 6*t\\nprint(ans)\", \"import sys\\nfrom math import ceil as C, floor as F, sqrt, gcd as G\\nfrom collections import defaultdict as D, Counter as CNT\\nfrom functools import reduce as R\\nimport heapq as HQ\\n\\nclass Heap:\\n  def __init__(self, data, reverse=False):\\n    self.reverse = -1 if reverse else 1\\n    self.data = [self.reverse * d for d in data]\\n    HQ.heapify(self.data)\\n  def push(self, x): return HQ.heappush(self.data, self.reverse * x)\\n  def pop(self): return self.reverse * HQ.heappop(self.data) \\n\\nALP = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'\\nalp = 'abcdefghijklmnopqrstuvwxyz'\\ndef _X(): return sys.stdin.readline().rstrip().split(' ')\\ndef _S(ss): return tuple(ss) if len(ss) > 1 else ss[0]\\ndef S(): return _S(_X())\\ndef Ss(): return list(S())\\ndef _I(ss): return tuple([int(s) for s in ss]) if isinstance(ss, tuple) else int(ss)\\ndef I(): return _I(S())\\ndef _Is(ss): return list(ss) if isinstance(ss, tuple) else [ss]\\ndef Is(): return _Is(I())\\n\\nn = I()\\n\\nans = 0\\nfor i in range(1, n+1):\\n    for j in range(1, n+1):\\n        for k in range(1, n+1):\\n           ans += G(i, G(j, k))\\n\\nprint(ans)\\n            \\n\", \"from math import gcd\\n\\nk = int(input())\\n\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    K = int(input())\\n    ans = 0\\n\\n    for i in range(1, K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for l in range(1, K+1):\\n                ans += gcd(temp, l)\\n                \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def main():\\n    import math\\n    k = int(input())\\n    cand = [int(v) for v in range(1, k + 1)]\\n    ans = 0\\n    for i in range(1, k + 1):\\n        for j in range(1, k + 1):\\n            for l in range(1, k + 1):\\n                temp = math.gcd(i, j)\\n                res = math.gcd(temp, l)\\n                ans += res\\n    return ans\\n\\n\\ndef __starting_point():\\n    print((main()))\\n\\n__starting_point()\", \"from math import *\\n\\nK=int(input())\\nans=0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    for k in range(1,K+1):\\n      ans+=gcd(k,gcd(i,j))\\nprint(ans)\\n\", \"import math\\n\\nk = int(input())\\ntotal = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1,k+1):\\n            total += math.gcd(tmp,k)\\nprint(total)\", \"from math import gcd \\nk = int(input())\\ncnt = 0\\nfor i in range(1, k+1):\\n  for j in range(1, k+1):\\n    for l in range(1, k+1):\\n      cnt += gcd(gcd(i, j), l)\\nprint(cnt)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"K = int(input())\\n\\nans = 0\\n\\ndef gcd(x, y):\\n    if x == 1 or y == 1:\\n        return 1\\n    else:\\n        while True:\\n            if x >= y:\\n                x %= y\\n            else:\\n                y %= x\\n            if x == 0 or y == 0:\\n                break\\n        return x + y\\n\\nfor i in range(1, K + 1):\\n    for j in range(i, K + 1):\\n        for k in range(j, K + 1):\\n            if i == j == k:\\n                ans += gcd(i, gcd(j, k))\\n            elif i == j or j == k or k == i:\\n                ans += gcd(i, gcd(j, k)) * 3\\n            else:\\n                ans += gcd(i, gcd(j, k)) * 6\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    for c in range(1,k+1):\\n      ans += gcd(gcd(a,b),c)\\nprint(ans)\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        temp = math.gcd(i, j)\\n        for l in range(1, k + 1):\\n            ans += math.gcd(temp, l)\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = gcd(i, j)\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += gcd(temp,k) * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# -*- coding: utf-8 -*-\\nimport sys\\n\\n\\ndef main():\\n    N = int( sys.stdin.readline() )\\n\\n\\n    def gcd(a,b):\\n        if b == 0: \\n            return a\\n        return gcd(b, a % b)\\n    \\n\\n    num_cnt_dic = {}\\n    num_gcd = [ [0]*(N+1) for _ in range(N+1) ]\\n\\n    for i in range(1, N+1):\\n        for j in range(1, N+1):\\n            g = gcd(i, j)\\n\\n            num_cnt_dic[g] = num_cnt_dic.get(g, 0) + 1\\n            num_gcd[i][j] = g\\n    \\n\\n    ans = 0\\n\\n    for i in list(num_cnt_dic.keys()):\\n        for j in range(1, N+1):\\n            ans += (num_gcd[i][j] * num_cnt_dic[i])\\n    \\n    \\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nread = sys.stdin.read\\nreadlines = sys.stdin.readlines\\nimport numpy as np\\ndef main():\\n    k = int(input())\\n\\n    k2 = np.arange(1, k+1)\\n    k2gcd = np.gcd.outer(k2, np.gcd.outer(k2, k2))\\n    print(k2gcd.sum())\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"#!/usr/env python3\\n#encoding:utf8\\n\\nimport math\\nfrom itertools import combinations_with_replacement as comb\\nfrom functools import reduce\\n\\ndef main():\\n    K = int(input())\\n\\n    ans = 0\\n    for abc in comb(range(1, K+1), 3):\\n        gcd = reduce(math.gcd, abc)\\n        s = len(set(abc))\\n        if s == 1:\\n            k = 1\\n        elif s == 2:\\n            k = 3\\n        else:\\n            k = 6\\n        ans += gcd * k\\n        #print(f\\\"abc={abc} gcd={gcd} k={k}\\\")\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1):\\n  for b in range(1,k+1):\\n    tmp=math.gcd(a,b)\\n    for c in range(1,k+1):\\n      ans+=math.gcd(tmp,c)\\n\\nprint(ans)\", \"import math\\nk=int(input())\\nans=0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        if math.gcd(a,b)==1:\\n            ans+=k\\n        else :\\n            for c in range(1,k+1):\\n                ans+=math.gcd(math.gcd(a,b),c)\\nprint(ans)\\n\", \"from collections import defaultdict\\nfrom math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    dd = defaultdict(int)\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            dd[gcd(a, b)] += 1\\n\\n    gcd_sum = 0\\n    for c in range(1, k + 1):\\n        for gcd_ab in dd.keys():\\n            gcd_sum += gcd(gcd_ab, c) * dd[gcd_ab]\\n\\n    return gcd_sum\\n\\n\\ndef main():\\n    k = int(input())\\n    print(answer(k))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections\\ndef gcd(a,b):\\n  if b==0:\\n    return a\\n  else:\\n    return gcd(b,a%b)\\nK=int(input())\\ncnt=collections.defaultdict(int)\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    cnt[gcd(a,b)]+=1\\nans=0\\nfor c in range(1,K+1):\\n  for gcd_of_ab in cnt.keys():\\n  \\tans+=gcd(gcd_of_ab,c)*cnt[gcd_of_ab]\\nprint(ans)\", \"k = int(input())\\nans = 0\\ndef gcd(a,b):\\n    if a % b == 0:\\n        return b\\n    c = a % b\\n    return gcd(b,c)\\n\\nfor l in range(1,k+1):\\n    for m in range(l,k+1):\\n        for n in range(m,k+1):\\n            tmp1 = gcd(l,n)\\n            tmp2= gcd(tmp1,m)\\n            if (l==m==n):\\n                ans+=tmp2\\n            elif(l==m or m==n):\\n                ans+= 3*tmp2\\n            else:\\n                ans += 6*tmp2\\n\\n\\nprint(ans)\", \"import math\\n\\nN=int(input())\\nans=0\\nfor i in range(1,N+1):\\n    for j in range(i,N+1):\\n        for k in range(j,N+1):\\n            if i==j and j==k:\\n                ans+=i\\n            elif i<j and j<k:\\n                ans+=6*math.gcd(i,math.gcd(j,k))\\n            else:\\n                ans+=3*math.gcd(i,math.gcd(j,k))\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = 0\\n\\nfor a in range(1, K+1):\\n    for b in range(1, K+1):\\n        tmp = math.gcd(a, b)\\n        for c in range(1, K+1):\\n            ans += math.gcd(tmp, c)\\n\\nprint(ans)\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        for k in range(1,k+1):\\n            ans += gcd(gcd(i,j),k)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(k):\\n        tmp = gcd1(i + 1, j + 1)\\n        if (tmp == 1):\\n            count = count + k\\n        else:\\n            for l in range(k):\\n                tmp2 = gcd1(tmp, l + 1)\\n                count = count + tmp2\\n\\nprint(count)\\n\", \"from math import gcd\\n\\n\\ndef main():\\n    k = int(input())\\n\\n    answer = 0\\n    for a in range(1, k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                answer += gcd(gcd(a, b), c)\\n\\n    print(answer)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import math\\nfrom functools import reduce\\n\\nk = int(input())\\n\\nans = 0\\n\\ngcd_sum = [0] * 201\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        gcd_sum[i] += math.gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(1, k + 1):\\n        _gcd = math.gcd(a,b)\\n        ans += gcd_sum[_gcd]\\n\\nprint(ans)\\n\", \"from math import gcd\\nk=int(input())\\n\\ncnt=0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a=gcd(i,j)\\n        for k in range(1,k+1):\\n            cnt+=gcd(a,k)\\nprint(cnt)\\n\", \"from math import gcd\\n\\nk = int(input())\\n\\nans = 0\\nfor x in range(1,k+1):\\n    for y in range(1,k+1):\\n        for z in range(1,k+1):\\n            ans += gcd(gcd(x,y),z)\\n\\nprint(ans)\\n\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    for i in range(1,K+1):\\n        for j in range(1, K+1):\\n            temp = gcd(i, j)\\n            for k in range(1, K+1):\\n                ans += gcd(temp,k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a+1, k+1):\\n        ans += math.gcd(a, b)\\n        for c in range(b+1, k+1):\\n            ans += math.gcd(math.gcd(a, b), c)\\nans *= 6\\nans += (k+1)*k/2\\nprint((int(ans)))\\n\", \"from math import gcd\\n\\nK = int(input())\\n\\n\\nans = 0\\nfor i in range(1, K+1):\\n    for j in range(1, K+1):\\n        tmp = gcd(i, j)\\n        if tmp == 1:\\n            ans += tmp*K\\n        else:\\n            for k in range(1, K+1):\\n                ans += gcd(tmp, k)\\n            \\n\\nprint(ans)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            if i == 1:\\n                D[i][j] = 1\\n            if i == j:\\n                D[i][j] = i\\n            else:\\n                D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from math import gcd\\nK=int(input())\\nresult=0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        for c in range(1,K+1):\\n            result+=gcd(gcd(a,b),c)\\nprint(result)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1, k+1):\\n    for j in range(1, k+1):\\n        a = math.gcd(i, j)\\n        for k in range(1, k+1):\\n            ans += math.gcd(a, k)\\nprint(ans)\", \"import math\\nk = int(input())\\nans = 0\\n\\n\\nfor a in range(1, k+1):\\n  for b in range(a, k+1):\\n    for c in range(b, k+1):\\n      gcd = math.gcd(math.gcd(a,b),c)\\n      if a == b == c:\\n        ans += gcd\\n      elif a == b or b == c:\\n        ans += gcd *3\\n      else:\\n        ans += gcd *6\\n      \\nprint(ans)\", \"import math\\nk = int(input())\\n\\nans = 0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        b = math.gcd(i,j)\\n        for k in range(1,k+1):\\n            ans += math.gcd(b,k)\\n\\nprint(ans)\\n\", \"from sys import stdin,stdout\\nimport math\\n\\ndef main():\\n    k = int(stdin.readline().rstrip())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1,k+1):\\n            ab = math.gcd(a,b)\\n            for c in range(1,k+1):\\n                ans = ans + math.gcd(ab,c)\\n\\n    stdout.write(str(ans))\\n    stdout.write('\\\\n')\\n\\nmain()\\n\", \"# C - Sum of gcd of Tuples (Easy)\\n\\nimport math\\n\\nk = int(input())\\ns = 0\\nfor a in range(1,k+1):\\n    for b in range(1,k+1):\\n        gcdab = math.gcd(a,b)\\n        if  gcdab== 1:\\n            s += k\\n        else:\\n            for c in range(1,k+1):\\n                s += math.gcd(gcdab,c)\\n\\nprint(s)\\n\\n\\n\", \"import math\\n\\nK = int(input())\\nsum = 0\\n\\nfor i in range(1, K + 1):\\n  for j in range(1, K + 1):\\n    tmp = math.gcd(i, j)\\n    for k in range(1, K + 1):\\n      sum += math.gcd(tmp, k)\\n      \\nprint(sum)\\n\", \"K = int(input())\\n\\ndef gcd(a, b):\\n    while b:\\n        a, b = b, a % b\\n    return a\\n\\nresult = 0\\n\\nfor a in range(1, K+1):\\n    result += a\\n    #print('{} add'.format(a))\\n    b = a + 1\\n    while b <= K:\\n        result += 6 * gcd(a, b)\\n        #print('{} and {} add'.format(a, b))\\n        l = gcd(a, b)\\n        c = b + 1\\n        while c <= K:\\n            result += 6 * gcd(l, c)\\n            #print('{} and {} and {} add'.format(a, b, c))\\n            c+= 1\\n        b += 1\\nprint(result)\", \"import math\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = math.gcd(a, b)\\n        for c in range(1,k+1):\\n            sum += math.gcd(x, c)\\n\\nprint(sum)\", \"import math\\n\\nK = int(input())\\n\\nsum = 0\\nfor i in range(1,K+1):\\n  for j in range(1,K+1):\\n    temp = math.gcd(i,j)\\n    for k in range(1,K+1):\\n      sum+=math.gcd(temp,k)\\nprint (sum)\", \"import math\\nK = int(input())\\nans = 0\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            ans += math.gcd(a,k)\\nprint(ans)\", \"#ABC162\\nK=int(input())\\nans=0\\nimport math\\nfor i in range(1,K+1):\\n    for j in range(1,K+1):\\n        a = math.gcd(i,j)\\n        for k in range(1,K+1):\\n            l=math.gcd(a,k)\\n            ans+=l\\nprint(ans)\", \"import math\\nn=int(input())\\n\\nans=0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"import math\\nk = int(input())\\n\\n\\ntotal = 0\\n\\nfor x in range(1,k+1):\\n    for y in range(x,k+1):\\n        for z in range(y,k+1):\\n            if x == y == z:\\n                total += x\\n            elif x == y or y == z:\\n                total += 3 * math.gcd(math.gcd(x,y),z)\\n            else:\\n                total += 6 * math.gcd(math.gcd(x,y),z)\\n\\nprint(total)\\n\", \"import math\\nfrom functools import reduce\\n\\ndef gcd(*numbers):\\n    return reduce(math.gcd, numbers)\\n\\ndef gcd_list(numbers):\\n    return reduce(math.gcd, numbers)\\nk=int(input())\\np=0\\nans=0\\nif k>2:\\n  for i in range(1,k-1):\\n    for j in range(i+1,k):\\n        for m in range(j+1,k+1):\\n            ans+=gcd(i,j,m)\\np+=6*ans\\nb=0\\nfor i in range(1,k):\\n    for j in range(i+1,k+1):\\n        b+=math.gcd(i,j)\\np+=6*b+k*(k+1)//2\\nprint(p)\", \"import math\\n\\nk = int(input())\\nwa = 0\\naa = 0\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    aa =math.gcd(i,j)\\n    for l in range(1,k+1):\\n      wa +=math.gcd(aa,l)\\nprint(wa)\", \"k=int(input())\\nimport math\\ns=0\\nresult=0\\nfor a in range(1,k+1,1):\\n    for b in range(a,k+1,1):\\n        for c in range(b,k+1,1):\\n            gcd=math.gcd(math.gcd(a,b),math.gcd(b,c))\\n            if a==b==c:\\n                pass\\n            elif (a==b and b!=c) or (b==c and b!=a) or (a==c and a!=c):\\n                gcd=3*gcd\\n            else:\\n                gcd=6*gcd\\n            \\n            result+=gcd\\nprint(result)\", \"import numpy as np\\n\\nK = int(input())\\nx = np.arange(1, K + 1)\\n\\nprint(np.sum(np.gcd.outer(np.gcd.outer(x, x), x)))\", \"import math\\nK=int(input())\\n\\nlist1=[]\\ns=1\\nwhile s<=K:\\n    list1.append(s)\\n    s=s+1\\n\\nlist3=[]\\nt=1\\nwhile t<=K*K:\\n    list3.append(t)\\n    t=t+1\\n\\nlist2=[]\\nk=1\\nfor i in list1:\\n    for j in list1:\\n        X=math.gcd(i,j)\\n        list2.append(X)\\n\\nY=0\\nfor k in list1:\\n    for l in list3:\\n        Y=Y+math.gcd(k,list2[l-1])\\n\\nprint(Y)\\n\", \"import sys\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, product, permutations, combinations\\nfrom operator import itemgetter\\nfrom bisect import bisect_left, bisect_right\\nfrom heapq import heappop, heappush\\nfrom math import ceil, floor, sqrt, gcd, inf\\nfrom copy import deepcopy\\nimport numpy as np\\nimport scipy as sp\\n\\nINF = inf\\nMOD = 1000000007\\n\\nk = int(input())\\n\\ntmp = [[0 for i in range(k + 1)] for j in range(k + 1)]\\nres = 0\\n\\nfor i in range(1, k + 1):\\n    for j in range(1, k + 1):\\n        tmp[i][j] = gcd(i, j)\\n\\nfor a in range(1, k + 1):\\n    for b in range(k + 1):\\n        for c in range(k + 1):\\n            res += tmp[tmp[a][b]][c]\\n\\nprint(res)\\n\", \"import math\\nn =int(input())\\nc=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        d= math.gcd(i,j)\\n        for k in range(1,n+1):\\n            c+=math.gcd(d,k)\\nprint(c)\", \"import math\\nn=int(input());ans=0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    r=math.gcd(i,j)\\n    for k in range(1,n+1):\\n      ans+=math.gcd(r,k)\\nprint(ans)\", \"from math import gcd\\n\\nk=int(input())\\nans=0\\n\\nfor a in range(1,k+1) :\\n    for b in range(1,k+1) :\\n        for c in range(1,k+1) :\\n            ans+=gcd(gcd(a,b),c)\\n\\nprint(ans)\\n\", \"k = int(input())\\ndef gcd1 (a, b):\\n    while True:\\n        if (a < b):\\n            a, b = b, a\\n        c = a%b\\n        if (c == 0):\\n            return (b)\\n        else:\\n            a = b\\n            b = c\\n\\ndef gcd2 (a, b, c):\\n    tmp = gcd1(a, b)\\n    ans = gcd1(tmp, c)\\n    return (ans)\\n\\ncount = 0\\nfor i in range(k):\\n    for j in range(i, k):\\n        for l in range(j, k):\\n            tmp = gcd2(i + 1, j + 1, l + 1)\\n            if (i == j == l):\\n                count = count + tmp\\n            elif (i == j or j == l):\\n                count = count + tmp*3\\n            else:\\n                count = count + tmp*6\\nprint(count)\\n\", \"k = int(input())\\nans = 0\\nfrom math import gcd\\nfor i in range(1,k+1):\\n  for j in range(1,k+1):\\n    for l in range(1,k+1):\\n      ans += gcd(gcd(i,j),l)\\nprint(ans)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\\n\", \"from math import gcd\\nK=int(input())\\ng=0\\nfor a in range(1,K-1):\\n  for b in range(a+1,K):\\n    for c in range(b+1,K+1):\\n      g+=gcd(gcd(a,b),c)*6\\nfor  d in range(1,K):\\n  for e in range(d+1,K+1):\\n    g+=gcd(d,e)*6\\ng+=K*(K+1)//2\\nprint(g)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import math\\n\\nk = int(input())\\nans = 0\\n\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        a = math.gcd(i,j)\\n        for l in range(1,k+1):\\n            ans += math.gcd(a,l)\\n\\nprint(ans)\", \"from math import gcd\\n\\nn = int(input())\\n\\nans = 0\\n\\n\\nfor i in range(1, n + 1):\\n    for j in range(1, n + 1):\\n        for k in range(1, n + 1):\\n            ans += gcd(gcd(i, j), k)\\n\\nprint(ans)\\n\", \"import math\\n\\nK = int(input())\\nans = sum(math.gcd(i, math.gcd(j, k)) for i in range(1, K+1) for j in range(1, K+1) for k in range(1, K+1))\\nprint(ans)\", \"from math import gcd\\n\\n\\ndef answer(k: int) -> int:\\n    result = 0\\n    for a in range(1, k + 1):\\n        for b in range(1, k + 1):\\n            temp = gcd(a, b)\\n            for c in range(1, k + 1):\\n                result += gcd(temp, c)\\n\\n    return result\\n\\n\\ndef main():\\n    k = int(input())\\n    print((answer(k)))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from math import gcd\\nk = int(input())\\nans = 0\\nfor i in range(1, k + 1):\\n  for j in range(1, k + 1):\\n    g = gcd(i, j)\\n    for k in range(1, k + 1):\\n      ans += gcd(g, k)\\nprint(ans)\", \"\\n\\nfrom math import gcd as g\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(1, k+1):\\n        temp = g(a, b)\\n        for c in range(1, k+1):\\n            ans += g(temp, c)\\n\\n\\nprint(ans)\\n\", \"import math\\nk=int(input())\\n\\nans=0\\nfor i in range(k):\\n  for j in range(k):\\n    q=math.gcd(i+1,j+1)\\n    for l in range(k):\\n      ans+=math.gcd(q,l+1)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        for c in range(1,k+1):\\n            sum += gcd(gcd(a, b), c)\\n\\nprint(sum)\", \"n = int(input())\\nmemo = [[0] * (n+1) for x in range(n+1)]\\n\\ndef gcd(x,y):\\n  if y == 0:\\n    return x\\n  if not memo[x][y] == 0:\\n    return memo[x][y]\\n  memo[x][y] = gcd(y,x % y)\\n  return gcd(y,x % y)\\n\\nres = 0\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    q = gcd(i,j)\\n    if q == 1:\\n      res += n\\n    else:\\n      for k in range(1,n+1):\\n        p = gcd(q,k)\\n        res += p\\nprint(res)\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"import sys\\nimport math\\nfrom collections import defaultdict, deque, Counter\\nfrom copy import deepcopy\\nfrom bisect import bisect, bisect_right, bisect_left\\nfrom heapq import heapify, heappop, heappush\\n    \\ninput = sys.stdin.readline\\ndef RD(): return input().rstrip()\\ndef F(): return float(input().rstrip())\\ndef I(): return int(input().rstrip())\\ndef MI(): return map(int, input().split())\\ndef MF(): return map(float,input().split())\\ndef LI(): return list(map(int, input().split()))\\ndef TI(): return tuple(map(int, input().split()))\\ndef LF(): return list(map(float,input().split()))\\ndef Init(H, W, num): return [[num for i in range(W)] for j in range(H)]\\n    \\ngcd = math.gcd\\n\\ndef main():\\n    K = I()\\n    ans = 0\\n    D = Init(K+1, K+1, 0)\\n    for i in range(1, K+1):\\n        for j in range(i, K+1):\\n            D[i][j] = gcd(i, j)\\n\\n    for i in range(1,K+1):\\n        for j in range(i, K+1):\\n            temp = D[i][j]\\n            for k in range(j, K+1):\\n                n = set([i,j,k])\\n                n = len(n)\\n                ans += D[temp][k] * (n+1)*(n)//2\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"def gcd_r(a, b):\\n  if memo[a][b] != 0:\\n    return memo[a][b]\\n\\n  if a < b:\\n    a, b = b, a\\n\\n  memo[a][b] = gcd(a, b)\\n  memo[b][a] = memo[a][b]\\n  return memo[a][b]\\n\\ndef gcd(a, b):\\n  r = a % b\\n  if r == 0:\\n    return b\\n\\n  return gcd(b, r)\\n\\nk = int(input())\\nmemo = [[0]*(k+1) for _ in range(k+1)]\\ntotal = 0\\n\\nfor i in range(1, k+1, 1):\\n  for c in range(1, k+1, 1):\\n    total += gcd_r(gcd_r(i, i), c)\\n\\nfor a in range(1, k+1, 1):\\n  for b in range(a+1, k+1, 1):\\n    for c in range(1, k+1, 1):\\n      total += gcd_r(gcd_r(a, b), c) * 2\\n\\nprint(total)\\n\", \"from math import gcd\\ndef main():\\n    ans = 0\\n    k = int(input())\\n    for i in range(k):\\n        for j in range(k):\\n            for l in range(k):\\n                ans += gcd(i+1,gcd(j+1,l+1))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nk = int(input())\\nans = 0\\nfor a in range(1, k+1):\\n    for b in range(a, k+1):\\n        for c in range(b, k+1):\\n            if a == b == c:\\n                ans += math.gcd(math.gcd(a,b),c)\\n            elif a == b or b == c or c == a:\\n                ans += 3 * math.gcd(math.gcd(a, b), c)\\n            else:\\n                ans += 6 * math.gcd(math.gcd(a, b), c)\\nprint(ans)\", \"from math import gcd\\nK = int(input())\\nans = 0\\nfor a in range(1,K+1):\\n    for b in range(1,K+1):\\n        r = gcd(a,b)\\n        for c in range(1,K+1):\\n            ans += gcd(r,c)\\nprint(ans)\", \"from math import gcd\\nk=int(input())\\nans=0\\nfor i in range(1,k+1):\\n    for j in range(1,k+1):\\n        ans_=gcd(i,j)\\n        for l in range(1,k+1):\\n            ans+=gcd(ans_,l)\\nprint(ans)\", \"import math\\nK=int(input())\\nans=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    g=math.gcd(a,b)\\n    for c in range(1,K+1):\\n      ans+=math.gcd(g,c)\\nprint(ans)      \", \"from math import gcd\\nK=int(input())\\nk=[1,1,1]\\ng=0\\nfor a in range(1,K+1):\\n  for b in range(1,K+1):\\n    for c in range(1,K+1):\\n      g+=gcd(gcd(a,b),c)\\n      \\n      \\nprint(g)\", \"from math import gcd\\n\\ndef main():\\n    k = int(input())\\n    ans = 0\\n    for i in range(1, k+1):\\n        for j in range(1, k+1):\\n            for k in range(1, k+1):\\n                ans += gcd(gcd(i, j), k)\\n    print(ans)\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import math\\nK =  int(input())\\nans = []\\nbc = [math.gcd(b, c) for b in range(1, K+1) for c in range(1, K+1)]\\nfor a in range(1, K+1):\\n    for i in bc:\\n        x = math.gcd(a, i)\\n        ans.append(x)\\nprint(sum(ans))\", \"import math\\ndef resolve():\\n    k = int(input())\\n    ans = 0\\n    for a in range(1,k+1):\\n        for b in range(1, k+1):\\n            for c in range(1, k+1):\\n                ans += math.gcd(math.gcd(a,b),c)\\n    print(ans)\\nresolve()\", \"import math\\nn=int(input())\\nans=0\\nfor i in range(1,n+1):\\n    for j in range(1,n+1):\\n        r=math.gcd(i,j)\\n        for k in range(1,n+1):\\n            ans+=math.gcd(r,k)\\nprint(ans)\", \"\\n\\nimport math\\nrst = 0\\nK = int(input())\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(k, tmp)\\nprint(rst)\\n\", \"import math\\nK = int(input())\\nrst = 0\\nfor i in range(1, K + 1):\\n    for j in range(1, K + 1):\\n        tmp = math.gcd(i, j)\\n        for k in range(1, K + 1):\\n            rst += math.gcd(tmp, k)\\nprint(rst)\", \"from math import gcd\\nn = int(input())\\nans = 0\\n\\nfor i in range(1,n+1):\\n  for j in range(1,n+1):\\n    x = gcd(i,j)\\n    for k in range(1,n+1):\\n        ans += gcd(x,k)\\nprint(ans)\\n\", \"from math import gcd\\n\\nk = int(input())\\nresult = 0\\n\\nfor a in range(1, k+1):\\n  for b in range(1, k+1):\\n    for c in range(1, k+1):\\n      result += gcd(gcd(a, b), c)\\n\\nprint(result)\", \"def gcd(a, b):\\n    if b == 0:\\n        return a\\n    else:\\n        return gcd(b, a % b)\\n\\n\\nk = int(input())\\nans = 0\\nfor a in range(1, k + 1):\\n    for b in range(a, k + 1):\\n        for c in range(b, k + 1):\\n            d = gcd(a, b)\\n            if len({a, b, c}) == 1:\\n                ans += gcd(c, d)\\n            elif len({a, b, c}) == 2:\\n                ans += 3 * gcd(c, d)\\n            else:\\n                ans += 6 * gcd(c, d)\\nprint(ans)\\n\", \"from math import gcd\\nk = int(input())\\nsum = 0\\n\\nfor a in range(1,k+1):\\n    for b in range(1,k+1): \\n        x = gcd(a,b)\\n        for c in range(1,k+1):\\n            sum += gcd(x, c)\\n\\nprint(sum)\", \"K = int(input())\\nans = 0\\ndef gcd(x,y):\\n    if(y == 0):\\n        return x\\n    if(x >= y):\\n        return gcd(y,x%y)\\n    if(x < y):\\n        return gcd(x,y%x)\\nfor i in range(1,K+1):\\n    ans += i\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        ans += gcd(i,j)*6\\nfor i in range(1,K+1):\\n    for j in range(i+1,K+1):\\n        for k in range(j+1,K+1):\\n            ans += gcd(gcd(i,j),k)*6  \\nprint(ans)\"]",
        "difficulty": "introductory",
        "input": "7\n",
        "output": "400\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc162/tasks/abc162_c"
    },
    {
        "id": 965,
        "task_id": 697,
        "test_case_id": 1,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "0 2\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 966,
        "task_id": 697,
        "test_case_id": 2,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "2 0\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 967,
        "task_id": 697,
        "test_case_id": 7,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "1 4\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 968,
        "task_id": 697,
        "test_case_id": 31,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "0 2000\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 969,
        "task_id": 697,
        "test_case_id": 32,
        "question": "Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \\ldots ,l}$ of length $l \\geq 0$. Then: \n\n$$f(a) = \\max (0, \\smash{\\displaystyle\\max_{1 \\leq i \\leq l}} \\sum_{j=1}^{i} a_j )$$\n\nNow they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\\: 244\\: 853$.\n\n\n-----Input-----\n\nThe only line contains two integers $n$ and $m$ ($0 \\le n,m \\le 2\\,000$).\n\n\n-----Output-----\n\nOutput the answer to the problem modulo $998\\: 244\\: 853$.\n\n\n-----Examples-----\nInput\n0 2\n\nOutput\n0\n\nInput\n2 0\n\nOutput\n2\n\nInput\n2 2\n\nOutput\n5\n\nInput\n2000 2000\n\nOutput\n674532367\n\n\n\n-----Note-----\n\nIn the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$. \n\nIn the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$. \n\nThere are $6$ possible arrays in the third example:\n\n[1,1,-1,-1], f([1,1,-1,-1]) = 2\n\n[1,-1,1,-1], f([1,-1,1,-1]) = 1\n\n[1,-1,-1,1], f([1,-1,-1,1]) = 1\n\n[-1,1,1,-1], f([-1,1,1,-1]) = 1\n\n[-1,1,-1,1], f([-1,1,-1,1]) = 0\n\n[-1,-1,1,1], f([-1,-1,1,1]) = 0\n\nSo the answer for the third example is $2+1+1+1+0+0 = 5$.",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\\n\", \"P = 998244853\\nN, M = list(map(int, input().split()))\\n\\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n\\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n\\nfainv = fainv[::-1]\\n\\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n\\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n\\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\\n\", \"MOD = 998244853\\nMAXN = 4000\\n\\nfact, inv_fact = [0] * (MAXN + 1), [0] * (MAXN + 1)\\nfact[0] = 1\\nfor i in range(MAXN):\\n    fact[i + 1] = fact[i] * (i + 1) % MOD\\n\\ninv_fact[-1] = pow(fact[-1], MOD - 2, MOD)\\nfor i in reversed(list(range(MAXN))):\\n    inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD\\n\\n\\ndef nCr_mod(n, r):\\n    res = 1\\n    while n or r:\\n        a, b = n % MOD, r % MOD\\n        if a < b:\\n            return 0\\n        res = res * fact[a] % MOD * inv_fact[b] % MOD * inv_fact[a - b] % MOD\\n        n //= MOD\\n        r //= MOD\\n    return res\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint(((k * nCr_mod(n + m, m) +\\n       sum(nCr_mod(n + m, m + i) for i in range(k + 1, n)) + min(1, n)) % MOD))\\n\", \"P = 998244853\\nN = 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0] = 1\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n\\nfi[-1] = pow(f[-1], P - 2, P)\\nfor i in reversed(list(range(N))):\\n    fi[i] = fi[i + 1] * (i + 1) % P\\n\\n\\ndef C(n, r):\\n    c = 1\\n    while n or r:\\n        a, b = n % P, r % P\\n        if a < b:\\n            return 0\\n        c = c * f[a] % P * fi[b] % P * fi[a - b] % P\\n        n //= P\\n        r //= P\\n    return c\\n\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P, N = 998244853, 4000\\n\\nf, fi = [0] * (N + 1), [0] * (N + 1)\\nf[0], fi[-1] = 1, 338887798\\nfor i in range(N):\\n    f[i + 1] = f[i] * (i + 1) % P\\n    fi[N - i - 1] = fi[N - i] * (N - i) % P\\n\\nC = lambda n, r: f[n] * fi[r] % P * fi[n - r] % P\\n\\nn, m = list(map(int, input().split()))\\nk = max(n - m - 1, 0)\\nprint((k * C(n + m, m) + sum(C(n + m, m + i) for i in range(k + 1, n)) + 1) % P if n else 0)\\n\", \"P = 998244853\\nN, M = map(int, input().split())\\n \\nfa = [1]\\nfor i in range(4040):\\n    fa.append(fa[-1]*(i+1)%P)\\n \\nfainv = [pow(fa[-1], P-2, P)]\\nfor i in range(4040)[::-1]:\\n    fainv.append(fainv[-1]*(i+1)%P)\\n \\nfainv = fainv[::-1]\\n \\ndef C(a, b):\\n    return fa[a]*fainv[a-b]*fainv[b]%P\\ndef calc(i):\\n    return C(N+M, M) if N-M > i else C(N+M, M+i)\\n \\nX = [0] * N + [1]\\nfor i in range(N):\\n    X[i] = calc(i) - calc(i+1)\\n \\nprint(sum([i*X[i] for i in range(1, N+1)]) % P)\", \"import sys\\nimport math\\n\\nMOD = 998244853\\n\\ndef prepare_c(n):\\n    result = [1]\\n    last = [1, 1]\\n    for i in range(2, n + 1):\\n        new = [1]\\n        for j in range(1, i):\\n            new.append((last[j - 1] + last[j]) % MOD)\\n        new.append(1)\\n        last = new\\n    return new\\n\\ndef main():\\n    (a, b) = tuple([int(x) for x in input().split()])\\n    if a + b == 0:\\n        print(0)\\n        return\\n\\n    c = prepare_c(a + b)\\n\\n    min_lv = max(0, a - b)\\n    max_lv = a\\n\\n    res = 0\\n    res += (min_lv * c[a]) % MOD\\n    for lv in range(min_lv + 1, max_lv + 1):\\n        t = 2 * lv - a + b\\n        res += c[(a + b + t) // 2]\\n        res = res % MOD\\n\\n    print(res)\\n\\n    \\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, m = map(int, input().split())\\nmod = 998244853\\nfact = [1]\\ninvfact = [1]\\ndef pw(x, y):\\n    ans = 1\\n    while (y):\\n        if (y & 1):\\n            ans = (ans * x) % mod\\n        x = x * x % mod\\n        y >>= 1\\n    return ans\\ndef inv(x):\\n    return pw(x, mod - 2)\\nfor i in range(1, n + m + 1):\\n    fact.append(fact[i - 1] * i % mod)\\n    invfact.append(invfact[i - 1] * inv(i) % mod)\\nmn = max(0, n - m)\\ndef ways_to(sub):\\n    inc = (n + m + sub) // 2\\n    return fact[n + m] * invfact[inc] * invfact[n + m - inc] % mod\\nans = 0\\nways = [0 for x in range(0, n + 2)]\\nfor i in range (mn, n + 1):\\n    ways[i] = ways_to(n - m) - ways_to(2 * i - n + m)\\nways[n + 1] = ways_to(n - m)\\nfor i in range(1, n + 1):\\n    ans += i * (ways[i + 1] - ways[i])\\n    ans %= mod\\nif (ans < 0) :\\n    ans += mod\\nprint(ans)\", \"mod = 998244853\\n\\ndef frac(limit):\\n    frac = [1]*limit\\n    for i in range(2,limit):\\n        frac[i] = i * frac[i-1]%mod\\n    fraci = [None]*limit\\n    fraci[-1] = pow(frac[-1], mod -2, mod)\\n    for i in range(-2, -limit-1, -1):\\n        fraci[i] = fraci[i+1] * (limit + i + 1) % mod\\n    return frac, fraci\\nfrac, fraci = frac(13413)\\ndef comb(a, b):\\n    if not a >= b >= 0:\\n        return 0\\n    return frac[a]*fraci[b]*fraci[a-b]%mod\\n\\nN, M = list(map(int, input().split()))\\nprint(sum(comb(N+M, min(i, M)) for i in range(N))%mod)\\n\"]",
        "difficulty": "interview",
        "input": "2000 0\n",
        "output": "2000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1204/E"
    },
    {
        "id": 970,
        "task_id": 893,
        "test_case_id": 7,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "2 8\n5 4 6 6 5 5 5 4\n2 3\n3 6\n2 5\n1 2\n7 8\n3 4\n3 7\n",
        "output": "71\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 971,
        "task_id": 893,
        "test_case_id": 29,
        "question": "As you know, an undirected connected graph with n nodes and n - 1 edges is called a tree. You are given an integer d and a tree consisting of n nodes. Each node i has a value a_{i} associated with it.\n\nWe call a set S of tree nodes valid if following conditions are satisfied: S is non-empty. S is connected. In other words, if nodes u and v are in S, then all nodes lying on the simple path between u and v should also be presented in S. $\\operatorname{max}_{u \\in S} a_{u} - \\operatorname{min}_{v \\in S} a_{v} \\leq d$.\n\nYour task is to count the number of valid sets. Since the result can be very large, you must print its remainder modulo 1000000007 (10^9 + 7).\n\n\n-----Input-----\n\nThe first line contains two space-separated integers d (0 ≤ d ≤ 2000) and n (1 ≤ n ≤ 2000).\n\nThe second line contains n space-separated positive integers a_1, a_2, ..., a_{n}(1 ≤ a_{i} ≤ 2000).\n\nThen the next n - 1 line each contain pair of integers u and v (1 ≤ u, v ≤ n) denoting that there is an edge between u and v. It is guaranteed that these edges form a tree.\n\n\n-----Output-----\n\nPrint the number of valid sets modulo 1000000007.\n\n\n-----Examples-----\nInput\n1 4\n2 1 3 2\n1 2\n1 3\n3 4\n\nOutput\n8\n\nInput\n0 3\n1 2 3\n1 2\n2 3\n\nOutput\n3\n\nInput\n4 8\n7 8 7 5 4 6 4 10\n1 6\n1 2\n5 8\n1 3\n3 5\n6 7\n3 4\n\nOutput\n41\n\n\n\n-----Note-----\n\nIn the first sample, there are exactly 8 valid sets: {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {3, 4} and {1, 3, 4}. Set {1, 2, 3, 4} is not valid, because the third condition isn't satisfied. Set {1, 4} satisfies the third condition, but conflicts with the second condition.",
        "solutions": "[\"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    u -= 1\\n    v -= 1\\n    p[u].append(v)\\n    p[v].append(u)\\n\\ndef g(u, x, a, b, q):\\n    k = 1\\n    for v in p[u]:\\n        if a < t[v] <= b or t[v] == a and v > q:\\n            if v != x: k += k * g(v, u, a, b, q) % m\\n    return k\\n\\n\\ns = 0\\nfor q in range(n):\\n    a = t[q]\\n    b = a + d\\n    s += g(q, -1, a, b, q)\\n\\nprint(s % m)\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: list(map(int, input().split()))\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\\n\", \"f = lambda: map(int, input().split())\\nm = 1000000007\\n\\nd, n = f()\\nt = list(f())\\np = [[] for i in range(n)]\\nfor j in range(n - 1):\\n    u, v = f()\\n    p[u - 1].append(v - 1)\\n    p[v - 1].append(u - 1)\\n\\ndef g(u, x, y):\\n    s = 1\\n    for v in p[u]:\\n        if 0 < t[v] - t[y] <= d or t[v] == t[y] and v > y:\\n            if v != x: s += s * g(v, u, y) % m\\n    return s\\n\\nprint(sum(g(y, -1, y) for y in range(n)) % m)\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\n# def print(x):\\n    # sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = readInts()\\n    adj: list = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u - 1].append(v - 1)\\n        adj[v - 1].append(u - 1)\\n\\n    vis = [False for _ in range(n)]\\n    f = [0 for _ in range(n)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(0, n):\\n        vis = [False for _ in range(n)]\\n        f = [0 for _ in range(n)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef readInts(): return [int(x) for x in sys.stdin.readline().split()]\\n\\n\\ndef readInt(): return int(sys.stdin.readline())\\n\\n\\ndef print(x):\\n    sys.stdout.write(str(x) + '\\\\n')\\n\\n\\ndef solve():\\n    MOD = int(1e9 + 7)\\n    d, n = readInts()\\n    a = [0] + readInts()\\n    adj: list = [[] for _ in range(n + 1)]\\n    for _ in range(n - 1):\\n        u, v = readInts()\\n        adj[u].append(v)\\n        adj[v].append(u)\\n\\n    vis = [False for _ in range(n + 1)]\\n    f = [0 for _ in range(n + 1)]\\n\\n    def dfs(cur, root):\\n        vis[cur] = True\\n\\n        f[cur] = 1\\n        for neigh in adj[cur]:\\n            if vis[neigh]:\\n                continue\\n            if not (a[root] <= a[neigh] <= a[root] + d):\\n                continue\\n            if a[neigh] == a[root] and neigh < root:\\n                continue\\n            dfs(neigh, root)\\n            f[cur] *= f[neigh] + 1\\n            f[cur] %= MOD\\n\\n    ans = 0\\n    for i in range(1, n + 1):\\n        vis = [False for _ in range(n + 1)]\\n        f = [0 for _ in range(n + 1)]\\n        dfs(i, i)\\n        ans += f[i]\\n        ans %= MOD\\n    print(ans)\\n\\n\\ndef main():\\n    t = 1\\n    # t = readInt()\\n    for _ in range(t):\\n        solve()\\n\\n\\nmain()\\n\", \"mod=10**9+7\\nd,n=map(int,input().split())\\na=[0]+list(map(int,input().split()))\\ntree=[[] for _ in range(n+1)]\\nfor _ in range(n-1):\\n    u,v=map(int,input().split())\\n    tree[u].append(v)\\n    tree[v].append(u)\\n\\ndef dfs(u,root):\\n    visited[u]=True\\n    f[u]=1\\n    for i in tree[u]:\\n        if visited[i]==False:\\n            if a[i]<a[root] or a[i]>a[root]+d:continue\\n            if a[i]==a[root] and i<root: continue  \\n            dfs(i,root)                           \\n            f[u]=(f[u]*(f[i]+1))%(mod)\\nans=0\\nfor i in range(1,n+1):\\n    visited = [False] * (n+1)\\n    f = [0] * (n+1)\\n    dfs(i,i)\\n    ans=(ans+f[i])%mod\\nprint(ans)\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if not ((a[j-1] < a[nach-1]) or (a[j-1] > a[nach-1] + d)) and not ((a[j-1] == a[nach-1]) and (j < nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u)    \\n    \\n# #mas = [[],[2,3,10],[1,6,7],[1],[7],[9],[2],[2,4,11],[9],[5,8,10],[1,9,12],[7],[10]]\\n# mas = [[],[2,3],[1],[1,4],[3]]\\n# a = [0,2,1,3,2]\\n\\n# d = 1\\n\\n# print('mas:',mas)\\n# print('a:',a)\\n\\nk = 0\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    nonlocal k\\n    f[nomer] = 1;\\n    # print(nomer ,\\\"--\\\", nach)\\n    tyt_yge_bili[nomer] = True\\n    # f.append(a[nomer-1])\\n    # # print(f)\\n    # if max(f)-min(f)<=d:\\n    #     k+=1\\n    #     print(f)\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                # print(a[j-1],a[nach-1])\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\", \" \\nd, n = list(map(int,input().split()))\\na = list(map(int, input().split()))\\n\\nmas = [[] for _ in range(n+1)]\\n\\nMOD = 1000000007\\nfor _ in range(n-1):\\n    u, v = list(map(int,input().split()))\\n    mas[u].append(v)\\n    mas[v].append(u) \\n\\n\\ndef dfs(nomer,mas,tyt_yge_bili,f,nach):\\n    f[nomer] = 1;\\n    tyt_yge_bili[nomer] = True\\n    for j in mas[nomer]:\\n        if tyt_yge_bili[j]!=True:\\n            if ((a[j-1] >= a[nach-1]) and (a[j-1] <= a[nach-1] + d)) and ((a[j-1] != a[nach-1]) or (j >= nach)):\\n                dfs(j,mas,tyt_yge_bili,f,nach)\\n                f[nomer] = (f[nomer] * (f[j] + 1)) % MOD\\n                \\n            \\nrez = 0\\nfor z in range(1,n+1):\\n    f = []\\n    tyt_yge_bili = []\\n    for _ in range(n+1):\\n        f.append(0)\\n        tyt_yge_bili.append(0)\\n    dfs(z,mas,tyt_yge_bili,f , z)\\n    rez = (rez + f[z]) % MOD\\n    \\n\\nprint(rez)\\n\\n\\n\\n\\n\\n    \\n\"]",
        "difficulty": "interview",
        "input": "65 6\n71 90 74 84 66 61\n2 6\n3 5\n1 4\n1 3\n1 2\n",
        "output": "25\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/486/D"
    },
    {
        "id": 972,
        "task_id": 1171,
        "test_case_id": 1,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "6 4\n-10 8 2 1 2 6\n",
        "output": "14\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 973,
        "task_id": 1171,
        "test_case_id": 2,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "6 4\n-6 -100 50 -2 -5 -3\n",
        "output": "44\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 974,
        "task_id": 1171,
        "test_case_id": 3,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "6 3\n-6 -100 50 -2 -5 -3\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 975,
        "task_id": 1171,
        "test_case_id": 4,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "25 22\n-4914956 3823889 1216217 -9302864 -1501144 -9443294 434483 -5719996 -687667 -3548437 5740256 3851980 -4631603 942858 4533097 1140983 -2849317 -6558335 -9825551 -6894413 -8391876 -4121113 828750 2790670 -6249249\n",
        "output": "19855113\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 976,
        "task_id": 1171,
        "test_case_id": 5,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "18 68\n114719 573006 5477661 6965010 3970483 -371248 -7480060 -8160596 2503405 4868392 -2014343 -7436533 -8424674 -9053788 -3443512 5156323 5940099 -513728\n",
        "output": "35569098\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 977,
        "task_id": 1171,
        "test_case_id": 6,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "22 80\n2434497 -8599097 6907816 7771164 7230404 -2908880 1754690 368348 8703086 5120382 168967 2421952 9501345 -7432622 -3698408 -3504067 -6489250 -8098116 1116615 6189321 7726685 -1750078\n",
        "output": "67415272\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 978,
        "task_id": 1171,
        "test_case_id": 7,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 100\n-764207 3102944 -2816278 3249966 5606865 -5997439 4365454 1776907 -1954736 1584162 -437416 -964246 -1828950 -9965518 8636253 -6232397 3421258 -7526570 -7457280 7482600 -1554803 1252 6781335 -3233072 9958581 2263691 -3088247 -2356190 7836991 -1918599 -164787 -9946145 -4801498 3171733 4929342 -6133400 1034826 3746419 -8038615 -7593315 -6791676 5749280 -6696702 6056324 1545511 -2745311 7686370 9321150 -140154 1709122\n",
        "output": "111018336\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 979,
        "task_id": 1171,
        "test_case_id": 8,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 100\n-8734657 2820319 -6460809 207497 -3387466 -5965195 6105861 -5401531 93483 -3264387 -5733913 -9659534 7499300 -5262180 1153040 512189 -4957457 -2085620 -3915834 7983796 -3837743 4771325 -5478816 -8343416 -8998426 3377954 8419644 723558 1015175 -3665877 -3809465 9549172 -4889921 -3756677 -5977731 9720140 -9268996 -196754 4003892 1560257 4091405 -7168956 -1832767 3958333 -9616921 -8659411 -4697881 -8603811 -5118948 5590497\n",
        "output": "83156837\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 980,
        "task_id": 1171,
        "test_case_id": 9,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 100\n10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000\n",
        "output": "500000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 981,
        "task_id": 1171,
        "test_case_id": 10,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 100\n-10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000 -10000000\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 982,
        "task_id": 1171,
        "test_case_id": 11,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "2 40\n770493 0\n",
        "output": "770493\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 983,
        "task_id": 1171,
        "test_case_id": 12,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "43 36\n-226248 550723 -330315 -412727 -767811 -822769 -250386 -1024352 -1068872 -640285 635885 142240 -487997 400033 443310 -326685 680938 -220159 -1168632 52976 361161 225561 -654332 139735 -797242 -1085003 246038 -911563 -228890 393078 541254 1194717 -32386 -1537374 -621550 924584 492140 -879298 109514 -256346 446281 1214087 -134327\n",
        "output": "6806487\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 984,
        "task_id": 1171,
        "test_case_id": 13,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "22 18\n115187 345785 137631 -308885 -92565 -95625 982856 -150112 662979 -1031543 -724679 300498 -165435 1040748 570599 -535482 -719162 253579 -732360 907300 -1059030 764784\n",
        "output": "4298480\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 985,
        "task_id": 1171,
        "test_case_id": 14,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 21\n-2966522 -1420421 -8159534 -453462 -6480943 1759044 -4567498 7483598 -8738883 5522786 -719331 -5588662 2484538 -8810884 4418197 -2655715 -9317196 511050 -5462973 7378458 -5701673 -2331559 -9300438 -4238807 5884549 -9377913 7896452 -5696269 -4415774 -9052837 1451067 4991535 -403555 1182232 9510992 -5443347 5117 -7416369 -4902282 -2124047 3380092 -4222761 1163302 7031725 -8406256 4035803 6442505 -7194613 -6435178 -4829904\n",
        "output": "22053427\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 986,
        "task_id": 1171,
        "test_case_id": 15,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 20\n-136648 434079 -483228 -561323 1114121 -399184 1802300 363432 345065 150999 -550071 1014776 241129 -332840 492951 35830 370208 344251 -434176 176300 -38423 449738 89248 163506 523309 169673 119970 21117 383205 -466591 -524266 -298168 -104714 144410 -155132 -956663 496569 464669 -1263614 -486383 513805 -1053992 -901552 -872411 -116108 15594 219371 164859 -371764 255046\n",
        "output": "5822204\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 987,
        "task_id": 1171,
        "test_case_id": 16,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 16\n360849 110595 822481 134472 1681303 1646656 1198921 164861 207413 528158 1015319 392356 -296562 245544 198779 1201365 98120 -281488 -191572 -185912 -732814 -953649 -1230575 811914 -506449 385280 640870 242038 -808864 271877 167816 435249 385045 42481 -552854 -26413 1034017 -561733 -310940 -29951 -1138487 748215 -672874 500945 859665 -904936 -822541 346625 383422 872134\n",
        "output": "9865565\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 988,
        "task_id": 1171,
        "test_case_id": 17,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 25\n690871 528197 -433146 1146123 472041 100562 -96209 1175338 -73870 -193119 561794 -114041 419743 23871 78933 437423 -1051691 300933 -616065 -47199 906262 76454 -153679 812748 133120 -1034052 -398154 157478 382605 -181900 307054 -871819 -543995 869532 -877766 201846 191492 -1258705 977025 -874824 -279810 418429 -124030 -644246 -205078 451385 597464 -512103 -733090 -1057819\n",
        "output": "6510772\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 989,
        "task_id": 1171,
        "test_case_id": 18,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 18\n124274 405562 515538 1029437 338031 -337493 413071 1006237 98997 414826 -600935 -349735 -619739 939087 514669 1000238 190025 -50243 631013 -897369 467488 816879 -651683 -294366 950162 799463 284882 -416754 -298081 -611853 138071 -341270 -487261 687932 -1244506 533808 18633 -137825 -443831 -524459 -287658 -174095 -45604 825670 -896240 -923764 -839266 727354 519093 -309662\n",
        "output": "6112739\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 990,
        "task_id": 1171,
        "test_case_id": 19,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "33 100\n-5469105 -3403763 2398626 -1410778 -6055065 1282336 -5634760 -7087202 -2228277 7812558 -247850 -8206943 5027048 -673409 1524394 4139756 -9408565 -7583575 -5472352 -2410571 3721020 8389408 4091741 -217839 8416269 -3090791 -8143966 6670759 9504264 7787867 -8156978 -208636 -1833521\n",
        "output": "70766046\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 991,
        "task_id": 1171,
        "test_case_id": 20,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "39 81\n-8885131 6107359 3329374 -3909653 4852248 4318185 -4699106 -7729774 -6765855 -4110937 6880398 6348861 861489 -2426676 -5859202 7753290 -5762987 3495780 -8356579 8303906 -2887351 7749443 -3917331 9359042 9911697 -169373 -1390876 -434841 9801982 6201738 977119 1161039 9738174 -5445932 -3052276 -4001330 -6526973 -3815757 1571036\n",
        "output": "108722160\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 992,
        "task_id": 1171,
        "test_case_id": 21,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "1 1\n1\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 993,
        "task_id": 1171,
        "test_case_id": 22,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "1 100\n4847850\n",
        "output": "4847850\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 994,
        "task_id": 1171,
        "test_case_id": 23,
        "question": "Your friend gave you a dequeue D as a birthday present.\nD is a horizontal cylinder that contains a row of N jewels.\nThe values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.\nIn the beginning, you have no jewel in your hands.\nYou can perform at most K operations on D, chosen from the following, at most K times (possibly zero):\n - Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.\n - Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.\n - Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.\nFind the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Constraints-----\n - All values in input are integers.\n - 1 \\leq N \\leq 50\n - 1 \\leq K \\leq 100\n - -10^7 \\leq V_i \\leq 10^7\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN K\nV_1 V_2 ... V_N\n\n-----Output-----\nPrint the maximum possible sum of the values of jewels in your hands after the operations.\n\n-----Sample Input-----\n6 4\n-10 8 2 1 2 6\n\n-----Sample Output-----\n14\n\nAfter the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.\n - Do operation A. You take out the jewel of value -10 from the left end of D.\n - Do operation B. You take out the jewel of value 6 from the right end of D.\n - Do operation A. You take out the jewel of value 8 from the left end of D.\n - Do operation D. You insert the jewel of value -10 to the right end of D.",
        "solutions": "[\"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nreverse_v = v[::-1]\\nlst = [0] * (k + 1)\\nleft = [[]]\\nright = [[]]\\nfor i in range(1, n+1):\\n  left.append(v[:i])\\n  right.append(reverse_v[:i])\\n\\nfor cnt in range(1, k+1):\\n  rest = k - cnt\\n  total = 0\\n  if cnt <= n:\\n    for j in range(cnt+1):\\n      lst_j = left[j] + right[cnt-j]\\n      lst_j.sort()\\n      l = cnt\\n      for idx in range(cnt):\\n        if lst_j[idx] >= 0:\\n          l = idx\\n          break\\n      if l == cnt:\\n        value = 0\\n      else:\\n        flg = min(l, rest)\\n        value = sum(lst_j[flg:])\\n      if value > total:\\n        total = value\\n    lst[cnt] = total\\n\\nans = max(lst)\\nprint(ans)\", \"N, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n            tmp.sort()\\n            ans = max(ans, sum(tmp) - sum([t for t in tmp if t < 0][: i - j]))\\nprint(ans)\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i + j <= k and i + j <= n:\\n            g = v[:i] + v[n-j:]\\n            g.sort()\\n            for l in range(min(k-i-j, i+j)+1):\\n                ans = max(ans, sum(g[l:]))\\n\\nprint(ans)\\n\", \"import heapq\\nimport itertools\\ndef main():\\n    n,k=list(map(int, input().split()))\\n    v=[int(i) for i in input().split()]\\n    #k\\u56deV\\u306e\\u4e21\\u7aef\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u308a\\u8a70\\u3081\\u305f\\u308a\\u3057\\u3066\\u6301\\u3063\\u3066\\u308b\\u3082\\u306e\\u306e\\u4fa1\\u5024\\u306e\\u6700\\u5927\\u5316\\u3092\\u3059\\u308b\\n    v_1 = v[::-1]\\n    #\\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092A\\u3001\\u53f3\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\u3092B\\u3068\\u3057\\u305f\\u3068\\u304d\\u3001\\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u6368\\u3066\\u308b\\u64cd\\u4f5c\\u306f\\uff08k-(A+B)\\uff09\\u56de\\u3067\\u304d\\u308b\\n    #\\u3053\\u308c\\u3092\\u5229\\u7528\\u3057\\u3066\\u5168\\u63a2\\u7d22\\n    res = 0\\n    for i in range(k+1):\\n        j = 0\\n        while(i+j<=min(n,k)):\\n\\n            q = []\\n            tmp = 0\\n            if(i>0):\\n                for l in range(i):\\n                    if(v[l]<0):\\n                        heapq.heappush(q,v[l])\\n                    tmp += v[l]\\n            if(j>0):\\n                for l in range(j):\\n                    if(v_1[l]<0):\\n                        heapq.heappush(q,v_1[l])\\n                    tmp += v_1[l]\\n            for l in range(k-(i+j)):\\n                if(len(q)>0):\\n                    z = heapq.heappop(q)\\n                    tmp -= z\\n            j+=1\\n            res = max(res,tmp)\\n\\n    print(res)\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def abc128_d():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n\\n    for a in range(max(1, k+1)):  # \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n        for b in range(max(1, k-a+1)):  # \\u53f3\\u304b\\u3089\\u53d6\\u308b\\u500b\\u6570\\n            if a+b > n: break\\n            arr = []\\n            if a > 0: arr += v[:a]\\n            if b > 0 and a < n: arr += v[max(a, 0, n-b):]\\n            arr.sort()\\n            for j in range(max(1, a+b)):  # \\u623b\\u3059\\u500b\\u6570\\n                if a+b+j > k: break\\n                score = sum(arr[j:])\\n                ans = max(ans, score)\\n                #print(a, b, j, score, arr)\\n    print(ans)\\n\\ndef __starting_point():\\n    abc128_d()\\n__starting_point()\", \"# \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u30d1\\u30bf\\u30fc\\u30f3\\u306f\\u6700\\u592750\\u901a\\u308a\\n# \\u305d\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\u306f2500\\u901a\\u308a\\n# \\u305d\\u306e\\u5168\\u901a\\u308a\\u306b\\u5bfe\\u3057\\u3066\\u3001\\u30de\\u30a4\\u30ca\\u30b9\\u306e\\u77f3\\u3092\\u5f15\\u3051\\u308b\\u3060\\u3051\\u5f15\\u304f\\u30d1\\u30bf\\u30fc\\u30f3\\u3092\\u8a66\\u3059\\u306850\\n# 125000\\n# \\u3059\\u3079\\u3066\\u8a66\\u3059\\n\\nimport sys\\nreadline = sys.stdin.readline\\n\\nN,K = map(int,readline().split())\\nV = [0] + list(map(int,readline().split())) + [0]\\nN += 2\\nK += 2\\n\\nleftsum = 0\\nleftminus = []\\nans = -(10 ** 9)\\nfor left in range(min(N,K)):\\n  leftsum += V[left]\\n  if V[left] < 0:\\n    leftminus += [V[left]]\\n  rightsum = 0\\n  rightminus = []\\n  limit = min(N - (left + 1) , K - (left + 1))\\n  for right in range(N - 1, N - limit - 1, -1):\\n    rightsum += V[right]\\n    if V[right] < 0:\\n      rightminus += [V[right]]\\n    allsum = leftsum + rightsum\\n    rest = max(K - ((left + 1) + (N - right)),0)\\n    allminus = sorted(leftminus + rightminus)\\n    allsum -= sum(allminus[:rest])\\n    if allsum > ans:\\n      ans = allsum\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nmax_value = -10**9\\nfor kL in range(0, K+1):\\n    for kR in range(0, K+1):\\n        if kL + kR > min(N, K):\\n            continue\\n        vL = V[:kL]\\n        vR = []\\n        if kR:\\n            vR = V[-kR:]\\n        hand = list(sorted(vL + vR, reverse=True))\\n        rest = K - (kL + kR)\\n        for kD in range(rest):\\n            if len(hand) == 0:\\n                break\\n            if hand[-1] < 0:\\n                hand.pop()\\n            else:\\n                break\\n        value = sum(hand[:K])\\n        if value > max_value:\\n            max_value = value\\nprint(max_value)\\n\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=v[:i]+v[::-1][:j]\\n    m=sorted([l[x] for x in range(len(l)) if l[x]<0])[:k-(i+j)]\\n    ans=max(ans, sum(l)-sum(m))\\n    \\nprint(ans)\", \"from heapq import heapify, heappop, heappush\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nINF = 10**9\\n\\ndef h(X, cnt):\\n  S = sum(X)\\n  heapify(X)\\n  limit = cnt\\n  while cnt < K and cnt < 2*limit:\\n    m = heappop(X)\\n    if m >= 0:\\n      break\\n    S -= m\\n    cnt += 1\\n  return S\\n\\nans = 0\\n\\nfor i in range(1, min(N, K)+1):\\n  if i != N:\\n    for j in range(i+1):\\n      X = V[:i-j] + V[N-j:]\\n      ans = max(ans, h(X, i))\\n  else:\\n    ans = max(ans, h(V, i))\\n\\nprint(ans)\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        l = min(k - (a + b), a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\ntarg = min(N, K)\\n\\nans = -float(\\\"inf\\\")\\n\\n#i: \\u5de6\\u304b\\u3089\\u53d6\\u308b\\u56de\\u6570\\nfor i in range(targ+1):\\n    #j: \\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n    for j in range(targ+1-i):\\n        get = sorted(V[:i] + V[N-j:])\\n        tmp = sum(get)\\n        for k in range(min(K-i-j, i+j)):\\n            tmp = max(tmp, tmp-get[k])\\n        \\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"#!/usr/bin/env python3\\nimport sys\\nfrom collections import deque, Counter\\nfrom heapq import heappop, heappush\\nfrom bisect import bisect_right\\nfrom itertools import accumulate\\n\\nsys.setrecursionlimit(10**6)\\nINF = 10**12\\nm = 10**9 + 7\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    # \\u65b9\\u91dd: \\u64cd\\u4f5cA\\u306e\\u56de\\u6570\\u30fb\\u64cd\\u4f5cB\\u306e\\u56de\\u6570\\u3067\\u5168\\u63a2\\u7d22\\n    ans = 0\\n    R = min(N,K)\\n\\n    for i in range(R+1):\\n        for j in range(R+1-i):\\n            tmp = 0\\n            q = []# \\u4fa1\\u5024\\u304c\\u8ca0\\u306e\\u5b9d\\u77f3\\u3092\\u3044\\u308c\\u308b(C)\\n            \\n            for k in range(i):\\n                tmp += V[k]\\n                if V[k] < 0:\\n                    q.append(V[k])\\n            for k in range(j):\\n                tmp += V[N-1-k]\\n                if V[N-1-k] < 0:\\n                    q.append(V[N-1-k])\\n\\n            q.sort()\\n            tmp -= sum(q[:min(len(q),K-i-j)])\\n            ans = max(ans,tmp)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    import copy\\n    n,k = list(map(int,input().split()))\\n    v = list(map(int,input().split()))\\n    ans = 0\\n    for i in range(min(n+1,k+1)):\\n        t = v[0:i]\\n        for j in range(min(n-i+1,k-i+1)):\\n            t_ = copy.deepcopy(t)\\n            t_ += v[n-j:n]\\n            \\n            r = k-i-j\\n            t_ = sorted(t_)\\n            cnt = 0\\n            for l in range(len(t_)):\\n                if t_[l]<0 and cnt<r:\\n                    t_[l] = 0\\n                    cnt += 1\\n                else:\\n                    break\\n            if ans<sum(t_):\\n                ans = sum(t_)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\ninput = sys.stdin.readline\\n\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0  # \\u4f55\\u3082\\u3057\\u306a\\u3044\\u30680\\u306a\\u306e\\u3067\\u3001ans\\u306e\\u521d\\u671f\\u5024\\u306f0\\n\\n\\n# pick->\\u53d6\\u308a\\u51fa\\u3059\\u64cd\\u4f5c\\u3092\\u3059\\u308b\\u56de\\u6570(list\\u3088\\u308a\\u591a\\u304f\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3088\\u3046\\u306b\\u6ce8\\u610f)\\nfor pick in range(min(K+1, N+1)):\\n    # \\u53f3\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n    for right in range(pick+1):\\n        left = pick-right  # \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u6570\\n        pick_list = V[:right]+V[N-left:]\\n\\n        # queue\\u306b\\u623b\\u3059\\u64cd\\u4f5c\\n        pick_list = sorted(pick_list)\\n        tmp_ans = sum(pick_list)  # \\u623b\\u3059\\u524d\\u306e\\u5408\\u8a08\\u5024\\n        rem = min(K-pick, len(pick_list))  # \\u623b\\u3059\\u64cd\\u4f5c\\u3092\\u3067\\u304d\\u308b\\u56de\\u6570\\n        for i in range(rem):  # \\u8ca0\\u306e\\u5024\\u306e\\u3082\\u306e\\u3092\\u3067\\u304d\\u308b\\u3060\\u3051\\u623b\\u3059\\n            if pick_list[i] < 0:\\n                tmp_ans -= pick_list[i]\\n            else:\\n                break\\n        ans = max(tmp_ans, ans)\\nprint(ans)\", \"n,k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = -10**9\\nfor back in range(n+1):\\n    for left in range(min(k-back+1, n+1)):\\n        for right in range(min(k-back-left+1, n-left+1)):\\n            hand = []\\n            hand.extend(v[:left])\\n            if right>0:\\n                hand.extend(v[-right:])\\n            hand.sort()\\n            ans = max(ans, sum(hand[back:]))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\n#\\u5de6\\u304b\\u3089l\\u500b\\u3001\\u53f3\\u304b\\u3089r\\u500b\\u53d6\\u308a\\u51fa\\u3059\\nfor l in range(K + 1):\\n    for r in range(K - l + 1):\\n        if l + r > N: continue\\n        d = K - l - r\\n        now = 0\\n        having = []\\n        for i in range(l):\\n            now += V[i]\\n            having.append(V[i])\\n        for i in range(N-r, N):\\n            now += V[i]\\n            having.append(V[i])\\n\\n        having.sort()\\n        #d\\u56de\\u307e\\u3067\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3(\\u4fa1\\u5024\\u304c\\u30de\\u30a4\\u30ca\\u30b9)\\u3092\\u623b\\u3059\\n        for i in range(d):\\n            if i >= len(having): break\\n            if having[i] >= 0: break\\n            now -= having[i]\\n        ans = max(ans, now)\\n\\nprint(ans)\\n\", \"import heapq\\nN, K = list(map(int, input().split()))\\nd = [int(x) for x in map(int, input().split())]\\ne = d[::-1]\\n\\nres = 0\\nnum = min(N, K)\\nfor i in range(num + 1):\\n    for j in range(num - i, -1, -1):\\n        s = d[:i] + e[:j]\\n        if not s:\\n            continue\\n        heapq.heapify(s)\\n        res = max(res, sum(s))\\n        l = min(i + j, K - (i + j))\\n        while l:\\n            heapq.heappop(s)\\n            res = max(res, sum(s))\\n            l -= 1\\nprint(res)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor a in range(num + 1):\\n    for b in range(num + 1 - a):\\n        trash = k - (a + b)\\n        s = d[:a] + d[-b:] if b != 0 else d[:a]\\n        heapify(s)\\n        for _ in range(min(trash, a + b)):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n                break\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp = v[:x]+v[len(v)-y:]\\n    if x+y >= n:\\n      temp = v\\n    temp1 = []\\n    temp2 = []\\n    for i in temp:\\n      if i >= 0:\\n        temp1.append(i)\\n      else:\\n        temp2.append(i)\\n\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"# equeue\\n\\nfrom collections import deque\\nimport heapq\\n\\n\\ndef main():\\n    N, K = map(int, input().split())\\n    V = list(map(int, input().split()))\\n    res = 0\\n    for pull in range(K+1):\\n        push = K - pull\\n        if push < 0 or push > N:\\n            continue\\n        for left_push in range(push+1):\\n            tmp_que = deque(V)\\n            tmp_heap = []\\n            heapq.heapify(tmp_heap)\\n            right_push = push - left_push\\n            for _ in range(left_push):\\n                heapq.heappush(tmp_heap, tmp_que.popleft())\\n            for _ in range(right_push):\\n                heapq.heappush(tmp_heap, tmp_que.pop())\\n\\n            tmp_res = sum(tmp_que)\\n            for i in range(min(push, pull)):\\n                x = heapq.heappop(tmp_heap)\\n                if x >= 0:\\n                    heapq.heappush(tmp_heap,x)\\n                    break\\n\\n\\n            res = max(res, sum(tmp_heap))\\n    print(res)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import sys, re\\nfrom collections import deque, defaultdict, Counter\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees, log2, gcd\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left, insort, insort_left\\nfrom heapq import heappush, heappop\\nfrom functools import reduce, lru_cache\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef TUPLE(): return tuple(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7 \\n#mod = 998244353\\n#from decimal import *\\n#import numpy as np\\n#decimal.getcontext().prec = 10\\n\\nN, K = MAP()\\nV = LIST()\\n\\nans = 0\\ntmp_sum = 0\\n\\nfor i in range(1, min(K, N)+1):\\n\\tleft = K-i\\n\\tfor j in range(i+1):\\n\\t\\ttmp = V[:j] + V[N-(i-j):]\\n\\t\\ttmp.sort()\\n\\t\\tidx = bisect_left(tmp, 0)\\n\\t\\tans = max(ans, sum(tmp) - sum(tmp[:min(idx, left)]))\\n\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = - 10 ** 19\\n# i: D\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u5b9d\\u77f3\\u306e\\u500b\\u6570\\nfor i in range(min(k + 1, n + 1)):\\n  # j: \\u5de6\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570(i-j\\u304c\\u53f3\\u304b\\u3089\\u3068\\u308b\\u500b\\u6570)\\n  for j in range(i + 1):\\n    inhands = V[:j] + V[n - i + j:]\\n    inhands.sort()\\n    remnum = k - i\\n    val = 0\\n    for h in inhands:\\n      if h < 0 and remnum > 0:\\n        remnum -= 1\\n      else:\\n        val += h\\n    ans = max(ans, val)\\nprint(ans)\", \"N,K=list(map(int,input().split()))\\nV=list(map(int,input().split()))\\nr=min(N,K)\\nans=0\\nfor a in range(r+1):\\n    for b in range(r-a+1):\\n        tmp,minus=0,[]\\n        for i in range(a):\\n            t=V[i]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        for j in range(b):\\n            t=V[N-1-j]\\n            tmp+=t\\n            if t<0:\\n                minus.append(t)\\n        tmp-=sum(sorted(minus)[:K-(a+b)])\\n        ans=max(ans,tmp)\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(min(n, k+1)):\\n    for right in range(min(n-left+1, k-left+1)):\\n        l = v[:left]\\n        l+=v[::-1][:right]\\n        l.sort()\\n        temp = sum(l)\\n        for t in range(max(0, min(len(l), k-left-right))):\\n            if l[t]<0:\\n                temp-=l[t]\\n        ans = max(ans, temp)\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = -1\\n\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        tmp = V[:l] + V[N-r:]\\n        if l+r > N:\\n            break\\n        res = K - (l+r)\\n        if res < 0:\\n            break\\n        tmp.sort()\\n        for i in range(min(res,len(tmp))):\\n            if tmp[0] < 0:\\n                tmp.pop(0)\\n            else:\\n                break\\n        ans = max(ans,sum(tmp))\\n            \\nprint(ans)\", \"import math\\nfrom collections import deque\\nn,k=map(int,input().split())\\nv=list(map(int,input().split()))\\n#k\\u56de\\u307e\\u3067\\u3060\\u3063\\u305f\\u30fc\\u8aa4\\u8aad\\nans=0\\nfor i in range(1,min(k,n)+1):\\n    #\\u53d6\\u308b\\u3068\\u3053\\u308d(\\u5927\\u304d\\u3044\\u65b9\\u3068\\u306f\\u9650\\u3089\\u306a\\u3044\\u306e\\u304b)\\n    for m in range(i+1):\\n        s=v[:m]+v[n-(i-m):]\\n        s.sort()\\n        #\\u53d6\\u3063\\u305f\\u500b\\u6570\\u3088\\u308a\\u591a\\u304f\\u8a70\\u3081\\u3088\\u3046\\u3068\\u3057\\u306a\\u3044\\u3088\\u3046\\u306b\\uff01\\n        #\\u6b8b\\u308ak-i\\u56de\\u3042\\u308b\\n        for j in range(i,min(k,2*i)):#\\u305d\\u308c\\u4ee5\\u4e0a\\u8a70\\u3081\\u3089\\u308c\\u306a\\u3044\\u5834\\u5408\\n            #0\\u3088\\u308a\\u5c0f\\u3055\\u3044\\u5834\\u5408\\u306f\\u66f8\\u304d\\u63db\\u3048\\u308b\\n            if s[j-i]<0:s[j-i]=0\\n        #\\u53d6\\u308a\\u51fa\\u3057\\u305f\\u3084\\u3064\\u306e\\u6700\\u5927(\\u623b\\u3059\\u5834\\u5408\\u306f0)\\n        ans=max(ans,sum(s))\\n        #\\u6b8b\\u308ak-j\\u56de\\u3042\\u308b(break\\u306e\\u6642\\u306ej\\u306f\\u3084\\u3063\\u3066\\u306a\\u3044\\u306e\\u3067)\\n        #\\u3053\\u3053\\u304b\\u3089\\u306f\\u3044\\u3044\\u611f\\u3058\\u306b\\u51fa\\u3059\\u306e\\u3068\\u5165\\u308c\\u308b\\u306e\\u3092\\u7e70\\u308a\\u8fd4\\u3059(\\u5076\\u6570\\u306e\\u5834\\u5408\\u306f\\u3053\\u306e\\u307e\\u307e\\u304b)\\n        #\\u5fc5\\u8981\\u3059\\u3089\\u306a\\u3044\\u3001k\\u56de\\u3092\\u4f7f\\u3044\\u5207\\u3089\\u306a\\u304f\\u3066\\u3044\\u3044\\u306e\\u3067\\n        #print(sum(s))\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n        for lef in range(tak+1):\\n            for rev in range(reverse +1):\\n            \\n                take =[]                    #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                take_V.append(sum(take[rev:]))\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nVrev = V[::-1]\\nans = 0\\nfor a in range(N+1):\\n    for b in range(min(K-a,N-a)+1):\\n        VV = V[:a] + Vrev[:b]\\n        VV.sort(reverse=True)\\n        for k in range(min(K-a-b,len(VV))):\\n            if VV[-1] < 0:\\n                VV.pop()\\n        ans = max(sum(VV),ans)\\nprint(ans)\\n\", \"n,k=[int(x) for x in input().rstrip().split()]\\nv=[int(x) for x in input().rstrip().split()]\\nvr=v[::-1]\\nans=0\\nmi=min(n,k)\\n\\nfor i in range(mi+1):\\n  for j in range(mi-i+1):\\n    minus=0\\n    now=v[:i]+vr[:j]\\n    d=k-(len(now))\\n    now.sort()\\n    md=min(d,len(now))\\n    for s in range(md):\\n      if 0<=now[s]:\\n        break\\n      minus+=now[s]\\n\\n    ans=max(ans,sum(now)-minus)\\nprint(ans)\", \"import bisect\\nN, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\nans = 0\\n\\nfor i in range(1, K + 1):#i\\u500b\\u4ee5\\u4e0b\\u53d6\\u308a\\u51fa\\u3057\\u3001K - i\\u500b\\u4ee5\\u4e0b\\u623b\\u3059\\uff03\\uff11\\uff10\\uff10\\n  nownow = 0\\n  for p in range(i + 1):#[0:i] \\u304b\\u3089 [N - i:N]\\u307e\\u3067\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\u3068\\u3053\\u308d\\u3092\\u9078\\u3079\\u308b#100\\n    #[p:p + N - i]\\u3092\\u53d6\\u308a\\u51fa\\u3055\\u306a\\u3044\\n    if i < N:\\n      now = sorted(V[:p] + V[p + N - i:])\\n    else:\\n      now = V\\n    #print(now, i, p)\\n    index = bisect.bisect_right(now, 0)\\n    #index\\u500b\\u306e\\u8ca0\\u306e\\u6570\\u304c\\u3042\\u308b\\u3002\\n    if K - i >= index:\\n      nownow = sum(now[index:])\\n    else:\\n      nownow = sum(now[K - i:])\\n    ans = max(ans, nownow)\\n    #print(now, nownow, ans)\\n\\nprint(ans)      \\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\n# \\u5de6\\u304b\\u3089\\u4f55\\u500b\\u3001\\u53f3\\u304b\\u3089\\u4f55\\u500b\\u3092\\u5168\\u63a2\\u7d22\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l + r > n:\\n            continue\\n        suteru = k - l - r\\n        get = []\\n        for i in range(l):\\n            get.append(v[i])\\n        for i in range(n - 1, n - 1 - r, -1):\\n            get.append(v[i])\\n        get.sort()\\n        minus = 0\\n        while minus < min(suteru, len(get)) and get[minus] < 0:\\n            minus += 1\\n        ans = max(ans, sum(get[minus:]))\\nprint(ans)\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nvv = [0] * (n + 1)\\nfor i in range(n):\\n    vv[i + 1] = vv[i] + v[i]\\nsumv = vv[n]\\nans = 0\\nfor l in range(n):\\n    for r in range(l - 1, n):\\n        c = r - l + 1\\n        if n - c > k:\\n            continue\\n        x = []\\n        value = sumv - (vv[r + 1] - vv[l])\\n        for i in range(n):\\n            if not l <= i <= r:\\n                x.append(v[i])\\n        x.sort()\\n        for i in range(max(0, min(k - (n - c), len(x)))):\\n            if x[i] < 0:\\n                value -= x[i]\\n            else:\\n                break\\n        ans = max(value, ans)\\nprint(ans)\", \"n, k = list(map(int, input().split()))\\nv = [int(i) for i in input().split()]\\nans = 0\\nfor a in range(min(n, k) + 1):\\n    pa, va = v[:a], v[a:]\\n    for b in range(min(n, k) - a + 1):\\n        pb = pa + va[-b:] if b > 0 else pa\\n        pb.sort()\\n        s = sum(pb)\\n        ans = max(s, ans)\\n        for c in range(min(k - a - b, a + b)):\\n            s -= pb[c]\\n            ans = max(s, ans)\\nprint(ans)\\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nW = V[::-1]\\nR = min(N, K)\\nans = -float('inf')\\nfor left in range(R+1):\\n    for right in range(R+1):\\n        if left+right > R: continue\\n        A = V[:left]+W[:right]\\n        tmp = sum(A)\\n        \\n        rest = K - left - right\\n        A = [k for k in A if k < 0]\\n        A.sort()\\n        if rest >= 0:\\n            tmp -= sum(A[:min(rest, len(A))])\\n        ans = max(tmp, ans)\\n\\nprint(ans)\", \"# \\u521d\\u671f\\u5165\\u529b\\nimport bisect\\nimport heapq\\nimport sys\\ninput = sys.stdin.readline\\nN,K = (int(i) for i in input().split())\\nV = list(map(int, input().split()))\\nV_sorted =sorted(V)\\n\\n#\\u53d6\\u308b\\u500b\\u6570take_num\\u3001\\u8fd4\\u3059\\u500b\\u6570reverse(t +r <=K)\\u500b\\u5168\\u3066\\u6c7a\\u3081\\u308b\\n\\nans =[]\\ntake_V =[] #\\u6700\\u5f8c\\u306b\\u6301\\u3063\\u3066\\u3044\\u308b\\u5b9d\\u77f3\\u306e\\u4fa1\\u5024\\n#V\\u304c\\u5168\\u3066\\u30de\\u30a4\\u30ca\\u30b9\\u306a\\u3089\\u53d6\\u3089\\u306a\\u3044\\nif V_sorted[-1] <=0:\\n    ans =0\\nelse:\\n    take_num =min(K,N)\\n    for tak in range(1,take_num +1):\\n        for lef in range(tak+1):\\n            #for rev in range(reverse +1):\\n            \\n                take =[]  #\\u53d6\\u3063\\u305f\\u5b9d\\u77f3\\n                rig =N-(tak -lef )          #lef:\\u5de6\\u5074\\u306e\\u53d6\\u308b\\u6570 rig:\\u53f3\\u5074\\u306e\\u53d6\\u308b\\u6570\\n                take.extend(V[    :lef])\\n                take.extend(V[rig:   ])\\n                take.sort()\\n                reverse =min(K -tak,tak) #\\u8fd4\\u3059\\u6570\\u306ftake_num\\u3067\\u306f\\u306a\\u304f\\u3001K\\u304b\\u3089\\u5f15\\u304f\\n                ind_0 =bisect.bisect_left(take,0)\\n                rev =min(ind_0,reverse,K -tak)\\n                take_V.append(sum(take[rev:])) #\\u8fd4\\u3059\\u5206\\u3092\\u9664\\u304f\\n        \\n    ans =max(take_V)\\nprint(ans)\", \"from heapq import heappop, heapify\\n\\n\\nN, K, *V = list(map(int, open(0).read().split()))\\n\\n# i: \\u64cd\\u4f5c\\u56de\\u6570\\n# j: \\u53d6\\u308a\\u51fa\\u3059\\u500b\\u6570\\n# k: \\u5de6\\u304b\\u3089\\u53d6\\u308a\\u51fa\\u3059\\u7bc4\\u56f2 [0, k)\\nans = 0\\nfor i in range(K + 1):\\n    for j in range(min(N + 1, i + 1)):\\n        for k in range(j + 1):\\n            right = N - j + k\\n            tmp = V[:k] + V[right:]\\n\\n            put_out_cnt = i - j\\n            heapify(tmp)\\n            while tmp and put_out_cnt > 0:\\n                heappop(tmp)\\n                put_out_cnt -= 1\\n            ans = max(ans, sum(tmp))\\nprint(ans)\\n\", \"from heapq import *\\nn, k = list(map(int, input().split()))\\nd = list(map(int, input().split()))\\n\\nres = 0\\nnum = min(k, n)\\nfor i in range(num + 1):\\n    for j in range(num + 1 - i):\\n        l = min(k - (i + j), i + j)\\n        s = d[:i] + d[-j:] if j != 0 else d[:i]\\n        heapify(s)\\n        for _ in range(l):\\n            tmp = heappop(s)\\n            if tmp > 0:\\n                heappush(s, tmp)\\n        res = max(res, sum(s))\\nprint(res)\\n\", \"n,k=map(int,input().split())\\nv=list(map(int, input().split()))\\nm=min(n,k)\\nans=0\\nfor i in range(m+1): #A\\n  for j in range(m+1-i): #B\\n    l=[]\\n    l=l+v[0:i]+v[n-j:n]\\n    l.sort()\\n    for kk in range(min(k-i-j, i+j)):\\n      if l[kk]<0:\\n        l[kk]=0\\n    ans=max(ans, sum(l))\\nprint(ans)\", \"#!/usr/bin/env python3\\n# -*- coding: utf-8 -*-\\n# \\u554f\\u984c\\uff1ahttps://atcoder.jp/contests/abc128/tasks/abc128_d\\n\\nn, k = list(map(int, input().strip().split()))\\nvalue = list(map(int, input().strip().split()))\\nres = 0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l + r > n:\\n            continue\\n        d = k - l - r\\n        now = 0\\n        s = []\\n        for i in range(l):\\n            now += value[i]\\n            s.append(value[i])\\n        for i in range(r):\\n            now += value[n-i-1]\\n            s.append(value[n-i-1])\\n        s.sort()\\n        for i in range(d):\\n            if i >= len(s):\\n                break\\n            if s[i] > 0:\\n                break\\n            now -= s[i]\\n        res = max(res, now)\\n\\nprint(res)\\n\\n\", \"import bisect\\n\\nN, K = [int(x) for x in input().split()]\\nV = [int(x) for x in input().split()]\\nM = min(N, K)\\n\\nans = 0\\njuwel = []\\nfor l in range(M + 1):\\n    for r in range(M - l + 1):\\n        juwel = sorted(V[:l] + V[N - r:])\\n        s = bisect.bisect_left(juwel, 0)\\n        s = min(s, K - l - r)\\n        juwel = juwel[s:]\\n        ans = max(ans, sum(juwel))\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\nN,K = (int(x) for x in input().rstrip('\\\\n').split())\\nvs = [int(x) for x in input().rstrip('\\\\n').split()]\\ng = [vs[0],vs[-1]]\\nrest = K\\nfor k in [K-1,K]:\\n  for b in range(K//2):\\n    t = k-b\\n    if t>N:\\n      t=N\\n    for i in range(t+1):\\n      v = []\\n      if i <t:\\n        v.extend(vs[:i])\\n        v.extend(vs[-(t-i):])\\n        for _ in range(b):\\n          if min(v)<0:\\n            v.remove(min(v))\\n          else:\\n            break\\n      else:\\n        v.extend(vs[:t])\\n        for _ in range(b):\\n          if min(v)<0:\\n          \\tv.remove(min(v))\\n      g.append(sum(v))\\nif max(g)<0:\\n  print(0)\\nelse:\\n  print(max(g))\", \"n,k=map(int,input().split())\\nv=[int(x) for x in input().split()]\\n\\nr=min(n,k)\\nans=-10**9\\nfor i in range(r+1):\\n  for j in range(r+1):\\n    if i+j > r: continue;\\n    l=sorted(v[:i]+v[::-1][:j])\\n    for a in range(k-(i+j)+1):\\n      ans=max(ans, sum(l[a:]))\\n    \\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nimport heapq\\nturn=min(N,K)\\nm=0\\nfor A in range(turn+1):\\n  for B in range(turn+1-A):\\n    h=[]\\n    heapq.heapify(h)\\n    if A>0:\\n      h=V[:A]+h\\n    if B>0:\\n      h=h+V[len(V)-B:]\\n    heapq.heapify(h)  \\n\\n    ans=sum(h)\\n    for i in range(K-A-B):\\n      if h!=[]:\\n        f=heapq.heappop(h)\\n \\n        if f<0:\\n          ans-=f\\n    m=max(ans,m)\\n\\n\\n    \\nprint(m)   \", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = V[0]\\nfor l in range(min(K, N)+1):\\n    for r in range(min(K, N)-l+1):\\n        v = V[:l] + V[N-r:]\\n        rest = K - (l + r)\\n        v.sort()\\n        tmp = 0\\n        for x in v:\\n            if rest and x < 0:\\n                rest -= 1\\n            else:\\n                tmp += x\\n        if ans < tmp:\\n            ans = tmp\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(min(n, k) + 1):\\n    for j in range(min(n, k) - i + 1):\\n        lis = sorted(v[:i] + v[n - j:])\\n        for l in range(k - i - j + 1):\\n            ans = max(ans, sum(lis[l:]))\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor lg in range(K+1):\\n    for rg in range(K-lg+1):\\n        if lg+rg > N:\\n            break\\n        for ls in range(K-lg-rg+1):\\n            if ls > lg:\\n                break\\n            for rs in range(K-lg-rg-ls+1):\\n                if rs > rg:\\n                    break\\n\\n                lv = V[:lg]\\n                rv = V[N-rg:]\\n\\n                \\\"\\\"\\\"\\n                print(lg, ls, rg, rs)\\n                print(lv, rv)\\n                print(lv[:lg-ls], rv[:rg-rs])\\n                print()\\n                \\\"\\\"\\\"\\n                lv.sort()\\n                lv.reverse()\\n                rv.sort()\\n                rv.reverse()\\n\\n                ans = max(sum(lv[:lg-ls])+sum(rv[:rg-rs]), ans)\\n\\nprint(ans)\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor left in range(n + 1):\\n    for right in range(n + 1):\\n        if left == 0 and right == 0:\\n            continue\\n        rest = k - left - right\\n        if left + right > n or rest < 0:\\n            break\\n        arr = v[:left] + v[n - right:]\\n        minus = 0\\n        neg = [x for x in arr if x < 0]\\n        if len(neg) > 0:\\n            neg.sort()\\n            minus = sum(neg[:rest])\\n        ans = max(ans, sum(arr) - minus)\\n\\nprint(ans)\", \"from heapq import heappush, heappop\\nfrom random import randint\\nfrom time import time\\n\\nclass PriorityQueue(object):\\n    def __init__(self):\\n        self.queue = []\\n    def push(self, value):\\n        heappush(self.queue, value)\\n    def pop(self):\\n        return heappop(self.queue)\\n    def __len__(self):\\n        return len(self.queue)\\n    def __contains__(self, item):\\n        return item in self.queue\\n\\nn,m = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\n#\\u53d6\\u5f97\\u3059\\u308b\\u64cd\\u4f5c\\u3092\\u4f55\\u56de\\u884c\\u3046\\u304b\\uff1f\\nfor i in range(0,m+1):\\n\\n\\t#\\u64cd\\u4f5c\\u6b8b\\u308a\\u56de\\u6570\\n\\tlef = m-i\\n\\n\\thp = PriorityQueue()\\n\\t#\\u5de6\\u5074\\u304b\\u3089\\u3069\\u308c\\u3060\\u3051\\u53d6\\u308b\\u304b\\uff1f\\n\\tfor j in range(0,i+1):\\n\\t\\tif j>n :\\n\\t\\t\\tbreak\\n\\n\\t\\t#\\u5de6\\u5074\\u304b\\u3089j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,j):\\n\\t\\t\\thp.push(v[k])\\n\\n\\t\\t#\\u53f3\\u5074\\u304b\\u3089 i-j\\u500b\\u53d6\\u308b\\n\\t\\tfor k in range(0,i-j):\\n\\t\\t\\tif n-1-k < j:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\thp.push(v[n-1-k])\\n\\n\\t\\t#minus\\u306f\\u623b\\u3059\\n\\t\\tfor k in range(0,lef):\\n\\t\\t\\tif len(hp) == 0:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\ttop = hp.pop()\\n\\t\\t\\tif top>=0 :\\n\\t\\t\\t\\thp.push(top)\\n\\t\\t\\t\\tbreak\\n\\n\\t\\tss = 0\\n\\t\\twhile len(hp)>0:\\n\\t\\t\\tss += hp.pop()\\n\\n\\t\\tans = max(ans,ss)\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nR = min(N,K)\\npoint = 0\\nmax_point = 0\\nhand = []\\n\\nfor i in range(R+1):\\n    for a in range(i+1):\\n        if i==a:\\n            hand = V[:a]\\n        else:\\n            hand = V[:a]+V[-(i-a):]\\n        negahand = [k for k in hand if k < 0]\\n        negahand.sort()\\n        if K-i > len(negahand):\\n            trash = -sum(negahand)\\n        else:\\n            trash = -sum(negahand[:K-i])\\n        point = sum(hand) + trash\\n        max_point = max(max_point,point)\\n\\nprint(max_point)\", \"n , k = list(map(int, input().split()))\\nv = list(map(int,input().split()))\\nma = float('INF')*(-1)\\nm = min(n,k)\\nfor i in range(m+1):\\n    for j in range(m+1-i):\\n        vl=v[:i]\\n        vr=v[n-j:]\\n        ans = sum(vl)+sum(vr)\\n        vb=vl+vr\\n        vb.sort()\\n        m2 = min(k-i-j, len(vb))\\n        for l in range(m2):\\n            if vb[l]<0:\\n                ans-=vb[l]\\n            else:\\n                break\\n        ma = max(ans,ma)\\nif ma<0:\\n    ma = 0\\nprint(ma)\\n\", \"jewel, query = map(int, input().split())\\nt = min(jewel, query)\\ndq = [int(i) for i in input().split()]\\nans = []\\n\\nfor i in range(t+1):\\n  for j in range(i+1):\\n    have = dq[:j] + dq[(jewel-i+j):]\\n    drop = query - len(have)\\n\\n    for k in range(drop):\\n      if not have:\\n        break\\n      \\n      cand = min(have)\\n      if cand < 0:\\n        have.remove(cand)\\n      else:\\n        break\\n        \\n    ans.append(sum(have))\\n    \\nprint(max(ans))\", \"# \\u89e3\\u8aac\\u3092\\u53c2\\u8003\\u306b\\u4f5c\\u6210\\n# import sys\\n# sys.setrecursionlimit(10 ** 6)\\n# import bisect\\nfrom collections import deque\\nimport heapq\\n\\n\\n# from decorator import stop_watch\\n#\\n#\\n# @stop_watch\\ndef solve(N, K, Vi):\\n    ans = 0\\n    for ab in range(min(N, K) + 1):\\n        for a in range(ab + 1):\\n            b = ab - a\\n            have = Vi[:a] + Vi[N - b:]\\n            have.sort()\\n            s = sum(have)\\n            for i in range(K - ab):\\n                if len(have) <= i:\\n                    break\\n                elif have[i] < 0:\\n                    s += abs(have[i])\\n                else:\\n                    break\\n            ans = max(ans, s)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    N, K = list(map(int, input().split()))\\n    Vi = [int(i) for i in input().split()]\\n    solve(N, K, Vi)\\n\\n    # # test\\n    # import random\\n    # from func import random_str\\n    # N, K = 100, 50\\n    # Vi = [random.randint(-(10 ** 7), 10 ** 7) for _ in range(N)]\\n    # solve(N, K, Vi)\\n\\n__starting_point()\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = 0\\n\\nfor gets in range(min(N, K)+1):\\n    for lgets in range(gets+1):\\n        rgets = gets - lgets\\n        haves = V[:lgets]+V[(N-rgets):]\\n        haves.sort()\\n        removes = K - gets\\n        for i in range(min(removes, len(haves))):\\n            if haves[i] < 0:\\n                haves[i] = 0\\n            else:\\n                break\\n        ans = max(ans, sum(haves))\\nprint(ans)\\n\\n\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        a = sorted(v[:l] + v[n - r:])\\n        m = min(k - l - r, len(a) - 1)\\n        ans = max(ans, sum(a[m:] if a[m] <\\n                           0 else filter(lambda x: x > 0, a)))\\nprint(ans)\", \"import sys\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, K = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n\\n    ans = -float(\\\"inf\\\")\\n    for t in range(min(N, K) + 1):\\n        s = K - t\\n        for l in range(t + 1):\\n            r = t - l\\n            gem = V[:l]\\n            gem += V[-r:] if r != 0 else []\\n            gem.sort()\\n            value = sum(gem)\\n            for i in range(min(s, t)):\\n                if gem[i] < 0:\\n                    value -= gem[i]\\n                else:\\n                    break\\n            ans = max(ans, value)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10000000)\\n\\nn,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans=0\\nfor l in range(k+1):\\n    for r in range(k-l+1):\\n        if l+r > n :break\\n\\n        d=k-l-r\\n        now=0\\n        s=[]\\n\\n        for i in range(l):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        for i in range(n-r,n):\\n            now+=v[i]\\n            s.append(v[i])\\n\\n        s.sort()\\n        for i in range(d):\\n            if(i >= len(s)):break\\n            if s[i] > 0 :break\\n\\n            now -= s[i]\\n\\n        ans=max(now,ans)\\nprint(ans)\", \"import heapq\\nfrom collections import deque\\nn , k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(n+1):\\n    for j in range(n+1):\\n        if i+j > n or i+j > k:\\n            continue\\n        p = v[:]\\n        d = deque(p)\\n        a = []\\n        heapq.heapify(a)\\n        for t in range(i):\\n            heapq.heappush(a,d.popleft())\\n        for l in range(j):\\n            heapq.heappush(a,d.pop())\\n        nokori = k-i-j\\n        for s in range(nokori):\\n            if a:\\n                now = heapq.heappop(a)\\n                if now > 0:\\n                    heapq.heappush(a,now)\\n                    break\\n            else:\\n                break\\n        ans = max(ans,sum(a))\\nprint(ans)\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\nr,ans=min(N,K),0\\nfor a in range(r+1):\\n  for b in range(r-a+1):\\n    tmp,minus=0,[]\\n    for i in range(a):\\n      tmp+=V[i]\\n      if V[i]<0:\\n        minus.append(V[i])\\n    for j in range(b):\\n      tmp+=V[N-j-1]\\n      if V[N-j-1]<0:\\n        minus.append(V[N-j-1])\\n    tmp-=sum(sorted(minus)[:K-(a+b)])\\n    ans=max(ans,tmp)\\nprint(ans)\", \"from collections import deque\\n\\nn, k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nw = deque(v)\\nans = 0\\nfor i in range(k+1):\\n    if i > n:\\n        break\\n    s = v[:i]\\n    for j in range(k+1-i):\\n        if j > n-i:\\n            break\\n        t = sorted(v[n-j:] + s, reverse=True)\\n        u = sum(t)\\n        for l in range(k+1-i-j):\\n            ans = max(ans, u)\\n            if not t:\\n                break\\n            u -= t.pop()\\nprint(ans)\", \"N, K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor a in range(min(N,K)+1):\\n    for b in range(min(N,K)-a+1):\\n        cur = V[:a] + V[N-b:]\\n        cur.sort()\\n\\n        idx = 0\\n        while idx < min(K-a-b, len(cur)):\\n            if cur[idx] > 0: break\\n            else: idx += 1\\n        \\n        ans = max(ans, sum(cur[idx:]))\\n    \\nprint(ans)\", \"n, k, *a = list(map(int, open(0).read().split()))\\n\\nans = 0\\nfor l in range(0, k + 1):\\n    for r in range(0, k + 1 - l):\\n        if l == r == 0 or l + r > n:\\n            continue\\n        b = sorted(a[:l] + a[n - r:])\\n        c = min(k - l - r, len(b) - 1)\\n        ans = max(ans, sum(b[c:] if b[c] < 0 else [x for x in b if x > 0]))\\nprint(ans)\\n\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\nM = min(N,K)\\nans = 0\\n\\nfor i in range(M+1):\\n  for j in range(M-i+1):\\n    G = V[:i]+V[N-j:]\\n    G.sort()\\n    for j in range(K-i-j+1):\\n      ans = max(ans,sum(G[j:]))\\n\\nprint(ans)\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nresult = 0\\nfor a in range(N + 1):\\n    for b in range(N + 1):\\n        if a + b > K or a + b > N:\\n            break\\n        out = V[:a] + V[N-b:]\\n        out.sort()\\n        sum_val = sum(out)\\n        for i in range(min(a + b, K - a - b)):\\n            if out[i] >= 0:\\n                break\\n            sum_val -= out[i]\\n        result = max(result, sum_val);\\nprint(result)\", \"import heapq\\nfrom collections import deque\\nfrom functools import lru_cache\\n\\nN,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nans = 0\\n\\nfor l in range(0,K + 1):\\n  for r in range(0,K - l + 1):\\n    d = K - r - l\\n    tmp = 0\\n    have = []\\n    if l + r > N:\\n      continue\\n    for i in range(0,l):\\n      tmp += V[i]\\n      have.append(V[i])\\n    for j in range(N - r,N):\\n      tmp += V[j]\\n      have.append(V[j])\\n    h = len(have)\\n    #print(have)\\n    have = sorted(have)\\n    for k in range(0,d):\\n      #print(have,sum(have))\\n      if k >= h:\\n        break\\n      if have[k] > 0:\\n        break\\n      tmp -= have[k]\\n    ans = max(tmp,ans)\\n    \\nprint(ans)\\n\", \"n, k = map(int,input().split())\\ntarg = min(n, k)\\nv = list(map(int,input().split()))\\n\\\"\\\"\\\"\\ni\\u2026\\u5de6\\u3092\\u53d6\\u308b\\u56de\\u6570\\nj\\u2026\\u53f3\\u3092\\u53d6\\u308b\\u56de\\u6570\\n\\u6b8b\\u308a\\u56de\\u6570 k-(i+j)\\n\\\"\\\"\\\"\\nans = -10**18\\nfor i in range(targ+1):\\n\\tfor j in range(targ-i+1):\\n\\t\\t#print(i, j)\\n\\t\\tnl = v[:i] + v[n-j:]\\n\\t\\tnl.sort()\\n\\t\\t#print(nl)\\n\\t\\ttmp = sum(nl)\\n\\t\\tfor t in range(min(k-(i+j), i+j)):\\n\\t\\t\\ttmp = max(tmp, tmp-nl[t])\\n\\t\\t\\t#print(\\\"\\u884c\\u3044\\u307e\\u3057\\u305f\\u3088!\\\")\\n\\t\\t#print(tmp)\\n\\t\\tans = max(tmp, ans)\\nprint(ans)\", \"def solve():\\n  ans = 0\\n  N, K = list(map(int, input().split()))\\n  V = list(map(int, input().split()))\\n  A = V[:]*2\\n  for k in range(K,-1,-1): #k\\u306f\\u64cd\\u4f5c\\u56de\\u6570\\n    if k>=N:\\n      suteru = k-N\\n      a = V[:]\\n      a.sort()\\n      ans = max(ans,sum(a[suteru:]))\\n    else:\\n      for i in range(k//2+1,k+1): #i\\u306f\\u53d6\\u308b\\u56de\\u6570\\n        suteru = k-i\\n        for j in range(i+1):\\n          a = A[N-j:N+i-j]\\n          a.sort()\\n          ans = max(ans,sum(a[suteru:]))\\n  return ans\\nprint((solve()))\\n\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    v = list(map(int, input().split()))\\n    reversed_v = [0] + v[::-1]\\n    v = [0] + v\\n    ans = -float(\\\"inf\\\")\\n    for i in range(n + 1):\\n        for j in range(n + 1):\\n            if i + j > min(n, k):\\n                continue\\n            now_k = k - i - j\\n            if now_k < 0:\\n                continue\\n            q = []\\n            for l in range(1, i + 1):\\n                q.append(v[l])\\n            for l in range(1, j + 1):\\n                q.append(reversed_v[l])\\n            q.sort()\\n            for l in range(len(q)):\\n                if q[l] < 0 and now_k > 0:\\n                    q[l] = 0\\n                    now_k -= 1\\n            ans = max(ans, sum(q))\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"N,K=map(int,input().split())\\nV=list(map(int,input().split()))\\n\\nans=0\\nfor l in range(N+1):\\n  for r in range(N+1):\\n    if l+r>N or l+r>K:break\\n    arr=V[:l]\\n    if r:arr+=V[-r:]\\n    arr.sort()\\n    rem=K-l-r\\n    tmp=0\\n    for a in arr:\\n      if a<0 and rem:\\n        rem-=1\\n      else:\\n        tmp+=a\\n    ans=max(ans,tmp)\\nprint(ans)\", \"N,K = map(int,input().split())\\nV = list(map(int,input().split()))\\n\\nans = 0\\nfor l in range(N+1):\\n    for r in range(N+1):\\n        if l+r > N or l+r > K: break\\n        arr = V[:l]\\n        if r: arr += V[-r:]\\n        arr.sort()\\n        rem = K - l - r\\n        tmp = 0\\n        for a in arr:\\n            if a < 0 and rem:\\n                rem -= 1\\n            else:\\n                tmp += a\\n        ans = max(ans, tmp)\\nprint(ans)\", \"import sys\\nfrom bisect import bisect_left\\n\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\n\\n\\ndef main():\\n    N, K, *V = list(map(int, read().split()))\\n\\n    ans = 0\\n    for k in range(min(N, K) + 1):\\n        for left in range(k + 1):\\n            right = k - left\\n            tmp = V[:left] + (V[-right:] if right else [])\\n            tmp.sort()\\n\\n            idx = bisect_left(tmp, 0)\\n            if idx > K - k:\\n                idx = K - k\\n\\n            this_ans = sum(tmp[idx:])\\n            if ans < this_ans:\\n                ans = this_ans\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n, k = map(int,input().split())\\nv = list(map(int,input().split()))\\n\\nans = 0\\n\\nfor l in range(n):\\n    for r in range(n+1)[::-1]:\\n        if l <= r and 0 <= (l+n-r) <= k: # l+n-r\\u500b\\u62fe\\u3046\\n            q = v[:l] + v[r:n]\\n            q.sort(reverse = True)\\n            \\n            for i in range(k-(l+n-r)):\\n                if q == []:\\n                    break\\n                if q[-1] < 0:\\n                    q.pop()\\n                else:\\n                    break\\n\\n            ans = max(ans, sum(q))\\n        else:\\n            continue\\n\\nprint(ans)\", \"#import math\\n#import itertools\\n#import numpy as np\\nfrom collections import deque\\n\\n\\nINT = lambda: int(input())\\nINTM = lambda: map(int,input().split())\\nSTRM = lambda: map(str,input().split())\\nSTR = lambda: str(input())\\nLIST = lambda: list(map(int,input().split()))\\nLISTS = lambda: list(map(str,input().split()))\\n\\ndef do():\\n    n,k=INTM()\\n    d=LIST()\\n    #q=deque(d)\\n    nkmin=min(n,k)\\n    stop=10**9\\n    ans=0\\n    for i1 in range(1,1+nkmin):\\n        for i2 in range(i1+1):\\n            temp=d[0:i2]+d[n-(i1-i2):]\\n            #print(temp,i1,i2)\\n            temp=sorted(temp)\\n            #print(temp,i1,i2)\\n            for i3 in range(i1):\\n                if temp[i3]>=0 or i3>=k-i1:\\n                    stop=i3\\n                    break\\n                stop=10**9\\n            if stop!=10**9:\\n                #print(temp[i3:],1)\\n                ans=max(sum(temp[i3:]),ans)\\n    \\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    do()\\n__starting_point()\", \"import heapq\\n\\nN, K = map(int, input().split())\\n\\nV = list(map(int, input().split()))\\n\\nans = 0\\nfor i in range(1, K+1):\\n    if i > N:\\n        continue\\n    for j in range(i+1):\\n        tmp = 0\\n        q = V[:j] + V[N-i+j:]\\n        heapq.heapify(q)\\n        for _ in range(min(i, K-i)):\\n            tmp_min = heapq.heappop(q)\\n            ans = max(ans, sum(q))\\n            if tmp_min > 0:\\n                ans = max(ans, sum(q)+tmp_min)\\n                break\\n        ans = max(ans, sum(q))\\n\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = [int(i) for i in input().split()]\\nans = 0\\nfor M in range(K+1):\\n    for m in range(M+1):\\n        ls = []\\n        for i, x in enumerate(V):\\n            if i < m:\\n                ls.append(x)\\n        for i, x in enumerate(V[::-1]):\\n            if i < min(M-m, N-m):\\n                ls.append(x)\\n        ls.sort()\\n        s = sum(ls)\\n        for i, x in enumerate(ls):\\n            if i < K-M and x < 0:\\n                s -= x\\n        ans = max(ans, s)\\nprint(ans)\\n\", \"import sys\\nimport copy\\nimport heapq\\nfrom collections import deque\\ninput = lambda: sys.stdin.readline().rstrip()\\nINF = 10**9 + 7\\n\\n\\ndef solve():\\n    N, K = list(map(int, input().split()))\\n    V = deque(list(map(int, input().split())))\\n\\n    ans = -INF\\n    for n in range(1, N + 1):\\n        if n > K:\\n            break\\n        for rn in range(n + 1):\\n            ln = n - rn\\n            tv = copy.copy(V)\\n            pv = 0\\n            mv = []\\n            heapq.heapify(mv)\\n            for _ in range(rn):\\n                v = tv.pop()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n            for _ in range(ln):\\n                v = tv.popleft()\\n                if v >= 0:\\n                    pv += v\\n                else:\\n                    heapq.heappush(mv, v)\\n\\n            mvlen = min(K - n, len(mv))\\n            for _ in range(0, mvlen):\\n                heapq.heappop(mv)\\n\\n            #print(ln, rn, pv, mv)\\n            ans = max(ans, pv + sum(mv))\\n\\n    print((max(0, ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"def readinput():\\n    n,k=list(map(int,input().split()))\\n    v=list(map(int,input().split()))\\n    return n,k,v\\n\\ndef main(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for kk in range(k+1):\\n        if n<kk:\\n            own=ruisekiL[n]\\n            m=0\\n            while m<n and vv[m][1]<0: \\n                own-=vv[i][1]\\n                m+=1\\n            #print(own)\\n            maxown=max(maxown,own)\\n        else:\\n            for l in range(kk//2+1):\\n                for i in range(kk-l+1):\\n                    j=kk-i-l\\n                    jj=n-j+1\\n                    own=ruisekiL[i]+ruisekiR[j]\\n                    #print(i,j,own)\\n                    m=0\\n                    mcnt=0\\n                    while(mcnt<l):\\n                        if i<vv[m][0] and vv[m][0]<jj:\\n                            m+=1\\n                            continue\\n                        #print(i,jj,vv[m])\\n                        own-=vv[m][1]\\n                        mcnt+=1\\n                        m+=1\\n                    #print(own)\\n                    maxown=max(own,maxown)\\n    return maxown\\n\\ndef main2(n,k,v):\\n    ruisekiL=[0]*(n+1)\\n    ruisekiR=[0]*(n+1)\\n    for i in range(1,n+1):\\n        ruisekiL[i]=ruisekiL[i-1]+v[i-1]\\n        ruisekiR[i]=ruisekiR[i-1]+v[n-i]\\n    vv=[]\\n    for i in range(n):\\n        vv.append((i+1,v[i]))\\n    vv.sort(key=lambda x:x[1])\\n    #print(vv)\\n\\n    maxown=-10**7\\n    for i in range(n+1):\\n        for j in range(n+1):\\n            if i+j>k or i+j>n:\\n                break\\n            jj=n-j+1\\n            own=ruisekiL[i]+ruisekiR[j]\\n            #print(i,j,own)\\n            l=min(i+j,k-i-j)\\n            m=0\\n            mcnt=0\\n            while(mcnt<l):\\n                if i<vv[m][0] and vv[m][0]<jj:\\n                    m+=1\\n                    continue\\n                if vv[m][1]>=0:\\n                    break\\n                own-=vv[m][1]\\n                mcnt+=1\\n                m+=1\\n            #print(own)\\n            maxown=max(own,maxown)\\n    return maxown\\n\\ndef __starting_point():\\n    n,k,v=readinput()\\n    ans=main2(n,k,v)\\n    print(ans)\\n\\n__starting_point()\", \"import heapq\\nimport copy\\nN,K = map(int, input().split())\\nV_list= list(map(int, input().split()))\\n\\nselected = []\\nleft = []\\n\\nmax_sum = 0\\nleft_sum = 0\\nfor l in range(N):\\n    selected = copy.deepcopy(left)\\n    lr_sum = left_sum\\n    for r in range(N-l+1):\\n        # skip\\n        stock = K-l-r\\n        if stock < 0:\\n            break\\n        ri = N-r-1\\n        select_cnt = len(selected)\\n        tmp_sum = lr_sum\\n        max_sum = max(max_sum, tmp_sum)\\n        popper = copy.deepcopy(selected)\\n\\n        for i in range(stock):\\n            if i >= select_cnt:\\n                break\\n            \\n            pop_v = heapq.heappop(popper)\\n            if pop_v > 0:\\n                break\\n            tmp_sum -= pop_v\\n            max_sum = max(max_sum, tmp_sum)\\n\\n        r_val = V_list[ri]\\n        lr_sum += r_val\\n        heapq.heappush(selected, r_val)\\n\\n    l_val = V_list[l]\\n    left_sum += l_val\\n    heapq.heappush(left, l_val)\\n\\nprint(max_sum)\", \"from collections import deque\\nn, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\nd = deque(v)\\n\\nt = min(n, k)\\nscore = 0\\nfor a in range(t+1):\\n    for b in range(t-a+1):\\n        tmp_d = d.copy()\\n        jems = []\\n        for i in range(a):\\n            jems.append(tmp_d.pop())\\n        for i in range(b):\\n            jems.append(tmp_d.popleft())\\n        jems.sort()\\n        for i in range(min(len(jems), k-b-a)):\\n            if jems[i] < 0:\\n                jems[i] = 0\\n            else:\\n                break\\n        tmp_score = sum(jems)\\n        if tmp_score > score:\\n            score = tmp_score\\nprint(score)\\n\", \"from collections import deque\\nimport heapq\\nn,k = map(int,input().split())\\nv = deque(list(map(int,input().split())))\\n\\nans = 0\\nfor i in range(min(n,k)+1):\\n    for j in range(min(n,k)-i+1):\\n        l = [0]\\n        r = [0]\\n        mm = [0]\\n        for p in range(i):\\n            if v[p] > 0:\\n                l.append(v[p])\\n            else:\\n                heapq.heappush(mm,v[p])\\n        for o in range(j):\\n            if v[-o-1] > 0:\\n                r.append(v[-o-1])\\n            else:\\n                heapq.heappush(mm,v[-o-1])\\n        for m in range(min(k-i-j,len(mm))):\\n            heapq.heappop(mm)\\n        ans = max(ans,sum(l)+sum(r)+sum(mm))\\nprint(ans)\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\n# input = sys.stdin.readline    ####\\ndef int1(x): return int(x) - 1\\ndef II(): return int(input())\\ndef MI(): return list(map(int, input().split()))\\ndef MI1(): return list(map(int1, input().split()))\\ndef LI(): return list(map(int, input().split()))\\ndef LI1(): return list(map(int1, input().split()))\\ndef LLI(rows_number): return [LI() for _ in range(rows_number)]\\ndef MS(): return input().split()\\ndef LS(): return list(input())\\ndef LLS(rows_number): return [LS() for _ in range(rows_number)]\\ndef printlist(lst, k=' '): print((k.join(list(map(str, lst)))))\\nINF = float('inf')\\n# from math import ceil, floor, log2\\nfrom collections import deque\\n# from itertools import combinations as comb, combinations_with_replacement as comb_w, accumulate, product, permutations\\nfrom heapq import heapify, heappop, heappush\\n# import numpy as np\\n# from numpy import cumsum  # accumulate\\n\\ndef solve():\\n    N, K = MI()\\n    V = LI()\\n    max_iter = min(N, K)\\n    ans = 0\\n    for a in range(max_iter+1):\\n        left = V[:a]\\n        for b in range(0, max_iter-a+1):\\n            right = V[-b:] if b != 0 else []\\n            k = K - a - b\\n            # print(a, b, k)\\n            # print(left, right)\\n            lst = sorted(left + right)\\n            # print(lst)\\n            sm = sum(lst)\\n            l = len(lst)\\n            for kk in range(1, k+1):\\n                if kk < l:\\n                    tmp = lst[kk-1]\\n                    if tmp >= 0: break\\n                    sm -= tmp\\n                    # print(tmp)\\n                else:\\n                    break\\n            ans = max(ans, sm)\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"n, k = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\n\\ndef get_jewelries(box, left_pop, right_pop):\\n    if left_pop + right_pop >= len(box):\\n        return box[:]\\n\\n    left = box[:left_pop]\\n    right = box[-right_pop:] if right_pop else []\\n\\n    return left + right\\n\\n\\ncandidates = []\\npop_max = min(k, n)\\nfor pop_count in range(pop_max+1):\\n    residue = k - pop_count\\n\\n    for left_pop in range(pop_count+1):\\n        right_pop = pop_count - left_pop\\n        jewelries = get_jewelries(V, left_pop, right_pop)\\n        jewelries.sort(reverse=True)\\n\\n        for _ in range(residue):\\n            if not jewelries:\\n                break\\n            if jewelries[-1] < 0:\\n                jewelries.pop()\\n\\n        value = sum(jewelries)\\n        candidates.append(value)\\n\\nprint((max(candidates)))\\n\", \"n,k = map(int, input().split())\\nv = list(map(int, input().split()))\\n\\nans = 0\\nfor x in range(k+1):# LeftHand\\n  for y in range(k-x+1):# RightHand\\n    temp1 = []\\n    temp2 = []\\n    if x+y >= n:\\n      for i in v:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    else:\\n      for i in v[:x]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n      for i in v[len(v)-y:]:\\n        if i >= 0:\\n          temp1.append(i)\\n        else:\\n          temp2.append(i)\\n    temp2.sort()\\n    ans = max(ans, sum(temp1) + sum(temp2[k-x-y:]))\\nprint(ans)\", \"n,k = list(map(int,input().split()))\\nv = list(map(int,input().split()))\\nans = 0\\nfor a in range(min(n,k)+1):\\n    for b in range(min(n,k)-a+1):\\n        c = v[:a] + v[n-b:]\\n        c = sorted(c)\\n        for i in range(k-a-b):\\n            if len(c) == 0:\\n                break\\n            if c[0] < 0:\\n                c = c[1:]\\n            else:\\n                break\\n        ans = max(ans,sum(c))\\nprint(ans)\\n\", \"import math,itertools,fractions,heapq,collections,bisect,sys,queue,copy\\n\\nsys.setrecursionlimit(10**7)\\ninf=10**20\\nmod=10**9+7\\ndd=[(-1,0),(0,1),(1,0),(0,-1)]\\nddn=[(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\n# def LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef LS(): return sys.stdin.readline().split()\\ndef S(): return input()\\n\\ndef main():\\n  n,k=LI()\\n  v=LI()\\n\\n  ans=0\\n  for i in range(k):\\n    if i+1<n:\\n      for l in range(i+1):\\n        r=i-l\\n  \\n        _ans=0\\n        _v=[]\\n        for x in range(l+1):\\n          _ans+=v[x]\\n          _v.append(v[x])\\n        for x in range(r):\\n          _ans+=v[n-x-1]\\n          _v.append(v[n-x-1])\\n\\n        _v.sort()\\n        for j in range(min(len(_v),k-i-1)):\\n          if _v[j]<0:\\n            _ans-=_v[j]\\n          else:\\n            break\\n        # print(_v,l,r,_ans)\\n        ans=max(ans,_ans)\\n    else:\\n      _v=v.sort()\\n      _ans=sum(v)\\n      _k=n-k-1\\n      for j in range(_k):\\n        if _v[j]<0:\\n          _ans-=_v[j]\\n        else:\\n          break\\n      ans=max(ans,_ans)\\n\\n  return ans\\n\\n# main()\\nprint((main()))\\n\", \"n, k = list(map(int, input().split()))\\nv = list(map(int, input().split()))\\n\\nres = 0\\nr = min(n, k)\\nfor i in range(r+1):\\n    for j in range(r-i+1):\\n        in_hand = v[:i] + v[n-j:] if i + j < n else v[:]\\n        in_hand.sort(reverse=True)\\n        for _ in range(k-len(in_hand)):\\n            if not in_hand or in_hand[-1] >= 0:\\n                break\\n            in_hand.pop()\\n        res = max(res, sum(in_hand))\\nprint(res)\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\nleft = [0]*(K+1)\\nright = [0]*(K+1)\\n\\nfor k in range(1,min(N+1,K+1)):\\n    max_vk = 0\\n    for j in range(k//2+1):        \\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(V[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    left[k] = max_vk\\n\\nVrev = V[::-1]\\n\\nfor k in range(1,K+1):\\n    max_vk = 0\\n    for j in range(k//2+1):\\n        push = j\\n        pop = k-j\\n        #print(k,pop,push)\\n        if pop>N:\\n            continue\\n        else:\\n            tmp = sum(sorted(Vrev[:pop])[push:])\\n        max_vk = max(max_vk,tmp)\\n    right[k] = max_vk\\n\\nfor i in range(len(right)-1):\\n    if right[i+1]<right[i]:\\n        right[i+1]= right[i]\\n    if left[i+1]<left[i]:\\n        left[i+1]=left[i]\\n\\nans = 0\\nNK = K\\nmax_ans = 0\\nfor v in V:\\n    if v>0:\\n        max_ans += v\\n    \\nfor i in range(NK+1):\\n    ans = max(ans,right[i]+left[NK-i])\\n\\nprint((min(max_ans,ans)))\\n\", \"N,K = list(map(int,input().split()))\\nV = list(map(int,input().split()))\\n#print(V)\\nmax_v = 0\\nhave_minus = []\\nfor left in range(N):\\n  for right in range(left,N + 1)[::-1]:\\n    if len(V[:left]) + len(V[right:]) > K: \\n      break\\n    #print (V[:left], V[right:])\\n    sum_ = sum(V[:left]) + sum(V[right:])\\n    have_minus = [num for num in V[:left]+V[right:] if num < 0]\\n    have_minus.sort()\\n    #print (f\\\"have_minus={have_minus}\\\" )\\n    d = max(0,K - (len(V[:left]) + len(V[right:])))\\n    for i in range(min(d, len(have_minus))):\\n        sum_ -= have_minus[i]   \\n    #print (f\\\"sum_ = {sum_}\\\")    \\n    \\n    max_v = max(max_v ,sum_)\\nprint(max_v)      \\n    \\n    \\n       \\n\", \"N, K = map(int, input().split())\\nV = list(map(int, input().split()))\\nans = 0\\nfor anum in range(min(N, K) + 1):\\n    owned = V[:anum]\\n    for bnum in range(min(N, K) - anum + 1):\\n        if bnum > 0:\\n            owned.append(V[-bnum])\\n        owned.sort()\\n        cdnum = K - (anum + bnum)\\n        total = 0\\n        for v in owned:\\n            if v < 0 and 0 < cdnum:\\n                cdnum -= 1\\n            else:\\n                total += v\\n        ans = max(ans, total)\\nprint(ans)\", \"N,M=list(map(int,input().split()))\\nl=list(map(int,input().split()))\\nS=min(M,N)\\nans=0\\nfor i in range(S+1):\\n   for j in range(S-i+1):\\n      mem=l[0:i]+l[N-j:]\\n      for k in range(min((M-i-j),len(mem))):\\n         if min(mem)<0:\\n            mem.remove(min(mem))\\n      ans=max(ans,sum(mem))\\nprint(ans)\", \"N, K = list(map(int, input().split()))\\nV = list(map(int, input().split()))\\n\\nans = []\\n# \\u304d\\u3063\\u3061\\u308aK\\u56de\\u3057\\u306a\\u304f\\u3066\\u3044\\u3044\\u307f\\u305f\\u3044\\nfor left in range(N + 1):\\n    for right in range(N + 1):\\n        if left + right > N:\\n            break\\n\\n        li = V[:left] + V[N - right :]\\n        li = sorted(li)\\n\\n        d = K - left - right\\n        if d < 0:\\n            break\\n\\n        c = sum(li)\\n        for i in range(min(d, len(li))):\\n            if li[i] < 0:\\n                c -= li[i]\\n            else:\\n                break\\n\\n        ans.append(c)\\n\\nprint((max(ans)))\\n\\n\", \"import sys\\nimport heapq\\ndef input(): return sys.stdin.readline().rstrip()\\n\\ndef main():\\n    n, k = map(int, input().split())\\n    v = list(map(int, input().split()))\\n    ans = 0\\n    for i in range(n):\\n        for j in range(i, n):\\n            h=[0]\\n            heapq.heapify(h)\\n            count = 0\\n            for p in range(i-1):\\n                heapq.heappush(h, v[p])\\n                count += 1\\n            for m in range(j+1, n):\\n                heapq.heappush(h, v[m])\\n                count += 1\\n            while h[0] < 0 and count < k:\\n                t = heapq.heappop(h)\\n                count += 1\\n            if count <= k:\\n                ans = max(ans,sum(h))\\n    counts = n\\n    heapq.heapify(v)\\n    while h[0] < 0 and counts < k:\\n        t = heapq.heappop(v)\\n        counts += 1\\n    if counts <= k:\\n        ans = max(ans,sum(v))\\n    print(ans)\\n   \\ndef __starting_point():\\n    main()\\n__starting_point()\", \"\\nN, K = map(int, input().split())\\nV = list(map(int, input().split()))\\n\\nMAX = 0\\nfor k in range(1, K+1):\\n    for ka in range(k+1):\\n        for kb in range(k+1-ka):\\n            kcd = k - ka - kb\\n            \\n            if (ka+kb <= kcd) or (ka+kb > N):\\n                continue\\n            \\n            #print(ka, kb, kcd)\\n            s = V[:ka] + V[-kb:] if kb != 0 else V[:ka]\\n            s.sort()\\n            #print(s[kcd:], sum(s[kcd:]))\\n            MAX = max(MAX, sum(s[kcd:]))\\n\\nprint(MAX)\", \"n,k=map(int,input().split())\\n\\na=list(map(int,input().split()))\\na_reverse=list(reversed(a))\\nb=sorted(a)\\ns=sum(a)\\n\\nans=[]\\n\\nfor i in range(k+1):\\n  if i>n:\\n    I=i-n\\n    for j in range(I): \\n      if j<n:\\n        ans1=s-b[j]\\n    ans.append(ans1)\\n  elif i==n:\\n    ans.append(s)\\n  elif i==0:\\n    ans.append(0)\\n  else:\\n    for h in range(i+1):\\n      H=i-h\\n      c=[]\\n      for x in range(h):\\n        c.append(a[x])\\n      for y in range(H):\\n        c.append(a_reverse[y])\\n        \\n      c.sort()\\n      S=sum(c)\\n      for g in range(1,k-i+1):\\n        if g<=i:\\n          if c[g-1]<0:\\n            S-=c[g-1]\\n\\n      ans.append(S)\\n      \\nprint(max(ans))\", \"n, k = map(int, input().split())\\nv = list(map(int, input().split()))\\nans = 0\\nfor i in range(k+1):\\n  for j in range(k+1):\\n    if i+j > k or i+j > n:\\n      continue\\n    t = k-(i+j)\\n    s = v[:i] + v[(n-j):]\\n    s.sort()\\n    u = 0\\n    # print (s,t)\\n    while u < t:\\n      u += 1\\n      if len(s) < 1:\\n        break\\n      if s[0] < 0:\\n        s.pop(0)\\n      else:\\n        break\\n    ans = max(ans, sum(s))\\nprint (ans)\", \"def main():\\n    n, k = list(map(int, input().split()))\\n    V = list(map(int, input().split()))\\n    ans = -1\\n    for i in range(c := min(n, k) + 1):\\n        for j in range(c - i):\\n            A = V[:i] + V[-j:] if j != 0 else V[:i]\\n            A.sort(reverse = True)\\n            r = k - (i + j)\\n            for _ in range(r):\\n                if len(A) == 0:\\n                    break\\n                if (a := A.pop()) >= 0:\\n                    A.append(a)\\n                    break\\n            ans = max(ans, sum(A))\\n    print(ans)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"n,k = map(int,input().split())\\nv = list(map(int,input().split()))\\nans = 0\\nfor i in range(min(n+1,k+1)):\\n    for j in range(min(n+1-i,k+1-i)):\\n        g = []\\n        x = 0\\n        while x < i:\\n            g.append(v[x])\\n            x += 1\\n        y = 0\\n        z = -1\\n        while y < j:\\n            g.append(v[z])\\n            y += 1\\n            z -= 1\\n        g.sort()\\n        x = k - i - j\\n        for z in range(min(len(g),x)):\\n            if g[z] < 0:\\n                g[z] = 0\\n            else:\\n                break\\n        ans = max(ans,sum(g))\\nprint(ans)\"]",
        "difficulty": "interview",
        "input": "50 1\n2181607 7433372 -6751221 -689200 2762222 -4691055 8206009 -5588713 -9216929 7794091 -1869709 1951499 -9505263 971829 8213878 1377199 4153242 374464 -2898685 4208676 -9283640 9407665 -4480562 -8217214 -7395423 6414348 -8987407 7082929 -2922863 -3222949 2600243 -7763396 7588135 -2908685 -1339773 -3050519 -4755005 3132111 7808523 8959464 -4912710 9473998 6605518 -2620197 -4899399 -5488291 5311978 2880755 4715803 -1713990\n",
        "output": "2181607\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc128/tasks/abc128_d"
    },
    {
        "id": 995,
        "task_id": 1194,
        "test_case_id": 1,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "2\n0 1\n0 1\n",
        "output": "0 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 996,
        "task_id": 1194,
        "test_case_id": 2,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "2\n0 1\n1 0\n",
        "output": "1 0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 997,
        "task_id": 1194,
        "test_case_id": 3,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "3\n1 2 0\n2 1 0\n",
        "output": "1 0 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 998,
        "task_id": 1194,
        "test_case_id": 4,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "2\n0 1\n1 0\n",
        "output": "1 0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 999,
        "task_id": 1194,
        "test_case_id": 5,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "5\n2 1 3 0 4\n2 0 4 3 1\n",
        "output": "4 2 0 3 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1000,
        "task_id": 1194,
        "test_case_id": 6,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "3\n0 2 1\n1 0 2\n",
        "output": "1 2 0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1001,
        "task_id": 1194,
        "test_case_id": 7,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "4\n2 0 1 3\n0 2 1 3\n",
        "output": "2 1 0 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1002,
        "task_id": 1194,
        "test_case_id": 8,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "1\n0\n0\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1003,
        "task_id": 1194,
        "test_case_id": 9,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "75\n71 69 34 23 13 68 19 45 40 6 74 11 53 24 27 7 50 5 70 47 4 21 25 54 62 30 17 33 52 16 67 15 14 57 38 18 48 29 58 1 8 36 2 35 56 43 44 39 20 10 0 64 3 61 32 22 37 28 26 55 63 60 49 42 59 51 66 46 73 41 9 65 12 72 31\n48 2 4 57 73 15 60 32 66 19 21 68 31 10 59 20 16 14 34 51 37 58 28 49 35 46 1 23 74 42 62 72 45 30 11 13 71 12 22 65 55 7 36 26 39 33 44 53 69 52 25 56 54 17 41 70 8 0 3 67 9 64 40 27 6 61 63 5 24 38 18 47 29 43 50\n",
        "output": "44 72 38 6 13 10 5 3 33 28 22 8 14 39 16 31 66 26 34 27 48 2 55 35 24 74 21 57 54 62 60 17 65 15 51 40 49 43 73 69 64 41 36 53 9 70 7 12 11 61 32 46 59 0 68 4 42 20 23 45 67 52 1 56 58 30 47 50 18 71 25 19 29 63 37\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1004,
        "task_id": 1194,
        "test_case_id": 10,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "84\n83 4 68 34 24 2 48 38 22 51 5 62 31 67 66 53 49 70 9 71 46 41 30 8 50 17 28 79 15 80 32 43 14 74 29 42 81 60 56 65 23 0 77 76 58 78 1 11 37 27 75 35 18 73 54 20 57 33 36 6 61 69 64 55 39 10 3 45 13 26 59 82 21 25 63 52 16 44 47 72 19 12 7 40\n63 41 80 52 36 45 17 69 22 66 37 21 46 44 64 9 48 74 58 81 10 32 0 78 68 35 26 83 14 25 79 33 13 29 75 61 6 11 49 1 31 71 59 47 62 54 2 55 30 3 53 4 16 34 77 12 43 8 28 56 18 42 5 76 82 73 27 20 70 40 23 51 38 39 7 67 50 19 60 72 24 65 57 15\n",
        "output": "62 46 66 3 61 47 68 21 44 30 41 0 78 27 45 65 13 56 70 64 58 80 31 4 32 54 57 77 28 20 24 81 29 17 22 19 6 75 15 69 55 74 52 39 40 49 1 67 76 33 43 34 26 23 50 35 12 38 71 53 82 16 79 59 36 5 14 72 2 83 7 37 51 60 73 25 42 63 10 48 8 9 18 11\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1005,
        "task_id": 1194,
        "test_case_id": 11,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "9\n8 5 0 1 6 7 4 2 3\n6 5 0 8 7 1 4 3 2\n",
        "output": "6 2 1 0 7 3 5 8 4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1006,
        "task_id": 1194,
        "test_case_id": 12,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "10\n1 7 8 0 2 5 4 6 3 9\n0 8 3 7 1 6 2 4 5 9\n",
        "output": "2 6 0 8 3 1 5 7 4 9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1007,
        "task_id": 1194,
        "test_case_id": 13,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "5\n4 3 0 1 2\n2 4 3 1 0\n",
        "output": "2 3 4 1 0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1008,
        "task_id": 1194,
        "test_case_id": 14,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "8\n5 2 4 6 1 0 3 7\n7 4 3 0 2 6 1 5\n",
        "output": "5 0 1 6 4 7 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1009,
        "task_id": 1194,
        "test_case_id": 15,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "7\n6 0 3 1 5 4 2\n6 0 2 4 3 5 1\n",
        "output": "5 0 4 6 2 1 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1010,
        "task_id": 1194,
        "test_case_id": 16,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "10\n5 2 9 1 8 6 7 4 3 0\n7 4 8 9 6 3 2 1 0 5\n",
        "output": "2 8 7 1 9 4 5 0 6 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1011,
        "task_id": 1194,
        "test_case_id": 17,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "10\n0 1 7 3 2 5 8 6 9 4\n9 5 2 7 1 4 0 6 8 3\n",
        "output": "9 5 8 7 1 4 6 0 2 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1012,
        "task_id": 1194,
        "test_case_id": 18,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "8\n2 3 0 5 4 7 6 1\n6 3 2 5 0 4 7 1\n",
        "output": "0 6 4 1 5 3 2 7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1013,
        "task_id": 1194,
        "test_case_id": 19,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "10\n7 4 6 1 0 9 2 8 5 3\n4 7 0 5 2 8 9 6 1 3\n",
        "output": "2 1 7 6 4 8 0 5 9 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1014,
        "task_id": 1194,
        "test_case_id": 20,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "10\n4 2 3 9 8 0 7 5 6 1\n7 3 1 2 9 8 6 4 0 5\n",
        "output": "1 6 5 2 9 0 7 8 4 3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1015,
        "task_id": 1194,
        "test_case_id": 21,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "10\n3 5 7 0 2 8 9 6 1 4\n4 3 8 7 9 6 0 5 2 1\n",
        "output": "7 9 3 8 1 5 0 4 6 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1016,
        "task_id": 1194,
        "test_case_id": 22,
        "question": "Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation [Image], where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order.\n\nFor example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0)\n\nMisha has two permutations, p and q. Your task is to find their sum.\n\nPermutation a = (a_0, a_1, ..., a_{n} - 1) is called to be lexicographically smaller than permutation b = (b_0, b_1, ..., b_{n} - 1), if for some k following conditions hold: a_0 = b_0, a_1 = b_1, ..., a_{k} - 1 = b_{k} - 1, a_{k} < b_{k}.\n\n\n-----Input-----\n\nThe first line contains an integer n (1 ≤ n ≤ 200 000).\n\nThe second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p.\n\nThe third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q.\n\n\n-----Output-----\n\nPrint n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces.\n\n\n-----Examples-----\nInput\n2\n0 1\n0 1\n\nOutput\n0 1\n\nInput\n2\n0 1\n1 0\n\nOutput\n1 0\n\nInput\n3\n1 2 0\n2 1 0\n\nOutput\n1 0 2\n\n\n\n-----Note-----\n\nPermutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0).\n\nIn the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is $\\operatorname{Perm}((0 + 0) \\operatorname{mod} 2) = \\operatorname{Perm}(0) =(0,1)$.\n\nIn the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is $\\operatorname{Perm}((0 + 1) \\operatorname{mod} 2) = \\operatorname{Perm}(1) =(1,0)$.\n\nPermutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).\n\nIn the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is $\\operatorname{Perm}((3 + 5) \\operatorname{mod} 6) = \\operatorname{Perm}(2) =(1,0,2)$.",
        "solutions": "[\"import sys\\n\\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n\\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n\\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nord_p = [0] * n\\nord_q = [0] * n\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q[i] = st.get_sum(0, val)\\n    st.add(val, -1)\\n\\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n\\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n\\nprint(*ord_q)\\n\", \"import sys\\n \\nclass SegmTree():\\n    def __init__(self, array=None):\\n        size = len(array)\\n        N = 1\\n        while N < size:\\n            N <<= 1\\n        self.N = N\\n        self.tree = [0] * (2*self.N)\\n        for i in range(size):\\n            self.tree[i+self.N] = array[i]\\n        self.build()\\n \\n    def build(self):\\n        for i in range(self.N - 1, 0, -1):\\n            self.tree[i] = self.tree[i<<1] + self.tree[i<<1|1]\\n \\n    def add(self, i, value=1):\\n        i += self.N\\n        while i > 0:\\n            self.tree[i] += value\\n            i >>= 1\\n    \\n    def get_sum(self, l, r):\\n        N = self.N\\n        l += N\\n        r += N\\n        result = 0\\n        while l < r:\\n            if l & 1:\\n                result += self.tree[l]\\n                l += 1\\n            if r & 1:\\n                r -= 1\\n                result += self.tree[r]\\n            l >>= 1\\n            r >>= 1\\n        return result\\n    \\n    def find_kth_nonzero(self, k):\\n        i = 1\\n        if k < 1 or k > self.tree[1]:\\n            return -1\\n        while i < self.N:\\n            i <<= 1\\n            if self.tree[i] < k:\\n                k -= self.tree[i]\\n                i |= 1\\n        return i - self.N\\n \\nreader = (line.rstrip() for line in sys.stdin)\\ninput = reader.__next__\\n \\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n \\nord_p = []\\nord_q = []\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(p):\\n    ord_p.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\nst = SegmTree([1] * n)\\nfor i, val in enumerate(q):\\n    ord_q.append(st.get_sum(0, val))\\n    st.add(val, -1)\\n \\ntransfer = 0\\nfor i in range(n-1, -1, -1):\\n    radix = n-i\\n    ord_p[i] = ord_p[i] + ord_q[i] + transfer\\n    if ord_p[i] < radix:\\n        transfer = 0\\n    else:\\n        transfer = 1\\n        ord_p[i] -= radix\\n \\nst = SegmTree([1] * n)\\nfor i in range(n):\\n    k = ord_p[i] + 1\\n    ord_q[i] = st.find_kth_nonzero(k)\\n    st.add(ord_q[i], -1)\\n \\nprint(*ord_q)\\n\", \"def sum(BIT, i):\\n    s = 0\\n    while i > 0:\\n        s += BIT[i]\\n        i -= i & (-i)\\n    return s\\n\\n\\ndef update(BIT, i, v):\\n    while i < len(BIT):\\n        BIT[i] += v\\n\\n        i += i & (-i)\\n\\n\\ndef find(fen, k):\\n    curr = 0\\n    ans = 0\\n    prevsum = 0\\n    for i in range(19, -1, -1):\\n        if ((curr + (1 << i) < n) and fen[curr + (1 << i)] + prevsum < k):\\n            ans = curr + (1 << i)\\n            curr = ans\\n            prevsum += fen[curr]\\n    return ans + 1\\n\\ndef Rank(x,BIT) :\\n\\n    return sum(BIT,x)\\n\\n\\n\\n\\nn = int(input())\\np = list(map(int, input().split()))\\nq = list(map(int, input().split()))\\n\\nfactp = []\\nfactq = []\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in p:\\n    factp.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\n\\n\\nfor val in q:\\n    factq.append(Rank(val+1,BIT)-1)\\n    update(BIT,val+1,-1)\\n\\n\\n\\n\\n\\ncarry = 0\\nfor i in range(n - 1, -1, -1):\\n    radix = n - i\\n    factp[i] = factp[i] + factq[i] + carry\\n    if factp[i] < radix:\\n        carry = 0\\n    else:\\n        carry = 1\\n        factp[i] -= radix\\n\\n\\nBIT = [0] * (n + 1)\\nfor j in range(n):\\n    update(BIT,j+1,1)\\nres=[]\\nfor i in range(n):\\n    k = factp[i]+1\\n    res.append(find(BIT,k)-1)\\n    update(BIT,res[-1]+1,-1)\\n\\nprint(*res)\"]",
        "difficulty": "interview",
        "input": "10\n1 2 0 3 4 8 6 5 7 9\n5 2 9 1 6 0 4 7 3 8\n",
        "output": "6 3 9 1 5 7 4 2 0 8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/501/D"
    },
    {
        "id": 1017,
        "task_id": 1246,
        "test_case_id": 3,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "1\ninsert 1\n",
        "output": "1\ninsert 1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1018,
        "task_id": 1246,
        "test_case_id": 4,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "1\ngetMin 31\n",
        "output": "2\ninsert 31\ngetMin 31\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1019,
        "task_id": 1246,
        "test_case_id": 6,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "2\ninsert 2\ngetMin 2\n",
        "output": "2\ninsert 2\ngetMin 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1020,
        "task_id": 1246,
        "test_case_id": 7,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "2\ninsert 31\nremoveMin\n",
        "output": "2\ninsert 31\nremoveMin\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1021,
        "task_id": 1246,
        "test_case_id": 8,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "2\ngetMin 31\nremoveMin\n",
        "output": "3\ninsert 31\ngetMin 31\nremoveMin\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1022,
        "task_id": 1246,
        "test_case_id": 10,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "8\ninsert 219147240\nremoveMin\ngetMin 923854124\nremoveMin\ngetMin -876779400\nremoveMin\ninsert 387686853\ngetMin 749998368\n",
        "output": "12\ninsert 219147240\nremoveMin\ninsert 923854124\ngetMin 923854124\nremoveMin\ninsert -876779400\ngetMin -876779400\nremoveMin\ninsert 387686853\nremoveMin\ninsert 749998368\ngetMin 749998368\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1023,
        "task_id": 1246,
        "test_case_id": 15,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "1\ngetMin 0\n",
        "output": "2\ninsert 0\ngetMin 0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1024,
        "task_id": 1246,
        "test_case_id": 19,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "2\ninsert 0\ngetMin 0\n",
        "output": "2\ninsert 0\ngetMin 0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1025,
        "task_id": 1246,
        "test_case_id": 20,
        "question": "Petya has recently learned data structure named \"Binary heap\".\n\nThe heap he is now operating with allows the following operations:   put the given number into the heap;  get the value of the minimum element in the heap;  extract the minimum element from the heap; \n\nThus, at any moment of time the heap contains several integers (possibly none), some of them might be equal.\n\nIn order to better learn this data structure Petya took an empty heap and applied some operations above to it. Also, he carefully wrote down all the operations and their results to his event log, following the format:   insert x — put the element with value x in the heap;  getMin x — the value of the minimum element contained in the heap was equal to x;  removeMin — the minimum element was extracted from the heap (only one instance, if there were many). \n\nAll the operations were correct, i.e. there was at least one element in the heap each time getMin or removeMin operations were applied.\n\nWhile Petya was away for a lunch, his little brother Vova came to the room, took away some of the pages from Petya's log and used them to make paper boats.\n\nNow Vova is worried, if he made Petya's sequence of operations inconsistent. For example, if one apply operations one-by-one in the order they are written in the event log, results of getMin operations might differ from the results recorded by Petya, and some of getMin or removeMin operations may be incorrect, as the heap is empty at the moment they are applied.\n\nNow Vova wants to add some new operation records to the event log in order to make the resulting sequence of operations correct. That is, the result of each getMin operation is equal to the result in the record, and the heap is non-empty when getMin ad removeMin are applied. Vova wants to complete this as fast as possible, as the Petya may get back at any moment. He asks you to add the least possible number of operation records to the current log. Note that arbitrary number of operations may be added at the beginning, between any two other operations, or at the end of the log.\n\n\n-----Input-----\n\nThe first line of the input contains the only integer n (1 ≤ n ≤ 100 000) — the number of the records left in Petya's journal.\n\nEach of the following n lines describe the records in the current log in the order they are applied. Format described in the statement is used. All numbers in the input are integers not exceeding 10^9 by their absolute value.\n\n\n-----Output-----\n\nThe first line of the output should contain a single integer m — the minimum possible number of records in the modified sequence of operations.\n\nNext m lines should contain the corrected sequence of records following the format of the input (described in the statement), one per line and in the order they are applied. All the numbers in the output should be integers not exceeding 10^9 by their absolute value.\n\nNote that the input sequence of operations must be the subsequence of the output sequence.\n\nIt's guaranteed that there exists the correct answer consisting of no more than 1 000 000 operations.\n\n\n-----Examples-----\nInput\n2\ninsert 3\ngetMin 4\n\nOutput\n4\ninsert 3\nremoveMin\ninsert 4\ngetMin 4\n\nInput\n4\ninsert 1\ninsert 1\nremoveMin\ngetMin 2\n\nOutput\n6\ninsert 1\ninsert 1\nremoveMin\nremoveMin\ninsert 2\ngetMin 2\n\n\n\n-----Note-----\n\nIn the first sample, after number 3 is inserted into the heap, the minimum number is 3. To make the result of the first getMin equal to 4 one should firstly remove number 3 from the heap and then add number 4 into the heap.\n\nIn the second sample case number 1 is inserted two times, so should be similarly removed twice.",
        "solutions": "[\"from heapq import *\\nn=int(input())\\nq,ans,k=[],[],0\\nfor i in range(n):\\n    ss=input()\\n    if ss!=\\\"removeMin\\\": \\n        s,mm=ss.split(); m=int(mm)\\n        if s=='insert':\\n            k+=1\\n            heappush(q,m)\\n        else:\\n            while k==0 or q[0]!=m:\\n                if k==0:\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n                    k+=1\\n                elif q[0]<m: \\n                    k-=1\\n                    t=heappop(q)\\n                    ans+=['removeMin']\\n                else: \\n                    k+=1\\n                    heappush(q,m)\\n                    ans+=['insert '+mm]\\n    else: \\n        if k==0:\\n            ans+=['insert 1']\\n        else: \\n            heappop(q)\\n            k-=1\\n    ans+=[ss]\\nprint(len(ans))\\nprint('\\\\n'.join(ans))\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        #else: heappop(heap)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\\n\", \"from heapq import *\\n\\nn = int(input())\\nheap = []\\nops = []\\nfor i in range(n):\\n    ln = input().rstrip()\\n    #print(\\\">\\\", ln, \\\"<=>\\\", heap)\\n    if ln[0] == 'i': # insert X\\n        X = int(ln.split()[1])\\n        heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n    if ln[0] == 'g': # getMin X\\n        X = int(ln.split()[1])\\n        while heap and heap[0] < X:\\n            heappop(heap)\\n            ops.append(\\\"removeMin\\\")\\n        if not heap or heap[0] > X:\\n            ops.append(\\\"insert \\\" + str(X))\\n            heappush(heap, X)\\n        ops.append(ln)\\n        continue\\n\\n    # removeMin\\n    if not heap:\\n        ops.append(\\\"insert 1\\\")\\n    else: heappop(heap)\\n    ops.append(ln)\\nprint(len(ops))\\nprint(\\\"\\\\n\\\".join(ops))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heappop,heappush\\n\\ndef main():\\n    n = int(sys.stdin.readline())\\n    h = []\\n    d = sys.stdin.readlines()\\n    res = []\\n    for i in range(n):\\n        x = d[i].split()\\n        if x[0]==\\\"insert\\\":\\n            t = int(x[1])\\n            heappush(h,t)\\n        elif x[0] == \\\"getMin\\\":\\n            t = int(x[1])\\n            while len(h)>0 and h[0]<t:\\n                res.append(\\\"removeMin\\\")\\n                heappop(h)\\n                \\n            if len(h)==0 or h[0]>t:\\n                res.append(\\\"insert \\\"+x[1])\\n                heappush(h,t)\\n            \\n        elif x[0] == \\\"removeMin\\\":\\n            if len(h)==0:\\n                res.append(\\\"insert 0\\\")  \\n            else:\\n                heappop(h)              \\n                \\n        res.append(d[i].rstrip())\\n\\n    print(len(res))\\n    print(\\\"\\\\n\\\".join(res))\\n\\nmain()\\n\", \"import heapq\\n\\nn = int(input())\\nm = n\\noperations = []\\nheap = []\\nfor i in range(n):\\n    s = input() \\n    a = s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(heap, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if heap:\\n            heapq.heappop(heap)\\n        else:\\n            m += 1\\n            operations.append('insert 1')\\n    elif a[0] == 'getMin':\\n        b = int(a[1])\\n        while heap:\\n            if heap[0] < b:\\n                operations.append('removeMin')\\n                m += 1\\n                heapq.heappop(heap)\\n            else:\\n                break\\n        else:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n        if heap[0] > b:\\n            operations.append('insert {}'.format(b))\\n            m += 1\\n            heapq.heappush(heap, b)\\n    operations.append(s)\\n\\nprint(m)\\nprint('\\\\n'.join(operations))\", \"import heapq\\n\\na = []\\nh = []\\nheapq.heapify(h)\\n\\nfor _ in range(int(input())):\\n    l = input()\\n\\n    if l.startswith(\\\"insert\\\"):\\n        heapq.heappush(h, int(l.split()[1]))\\n\\n    elif l.startswith(\\\"getMin\\\"):\\n        m = int(l.split()[1])\\n        while len(h) > 0 and h[0] != m:\\n            if h[0] < m:\\n                heapq.heappop(h)\\n                a.append(\\\"removeMin\\\")\\n            else:\\n                heapq.heappush(h, m)\\n                a.append(\\\"insert \\\" + str(m))\\n        if len(h) == 0:\\n            heapq.heappush(h, m)\\n            a.append(\\\"insert \\\" + str(m))\\n\\n    elif l.startswith(\\\"removeMin\\\"):\\n        if len(h) == 0:\\n            a.append(\\\"insert 1\\\")\\n        else:\\n            heapq.heappop(h)\\n\\n    a.append(l)\\n\\nprint(len(a))\\nprint(\\\"\\\\n\\\".join(a))\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans.append('removeMin')\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans.append('insert ' + bb)\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans.append('insert 0')\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans.append(ss)\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\n\\n\\n\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from heapq import heappush , heappop\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n,hsize = int(input()),0\\n\\n    for i in range(n):\\n        ss = input()\\n        if ss != \\\"removeMin\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                hsize += 1\\n                heappush(heap,b)\\n            else:\\n                while hsize > 0 and heap[0] < b:\\n                    ans += ['removeMin']\\n                    hsize -= 1\\n                    heappop(heap)\\n                if hsize == 0 or heap[0] != b:\\n                    ans += ['insert ' + bb]\\n                    heappush(heap,b)\\n                    hsize += 1\\n\\n        else:\\n            if hsize == 0:\\n                ans += ['insert 0']\\n            else:\\n                heappop(heap)\\n                hsize -= 1\\n        ans += [ss]\\n    print(len(ans))\\n    print(\\\"\\\\n\\\".join(ans))\\n\\nmain()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(inp())\\n\\n    for i in range(n):\\n        ss = inp()\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\nmain()\", \"import sys\\n\\n#@profile\\ndef insert(heap, val):\\n    idx = len(heap)\\n    heap.append(val)\\n\\n    while idx > 0:\\n        p = (idx - 1) >> 1\\n\\n        if heap[p] <= heap[idx]: break\\n\\n        heap[p], heap[idx] = heap[idx], heap[p]\\n\\n        idx = p\\n\\n#@profile\\ndef removeMin(heap):\\n    last = heap.pop()\\n\\n    L = len(heap)\\n\\n    if L == 0: return\\n    heap[0] = last\\n    idx = 0\\n    while True:\\n        l = (idx << 1) + 1\\n        if l >= L: return\\n\\n        r = l + 1\\n        best = r if r < L and heap[r] < heap[l] else l\\n\\n        if heap[best] >= heap[idx]:\\n            return\\n\\n        heap[best], heap[idx] = heap[idx], heap[best]\\n\\n        idx = best\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            insert(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                removeMin(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                insert(h, v)\\n        elif h:\\n            removeMin(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"from heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n__starting_point()\", \"from heapq import heappush , heappop\\nimport sys\\ninp = sys.stdin.readline\\n\\ndef main():\\n\\n    heap = []\\n    ans = []\\n    n = int(input())\\n\\n    all = sys.stdin.readlines()\\n    for ss in all:\\n        if ss != \\\"removeMin\\\\n\\\":\\n            a,bb = ss.split(); b = int(bb)\\n\\n            if a == \\\"insert\\\":\\n                heappush(heap,b)\\n            else:\\n                while heap and heap[0] < b:\\n                    ans += ['removeMin\\\\n']\\n                    heappop(heap)\\n                if not heap or heap[0] != b:\\n                    ans += ['insert %s\\\\n'%bb]\\n                    heappush(heap,b)\\n\\n        else:\\n            if not heap:\\n                ans += ['insert 0\\\\n']\\n            else:\\n                heappop(heap)\\n\\n        ans += [ss]\\n\\n    print(len(ans))\\n    print(\\\"\\\".join(ans))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    elif a[0] == 'removeMin':\\n        if hp_dgt:\\n            heapq.heappop(hp_dgt)\\n        else:\\n            m += 1; ans.append('insert 1')\\n    elif a[0] == 'getMin':\\n        x = int(a[1])\\n        while hp_dgt:\\n            if hp_dgt[0] < x:\\n                m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n            else:\\n                break\\n        else:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n        if hp_dgt[0] > x:\\n            m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import heapq\\nn = int(input()); m = n; ans, hp_dgt = [], [] \\nfor i in range(n):\\n    s=input(); a=s.split()\\n    if a[0] == 'insert':\\n        heapq.heappush(hp_dgt, int(a[1])) \\n    else:\\n        if a[0] == 'removeMin':\\n            if hp_dgt:\\n                heapq.heappop(hp_dgt)\\n            else:\\n                m += 1; ans.append('insert 1')\\n        else:\\n            x = int(a[1])\\n            while hp_dgt:\\n                if hp_dgt[0] < x:\\n                    m += 1; ans.append('removeMin'); heapq.heappop(hp_dgt)\\n                else:\\n                    break\\n            else:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n            if hp_dgt[0] > x:\\n                m += 1; ans.append('insert '+str(x)); heapq.heappush(hp_dgt, x)\\n    ans.append(s)\\nprint(m)\\nprint('\\\\n'.join(ans))\\n\", \"import sys\\nfrom heapq import *\\n\\nanswer = []\\nn = int(input())\\n\\ndef main():\\n    h = []\\n\\n    all_input = sys.stdin.readlines()\\n\\n    for q in all_input:\\n        raw = q.split()\\n\\n        if raw[0] == \\\"insert\\\":\\n            v = int(raw[1])\\n            heappush(h, v)\\n        elif raw[0] == \\\"getMin\\\":\\n            v = int(raw[1])\\n\\n            while h and h[0] < v:\\n                answer.append(\\\"removeMin\\\\n\\\")\\n                heappop(h)\\n\\n            if not h or h[0] > v:\\n                answer.append(\\\"insert \\\" + raw[1] + '\\\\n')\\n                heappush(h, v)\\n        elif h:\\n            heappop(h)\\n        else:\\n            answer.append(\\\"insert 0\\\\n\\\")\\n\\n        answer.append(q)\\n\\ndef print_answer():\\n    print(len(answer))\\n    print(\\\"\\\".join(answer))\\n\\nmain()\\nprint_answer()\", \"\\nfrom heapq import heappush, heappop\\n\\nh = []\\nans = []\\n\\ndef main():\\n\\tnonlocal h\\n\\n\\tn = int(input())\\n\\tfor i in range(n):\\n\\t\\ts = input()\\n\\t\\tif s[0] == 'i':\\n\\t\\t\\tdata = int(s.split()[1])\\n\\t\\t\\theappush(h, data)\\n\\t\\t\\tans.append(s)\\n\\t\\telif s[0] == 'g':\\n\\t\\t\\tdata = int(s.split()[1]) \\n\\t\\t\\twhile h and h[0] < data:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\t\\tans.append('removeMin')\\n\\t\\t\\tif (not h) or (h[0] > data):\\n\\t\\t\\t\\theappush(h, data)\\n\\t\\t\\t\\tans.append('insert ' + str(data))\\n\\t\\t\\tans.append(s)\\n\\t\\telse:\\n\\t\\t\\tif not h:\\n\\t\\t\\t\\tans.append('insert 0')\\n\\t\\t\\telse:\\n\\t\\t\\t\\theappop(h)\\n\\t\\t\\tans.append('removeMin')\\n\\n\\tprint(len(ans))\\n\\tprint(\\\"\\\\n\\\".join(ans))\\n\\n\\ndef __starting_point():\\n    main()\\n    \\n\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    h, res = [], []\\n    for _ in range(int(input())):\\n        s = input()\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from heapq import heappop, heappush\\n    from sys import stdin\\n    _, *l = stdin.read().splitlines()\\n    h, res = [], []\\n    for s in l:\\n        if s == (\\\"removeMin\\\"):\\n            if h:\\n                heappop(h)\\n            else:\\n                res.append(\\\"insert 1\\\")\\n        else:\\n            c, x = s.split()\\n            x = int(x)\\n            if c == \\\"insert\\\":\\n                heappush(h, x)\\n            else:\\n                while h and h[0] < x:\\n                    heappop(h)\\n                    res.append(\\\"removeMin\\\")\\n                if not h or h[0] > x:\\n                    heappush(h, x)\\n                    res.append(\\\"insert %d\\\" % x)\\n        res.append(s)\\n    print(len(res))\\n    print('\\\\n'.join(res))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from heapq import heappop, heappush\\nfrom sys import stdin\\nn, *l = stdin.read().splitlines()\\nheap, res = [], []\\nfor s in l:\\n    array = s.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            res.append('removeMin')\\n        if not heap or heap[0] != key:\\n            heappush(heap, key)\\n            res.append('insert ' + array[1])\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            res.append('insert 0')\\n    res.append(s)\\nprint(len(res))\\nprint('\\\\n'.join(res))\\n\", \"from heapq import heappush, heappop\\nfrom sys import stdin\\nheap = []\\nL = []\\n\\nn, *l = stdin.read().splitlines()\\nfor string in l:\\n    array = string.split()\\n    if array[0] == 'insert':\\n        heappush(heap, int(array[1]))\\n    elif array[0] == 'getMin':\\n        key = int(array[1])\\n        while heap and heap[0] < key:\\n            heappop(heap)\\n            L.append('removeMin')\\n        if not heap or heap[0] > key:\\n            heappush(heap, key)\\n            L.append('insert ' + str(key))\\n    else:\\n        if heap:\\n            heappop(heap)\\n        else:\\n            L.append('insert 0')\\n    L.append(string)\\n\\nprint(len(L))\\nprint('\\\\n'.join(L))\\n\"]",
        "difficulty": "interview",
        "input": "1\ninsert -1\n",
        "output": "1\ninsert -1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/681/C"
    },
    {
        "id": 1026,
        "task_id": 1388,
        "test_case_id": 16,
        "question": "Ashish has a tree consisting of $n$ nodes numbered $1$ to $n$ rooted at node $1$. The $i$-th node in the tree has a cost $a_i$, and binary digit $b_i$ is written in it. He wants to have binary digit $c_i$ written in the $i$-th node in the end.\n\nTo achieve this, he can perform the following operation any number of times:   Select any $k$ nodes from the subtree of any node $u$, and shuffle the digits in these nodes as he wishes, incurring a cost of $k \\cdot a_u$. Here, he can choose $k$ ranging from $1$ to the size of the subtree of $u$. \n\nHe wants to perform the operations in such a way that every node finally has the digit corresponding to its target.\n\nHelp him find the minimum total cost he needs to spend so that after all the operations, every node $u$ has digit $c_u$ written in it, or determine that it is impossible.\n\n\n-----Input-----\n\nFirst line contains a single integer $n$ $(1 \\le n \\le 2 \\cdot 10^5)$ denoting the number of nodes in the tree.\n\n$i$-th line of the next $n$ lines contains 3 space-separated integers $a_i$, $b_i$, $c_i$ $(1 \\leq a_i \\leq 10^9, 0 \\leq b_i, c_i \\leq 1)$  — the cost of the $i$-th node, its initial digit and its goal digit.\n\nEach of the next $n - 1$ lines contain two integers $u$, $v$ $(1 \\leq u, v \\leq n, \\text{ } u \\ne v)$, meaning that there is an edge between nodes $u$ and $v$ in the tree.\n\n\n-----Output-----\n\nPrint the minimum total cost to make every node reach its target digit, and $-1$ if it is impossible.\n\n\n-----Examples-----\nInput\n5\n1 0 1\n20 1 0\n300 0 1\n4000 0 0\n50000 1 0\n1 2\n2 3\n2 4\n1 5\n\nOutput\n4\nInput\n5\n10000 0 1\n2000 1 0\n300 0 1\n40 0 0\n1 1 0\n1 2\n2 3\n2 4\n1 5\n\nOutput\n24000\nInput\n2\n109 0 1\n205 0 1\n1 2\n\nOutput\n-1\n\n\n-----Note-----\n\nThe tree corresponding to samples $1$ and $2$ are: [Image]\n\nIn sample $1$, we can choose node $1$ and $k = 4$ for a cost of $4 \\cdot 1$ = $4$ and select nodes ${1, 2, 3, 5}$, shuffle their digits and get the desired digits in every node.\n\nIn sample $2$, we can choose node $1$ and $k = 2$ for a cost of $10000 \\cdot 2$, select nodes ${1, 5}$ and exchange their digits, and similarly, choose node $2$ and $k = 2$ for a cost of $2000 \\cdot 2$, select nodes ${2, 3}$ and exchange their digits to get the desired digits in every node.\n\nIn sample $3$, it is impossible to get the desired digits, because there is no node with digit $1$ initially.",
        "solutions": "[\"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nfrom collections import deque\\nN = int(input())\\nC, Y = [], []\\nfor _ in range(N):\\n    a, b, c = list(map(int, input().split()))\\n    C.append(a)\\n    Y.append(c - b)\\n\\nif sum(Y):\\n    print(-1)\\n    return\\n\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = list(map(int, input().split()))\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\nP = [-1] * N\\nQ = deque([0])\\nR = []\\nwhile Q:\\n    i = deque.popleft(Q)\\n    R.append(i)\\n    for a in X[i]:\\n        if a != P[i]:\\n            P[a] = i\\n            X[a].remove(i)\\n            deque.append(Q, a)\\n\\nfor i in R[1:]:\\n    C[i] = min(C[i], C[P[i]])\\n\\nans = 0\\nfor i in R[1:][::-1]:\\n    if Y[i] * Y[P[i]] < 0:\\n        ans += C[P[i]] * min(abs(Y[i]), abs(Y[P[i]]))\\n    Y[P[i]] += Y[i]\\n\\nprint(ans * 2)\\n\", \"import sys\\ninput = sys.stdin.readline\\nn = int(input())\\ncbc = [list(map(int,input().split())) for i in range(n)]\\nab = [list(map(int,input().split())) for i in range(n-1)]\\ngraph = [[] for i in range(n+1)]\\nif n == 1:\\n  if cbc[0][1] != cbc[0][2]:\\n    print(-1)\\n  else:\\n    print(0)\\n  return\\ndeg = [0]*(n+1)\\nfor a,b in ab:\\n  graph[a].append(b)\\n  graph[b].append(a)\\n  deg[a] += 1\\n  deg[b] += 1\\ndeg[1] += 1\\nstack = [1]\\npar = [0]*(n+1)\\npar[1] = -1\\nleaf = []\\nwhile stack:\\n  x = stack.pop()\\n  if x != 1 and len(graph[x]) == 1:\\n    leaf.append(x)\\n  for y in graph[x]:\\n    if par[y]:\\n      continue\\n    par[y] = x\\n    cbc[y-1][0] = min(cbc[y-1][0],cbc[x-1][0])\\n    stack.append(y)\\ndp = [[0,0] for i in range(n+1)]\\nans = 0\\nwhile leaf:\\n  x = leaf.pop()\\n  p = par[x]\\n  if cbc[x-1][1] != cbc[x-1][2]:\\n    if cbc[x-1][1] == 1:\\n      dp[x][0] += 1\\n    else:\\n      dp[x][1] += 1\\n  if min(dp[x][0],dp[x][1]):\\n    if dp[x][0] > dp[x][1]:\\n      dp[x][0] -= dp[x][1]\\n      ans += cbc[x-1][0]*dp[x][1]*2\\n      dp[x][1] = 0\\n    else:\\n      dp[x][1] -= dp[x][0]\\n      ans += cbc[x-1][0]*dp[x][0]*2\\n      dp[x][0] = 0\\n  dp[p][0] += dp[x][0]\\n  dp[p][1] += dp[x][1]\\n  deg[p] -= 1\\n  if deg[p] == 1:\\n    leaf.append(p)\\nif dp[1][0] != dp[1][1]:\\n  print(-1)\\nelse:\\n  print(ans)\", \"#!usr/bin/env python3\\nfrom collections import defaultdict, deque\\nfrom heapq import heappush, heappop\\nfrom itertools import permutations, accumulate\\nimport sys\\nimport math\\nimport bisect\\ndef LI(): return [int(x) for x in sys.stdin.readline().split()]\\ndef I(): return int(sys.stdin.readline())\\ndef LS():return [list(x) for x in sys.stdin.readline().split()]\\ndef S():\\n    res = list(sys.stdin.readline())\\n    if res[-1] == \\\"\\\\n\\\":\\n        return res[:-1]\\n    return res\\ndef IR(n):\\n    return [I() for i in range(n)]\\ndef LIR(n):\\n    return [LI() for i in range(n)]\\ndef SR(n):\\n    return [S() for i in range(n)]\\ndef LSR(n):\\n    return [LS() for i in range(n)]\\n\\nsys.setrecursionlimit(10000000)\\nmod = 1000000007\\ndef solve():\\n    n = I()\\n    a = []\\n    b = []\\n    c = []\\n    for _ in range(n):\\n        x,y,z = LI()\\n        a.append(x)\\n        b.append(y)\\n        c.append(z)\\n    v = [[] for i in range(n)]\\n    for _ in range(n-1):\\n        x,y = LI()\\n        x -= 1\\n        y -= 1\\n        v[x].append(y)\\n        v[y].append(x)\\n    if b.count(1) != c.count(1):\\n        print(-1)\\n        return\\n    q = deque([0])\\n    q2 = deque()\\n    d = [1]*n\\n    d[0] = 0\\n    ans = 0\\n    p = [[0]*2 for i in range(n)]\\n    while q:\\n        x = q.popleft()\\n        ax = a[x]\\n        if b[x] != c[x]:\\n            p[x][b[x]] = 1\\n        for y in v[x]:\\n            if d[y]:\\n                d[y] = 0\\n                if ax < a[y]:\\n                    a[y] = ax\\n                q.append(y)\\n                q2.append((x,y))\\n    while q2:\\n        x,y = q2.pop()\\n        p[x][0] += p[y][0]\\n        p[x][1] += p[y][1]\\n        m = min(p[x])\\n        ans += m*a[x]\\n        p[x][0] -= m\\n        p[x][1] -= m\\n    print(ans*2)\\n    return\\n\\n#Solve\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "3\n82 0 0\n2 1 1\n88 0 0\n3 1\n3 2\n",
        "output": "0",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1363/E"
    },
    {
        "id": 1027,
        "task_id": 1421,
        "test_case_id": 3,
        "question": "Generous sponsors of the olympiad in which Chloe and Vladik took part allowed all the participants to choose a prize for them on their own. Christmas is coming, so sponsors decided to decorate the Christmas tree with their prizes. \n\nThey took n prizes for the contestants and wrote on each of them a unique id (integer from 1 to n). A gift i is characterized by integer a_{i} — pleasantness of the gift. The pleasantness of the gift can be positive, negative or zero. Sponsors placed the gift 1 on the top of the tree. All the other gifts hung on a rope tied to some other gift so that each gift hung on the first gift, possibly with a sequence of ropes and another gifts. Formally, the gifts formed a rooted tree with n vertices.\n\nThe prize-giving procedure goes in the following way: the participants come to the tree one after another, choose any of the remaining gifts and cut the rope this prize hang on. Note that all the ropes which were used to hang other prizes on the chosen one are not cut. So the contestant gets the chosen gift as well as the all the gifts that hang on it, possibly with a sequence of ropes and another gifts.\n\nOur friends, Chloe and Vladik, shared the first place on the olympiad and they will choose prizes at the same time! To keep themselves from fighting, they decided to choose two different gifts so that the sets of the gifts that hang on them with a sequence of ropes and another gifts don't intersect. In other words, there shouldn't be any gift that hang both on the gift chosen by Chloe and on the gift chosen by Vladik. From all of the possible variants they will choose such pair of prizes that the sum of pleasantness of all the gifts that they will take after cutting the ropes is as large as possible.\n\nPrint the maximum sum of pleasantness that Vladik and Chloe can get. If it is impossible for them to choose the gifts without fighting, print Impossible.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 2·10^5) — the number of gifts.\n\nThe next line contains n integers a_1, a_2, ..., a_{n} ( - 10^9 ≤ a_{i} ≤ 10^9) — the pleasantness of the gifts.\n\nThe next (n - 1) lines contain two numbers each. The i-th of these lines contains integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the description of the tree's edges. It means that gifts with numbers u_{i} and v_{i} are connected to each other with a rope. The gifts' ids in the description of the ropes can be given in arbirtary order: v_{i} hangs on u_{i} or u_{i} hangs on v_{i}. \n\nIt is guaranteed that all the gifts hang on the first gift, possibly with a sequence of ropes and another gifts.\n\n\n-----Output-----\n\nIf it is possible for Chloe and Vladik to choose prizes without fighting, print single integer — the maximum possible sum of pleasantness they can get together.\n\nOtherwise print Impossible.\n\n\n-----Examples-----\nInput\n8\n0 5 -1 4 3 2 6 5\n1 2\n2 4\n2 5\n1 3\n3 6\n6 7\n6 8\n\nOutput\n25\nInput\n4\n1 -5 1 1\n1 2\n1 4\n2 3\n\nOutput\n2\nInput\n1\n-1\n\nOutput\nImpossible",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n\\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\ndef dfs(start):\\n    stack = [(start, -1)]\\n    visit = [False] * n\\n    while stack:\\n        u, p = stack[-1]\\n        if not visit[u]:\\n            for v in adj[u]:\\n                if v != p:\\n                    stack.append((v, u))\\n            visit[u] = True\\n        else:\\n            x = [-oo] * 3\\n            for v in adj[u]:\\n                if v != p:\\n                    sm[u] += sm[v]\\n                    mx[u] = max(mx[u], mx[v])\\n                    best[u] = max(best[u], best[v])\\n                    x[0] = mx[v]\\n                    x.sort()\\n            sm[u] += a[u]\\n            mx[u] = max(mx[u], sm[u])\\n            if x[1] > -oo and x[2] > -oo:\\n                cur = x[1] + x[2]\\n                best[u] = max(best[u], cur)\\n            stack.pop()\\n\\ndfs(0)\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\", \"import sys\\ninput = sys.stdin.readline\\n \\noo = 10**20\\nn = int(input())\\na = list(map(int, input().split()))\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    u, v = [int(i) - 1 for i in input().split()]\\n    adj[u].append(v)\\n    adj[v].append(u)\\nsm = [0] * n\\nmx = [-oo] * n\\nbest = [-oo] * n\\n\\n \\nstack = [(0, -1)]\\nvisit = [False] * n\\nwhile stack:\\n    u, p = stack[-1]\\n    if not visit[u]:\\n        for v in adj[u]:\\n            if v != p:\\n                stack.append((v, u))\\n        visit[u] = True\\n    else:\\n        x = [-oo] * 3\\n        for v in adj[u]:\\n            if v != p:\\n                sm[u] += sm[v]\\n                mx[u] = max(mx[u], mx[v])\\n                best[u] = max(best[u], best[v])\\n                x[0] = mx[v]\\n                x.sort()\\n        sm[u] += a[u]\\n        mx[u] = max(mx[u], sm[u])\\n        if x[1] > -oo and x[2] > -oo:\\n            cur = x[1] + x[2]\\n            best[u] = max(best[u], cur)\\n        stack.pop()\\n\\nans = max(best)\\nif ans <= -oo:\\n    print('Impossible')\\nelse:\\n    print(ans)\"]",
        "difficulty": "interview",
        "input": "1\n-1\n",
        "output": "Impossible",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/743/D"
    },
    {
        "id": 1028,
        "task_id": 1793,
        "test_case_id": 2,
        "question": "You are given a rooted tree on $n$ vertices, its root is the vertex number $1$. The $i$-th vertex contains a number $w_i$. Split it into the minimum possible number of vertical paths in such a way that each path contains no more than $L$ vertices and the sum of integers $w_i$ on each path does not exceed $S$. Each vertex should belong to exactly one path.\n\nA vertical path is a sequence of vertices $v_1, v_2, \\ldots, v_k$ where $v_i$ ($i \\ge 2$) is the parent of $v_{i - 1}$.\n\n\n-----Input-----\n\nThe first line contains three integers $n$, $L$, $S$ ($1 \\le n \\le 10^5$, $1 \\le L \\le 10^5$, $1 \\le S \\le 10^{18}$) — the number of vertices, the maximum number of vertices in one path and the maximum sum in one path.\n\nThe second line contains $n$ integers $w_1, w_2, \\ldots, w_n$ ($1 \\le w_i \\le 10^9$) — the numbers in the vertices of the tree.\n\nThe third line contains $n - 1$ integers $p_2, \\ldots, p_n$ ($1 \\le p_i < i$), where $p_i$ is the parent of the $i$-th vertex in the tree.\n\n\n-----Output-----\n\nOutput one number  — the minimum number of vertical paths. If it is impossible to split the tree, output $-1$.\n\n\n-----Examples-----\nInput\n3 1 3\n1 2 3\n1 1\n\nOutput\n3\nInput\n3 3 6\n1 2 3\n1 1\n\nOutput\n2\nInput\n1 1 10000\n10001\n\nOutput\n-1\n\n\n-----Note-----\n\nIn the first sample the tree is split into $\\{1\\},\\ \\{2\\},\\ \\{3\\}$.\n\nIn the second sample the tree is split into $\\{1,\\ 2\\},\\ \\{3\\}$ or $\\{1,\\ 3\\},\\ \\{2\\}$.\n\nIn the third sample it is impossible to split the tree.",
        "solutions": "[\"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in list(dp[c].items()):\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\\n\", \"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in dp[c].items():\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\", \"from collections import defaultdict\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, input().split())\\nfor w in map(int, input().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, input().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\", \"from collections import defaultdict\\nimport sys\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, sys.stdin.readline().split())\\nfor w in map(int, sys.stdin.readline().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, sys.stdin.readline().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\"]",
        "difficulty": "interview",
        "input": "3 3 6\n1 2 3\n1 1\n",
        "output": "2",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1059/E"
    },
    {
        "id": 1029,
        "task_id": 1793,
        "test_case_id": 5,
        "question": "You are given a rooted tree on $n$ vertices, its root is the vertex number $1$. The $i$-th vertex contains a number $w_i$. Split it into the minimum possible number of vertical paths in such a way that each path contains no more than $L$ vertices and the sum of integers $w_i$ on each path does not exceed $S$. Each vertex should belong to exactly one path.\n\nA vertical path is a sequence of vertices $v_1, v_2, \\ldots, v_k$ where $v_i$ ($i \\ge 2$) is the parent of $v_{i - 1}$.\n\n\n-----Input-----\n\nThe first line contains three integers $n$, $L$, $S$ ($1 \\le n \\le 10^5$, $1 \\le L \\le 10^5$, $1 \\le S \\le 10^{18}$) — the number of vertices, the maximum number of vertices in one path and the maximum sum in one path.\n\nThe second line contains $n$ integers $w_1, w_2, \\ldots, w_n$ ($1 \\le w_i \\le 10^9$) — the numbers in the vertices of the tree.\n\nThe third line contains $n - 1$ integers $p_2, \\ldots, p_n$ ($1 \\le p_i < i$), where $p_i$ is the parent of the $i$-th vertex in the tree.\n\n\n-----Output-----\n\nOutput one number  — the minimum number of vertical paths. If it is impossible to split the tree, output $-1$.\n\n\n-----Examples-----\nInput\n3 1 3\n1 2 3\n1 1\n\nOutput\n3\nInput\n3 3 6\n1 2 3\n1 1\n\nOutput\n2\nInput\n1 1 10000\n10001\n\nOutput\n-1\n\n\n-----Note-----\n\nIn the first sample the tree is split into $\\{1\\},\\ \\{2\\},\\ \\{3\\}$.\n\nIn the second sample the tree is split into $\\{1,\\ 2\\},\\ \\{3\\}$ or $\\{1,\\ 3\\},\\ \\{2\\}$.\n\nIn the third sample it is impossible to split the tree.",
        "solutions": "[\"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in list(dp[c].items()):\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\\n\", \"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in dp[c].items():\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\", \"from collections import defaultdict\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, input().split())\\nfor w in map(int, input().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, input().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\", \"from collections import defaultdict\\nimport sys\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, sys.stdin.readline().split())\\nfor w in map(int, sys.stdin.readline().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, sys.stdin.readline().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\"]",
        "difficulty": "interview",
        "input": "6 3 100\n1 100 1 1 1 1\n1 1 2 3 3\n",
        "output": "4",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1059/E"
    },
    {
        "id": 1030,
        "task_id": 1793,
        "test_case_id": 8,
        "question": "You are given a rooted tree on $n$ vertices, its root is the vertex number $1$. The $i$-th vertex contains a number $w_i$. Split it into the minimum possible number of vertical paths in such a way that each path contains no more than $L$ vertices and the sum of integers $w_i$ on each path does not exceed $S$. Each vertex should belong to exactly one path.\n\nA vertical path is a sequence of vertices $v_1, v_2, \\ldots, v_k$ where $v_i$ ($i \\ge 2$) is the parent of $v_{i - 1}$.\n\n\n-----Input-----\n\nThe first line contains three integers $n$, $L$, $S$ ($1 \\le n \\le 10^5$, $1 \\le L \\le 10^5$, $1 \\le S \\le 10^{18}$) — the number of vertices, the maximum number of vertices in one path and the maximum sum in one path.\n\nThe second line contains $n$ integers $w_1, w_2, \\ldots, w_n$ ($1 \\le w_i \\le 10^9$) — the numbers in the vertices of the tree.\n\nThe third line contains $n - 1$ integers $p_2, \\ldots, p_n$ ($1 \\le p_i < i$), where $p_i$ is the parent of the $i$-th vertex in the tree.\n\n\n-----Output-----\n\nOutput one number  — the minimum number of vertical paths. If it is impossible to split the tree, output $-1$.\n\n\n-----Examples-----\nInput\n3 1 3\n1 2 3\n1 1\n\nOutput\n3\nInput\n3 3 6\n1 2 3\n1 1\n\nOutput\n2\nInput\n1 1 10000\n10001\n\nOutput\n-1\n\n\n-----Note-----\n\nIn the first sample the tree is split into $\\{1\\},\\ \\{2\\},\\ \\{3\\}$.\n\nIn the second sample the tree is split into $\\{1,\\ 2\\},\\ \\{3\\}$ or $\\{1,\\ 3\\},\\ \\{2\\}$.\n\nIn the third sample it is impossible to split the tree.",
        "solutions": "[\"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in list(dp[c].items()):\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\\n\", \"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in dp[c].items():\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\", \"from collections import defaultdict\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, input().split())\\nfor w in map(int, input().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, input().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\", \"from collections import defaultdict\\nimport sys\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, sys.stdin.readline().split())\\nfor w in map(int, sys.stdin.readline().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, sys.stdin.readline().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\"]",
        "difficulty": "interview",
        "input": "3 3 2200000000\n1000000000 1000000000 1000000000\n1 2\n",
        "output": "2",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1059/E"
    },
    {
        "id": 1031,
        "task_id": 1793,
        "test_case_id": 9,
        "question": "You are given a rooted tree on $n$ vertices, its root is the vertex number $1$. The $i$-th vertex contains a number $w_i$. Split it into the minimum possible number of vertical paths in such a way that each path contains no more than $L$ vertices and the sum of integers $w_i$ on each path does not exceed $S$. Each vertex should belong to exactly one path.\n\nA vertical path is a sequence of vertices $v_1, v_2, \\ldots, v_k$ where $v_i$ ($i \\ge 2$) is the parent of $v_{i - 1}$.\n\n\n-----Input-----\n\nThe first line contains three integers $n$, $L$, $S$ ($1 \\le n \\le 10^5$, $1 \\le L \\le 10^5$, $1 \\le S \\le 10^{18}$) — the number of vertices, the maximum number of vertices in one path and the maximum sum in one path.\n\nThe second line contains $n$ integers $w_1, w_2, \\ldots, w_n$ ($1 \\le w_i \\le 10^9$) — the numbers in the vertices of the tree.\n\nThe third line contains $n - 1$ integers $p_2, \\ldots, p_n$ ($1 \\le p_i < i$), where $p_i$ is the parent of the $i$-th vertex in the tree.\n\n\n-----Output-----\n\nOutput one number  — the minimum number of vertical paths. If it is impossible to split the tree, output $-1$.\n\n\n-----Examples-----\nInput\n3 1 3\n1 2 3\n1 1\n\nOutput\n3\nInput\n3 3 6\n1 2 3\n1 1\n\nOutput\n2\nInput\n1 1 10000\n10001\n\nOutput\n-1\n\n\n-----Note-----\n\nIn the first sample the tree is split into $\\{1\\},\\ \\{2\\},\\ \\{3\\}$.\n\nIn the second sample the tree is split into $\\{1,\\ 2\\},\\ \\{3\\}$ or $\\{1,\\ 3\\},\\ \\{2\\}$.\n\nIn the third sample it is impossible to split the tree.",
        "solutions": "[\"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in list(dp[c].items()):\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\\n\", \"def solve(n, l, s, www, children):\\n    ans = 0\\n    dp = [{} for _ in range(n)]\\n    for v in range(n - 1, -1, -1):\\n        cv = children[v]\\n        if not cv:\\n            dp[v][1] = www[v]\\n            continue\\n        ans += len(cv) - 1\\n        wv = www[v]\\n        if wv > s:\\n            return -1\\n        dv = dp[v]\\n        for c in cv:\\n            for lc, wc in dp[c].items():\\n                if lc == l:\\n                    continue\\n                wt = wc + wv\\n                if wt > s:\\n                    continue\\n                if lc + 1 not in dv:\\n                    dv[lc + 1] = wt\\n                else:\\n                    dv[lc + 1] = min(dv[lc + 1], wt)\\n        if not dv:\\n            ans += 1\\n            dv[1] = wv\\n\\n    return ans + 1\\n\\n\\nn, l, s = list(map(int, input().split()))\\nwww = list(map(int, input().split()))\\nif n == 1:\\n    print(-1 if www[0] > s else 1)\\n    return\\nchildren = [set() for _ in range(n)]\\nfor i, p in enumerate(map(int, input().split())):\\n    children[p - 1].add(i + 1)\\nprint(solve(n, l, s, www, children))\", \"from collections import defaultdict\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, input().split())\\nfor w in map(int, input().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, input().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\", \"from collections import defaultdict\\nimport sys\\n\\n\\ndef rec(n, l, s):\\n    # dp[i] is a doctionary, where key is the chain length\\n    # and value is the weight at node i\\n    dp = [{} for _ in range(n)]\\n    ans = 0\\n    for cur in range(n - 1, -1, -1):\\n        child_of_cur = children.get(cur, None)\\n        if not child_of_cur:\\n            dp[cur][1] = weight[cur]\\n            continue\\n        ans += len(child_of_cur) - 1\\n        cur_weight = weight[cur]\\n        for ch in child_of_cur:\\n            for ln, child_weight in dp[ch].items():\\n                if ln == l:\\n                    continue\\n                agg_weight = cur_weight + child_weight\\n                if agg_weight > s:\\n                    continue\\n                if ln + 1 not in dp[cur]:\\n                    dp[cur][ln + 1] = agg_weight\\n                dp[cur][ln + 1] = min(dp[cur][ln + 1], agg_weight)\\n\\n        if not dp[cur]:\\n            ans += 1\\n            dp[cur][1] = weight[cur]\\n    \\n    return ans + 1\\n\\n\\nweight = []\\nchildren = defaultdict(set)\\nn, l, s = map(int, sys.stdin.readline().split())\\nfor w in map(int, sys.stdin.readline().split()):\\n    if w > s:\\n        print(-1)\\n        return\\n    weight.append(w)\\nif n == 1:\\n    print(-1 if weight[0] > s else 1)\\n    return\\nfor i, v in enumerate(map(int, sys.stdin.readline().split())):\\n    children[v -1].add(i + 1)\\nprint(rec(n,l,s))\"]",
        "difficulty": "interview",
        "input": "7 2 91648\n7 3 6 2 5 4 1\n1 1 1 1 3 2\n",
        "output": "4",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1059/E"
    },
    {
        "id": 1032,
        "task_id": 1844,
        "test_case_id": 1,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "3\n10 6 15\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1033,
        "task_id": 1844,
        "test_case_id": 2,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "3\n2 4 6\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1034,
        "task_id": 1844,
        "test_case_id": 3,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "7\n30 60 21 42 70 15 30\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1035,
        "task_id": 1844,
        "test_case_id": 4,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "10\n91630 147840 12600 52206 270270 48510 16170 33495 25080 60060\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1036,
        "task_id": 1844,
        "test_case_id": 5,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "30\n13469 121221 107752 94283 107752 296318 67345 296318 161628 215504 121221 215504 296318 134690 282849 161628 148159 202035 242442 107752 80814 80814 107752 255911 67345 53876 255911 215504 202035 175097\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1037,
        "task_id": 1844,
        "test_case_id": 8,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "7\n255255 170170 102102 72930 46410 39270 30030\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1038,
        "task_id": 1844,
        "test_case_id": 9,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "10\n30030 4290 120120 105105 240240 80080 60060 12012 9240 21840\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1039,
        "task_id": 1844,
        "test_case_id": 10,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "10\n735 209585 127695 85795 28815 52080 53555 157060 133665 22010\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1040,
        "task_id": 1844,
        "test_case_id": 11,
        "question": "Janusz is a businessman. He owns a company \"Januszex\", which produces games for teenagers. Last hit of Januszex was a cool one-person game \"Make it one\". The player is given a sequence of $n$ integers $a_i$.\n\nIt is allowed to select any subset of them, and the score is equal to the greatest common divisor of selected elements. The goal is to take as little elements as it is possible, getting the score $1$. Now Janusz wonders, for given sequence, how much elements should the player choose?\n\n\n-----Input-----\n\nThe first line contains an only integer $n$ ($1 \\le n \\le 300\\,000$) — the number of integers in the sequence.\n\nThe second line contains $n$ integers $a_1, a_2, \\ldots, a_n$ ($1 \\le a_i \\le 300\\,000$).\n\n\n-----Output-----\n\nIf there is no subset of the given sequence with gcd equal to $1$, output -1.\n\nOtherwise, output exactly one integer — the size of the smallest subset with gcd equal to $1$.\n\n\n-----Examples-----\nInput\n3\n10 6 15\n\nOutput\n3\n\nInput\n3\n2 4 6\n\nOutput\n-1\n\nInput\n7\n30 60 21 42 70 15 30\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first example, selecting a subset of all numbers gives a gcd of $1$ and for all smaller subsets the gcd is greater than $1$.\n\nIn the second example, for all subsets of numbers the gcd is at least $2$.",
        "solutions": "[\"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\n\\nfre = [0 for i in range(maxn)]\\nisprime = [1 for i in range(maxn)]\\nprime = []\\ndivi = [0 for i in range(maxn)]\\nfact = [1] * 10\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n == r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\nn = int(stdin.readline())\\narr = list(map(int, stdin.readline().split()))\\nfor i in arr:\\n\\tif i is 1:\\n\\t\\tprint(1)\\n\\t\\treturn\\n\\tfre[i] += 1\\n\\ndivi[1] = n\\nfor i in range(2, maxn):\\n\\tif isprime[i] is 1:\\n\\t\\tprime.append(i)\\n\\tfor j in range(1, maxn):\\n\\t\\tif i * j >= maxn:\\n\\t\\t\\tbreak\\n\\t\\tisprime[i * j] = 0\\n\\t\\tdivi[i] += fre[i * j]\\n\\nfor i in range(1, 10):\\n\\tfact[i] = fact[i - 1] * i\\n\\nmobius = [0 for i in range(maxn)]\\n\\nfor i in range(1, maxn):\\n\\tmobius[i] = 1\\nfor p in prime:\\n\\tif p * p >= maxn:\\n\\t\\tbreak\\n\\tx = p * p\\n\\tfor j in range(x, maxn, x):\\n\\t\\tmobius[j] = 0\\nfor p in prime:\\n\\tfor j in range(p, maxn, p):\\n\\t\\tmobius[j] *= -1 \\n\\t\\t\\nfor r in range(2, 10):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\tif coprime > 0:\\n\\t\\tprint(r)\\n\\t\\treturn\\nprint(-1)\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 10):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\", \"from sys import *\\n\\nmaxn = 3 * 10 ** 5 + 5\\nfre = [0] * maxn\\ndivi = [0] * maxn\\nisprime = [0] * maxn\\nprime = []\\n\\ndef seive(n):\\n\\tfor i in range(n):\\n\\t\\tisprime[i] = True\\n\\tfor i in range(2, n):\\n\\t\\tif isprime[i] is True:\\n\\t\\t\\tprime.append(i)\\n\\t\\tfor j in range(1, n):\\n\\t\\t\\tif i * j >= n:\\n\\t\\t\\t\\tbreak\\n\\t\\t\\tisprime[i * j] = False\\n\\t\\t\\tdivi[i] += fre[i * j]\\n\\nmobius = [0] * maxn\\n\\ndef calc_mobius(n):\\n\\tfor i in range(1, n):\\n\\t\\tmobius[i] = 1\\n\\tfor p in prime:\\n\\t\\tif p * p >= n:\\n\\t\\t\\tbreak\\n\\t\\tx = p * p\\n\\t\\tfor j in range(x, n, x):\\n\\t\\t\\tmobius[j] = 0\\n\\tfor p in prime:\\n\\t\\tfor j in range(p, n, p):\\n\\t\\t\\tmobius[j] *= -1\\n\\nfact = [1] * 10\\n\\ndef calc_fact():\\n\\tfact[0] = 1\\n\\tfor i in range(1, 10):\\n\\t\\tfact[i] = i * fact[i - 1]\\n\\ndef nCr(n, r):\\n\\tif n < r:\\n\\t\\treturn 0\\n\\tif n is r:\\n\\t\\treturn 1\\n\\tpro = 1\\n\\tfor i in range(r):\\n\\t\\tpro *= (n - i)\\n\\tpro //= fact[r]\\n\\treturn pro\\n\\ndef count_coprime(r):\\n\\tcoprime = 0\\n\\tfor d in range(1, maxn):\\n\\t\\tncr = nCr(divi[d], r)\\n\\t\\tcoprime += mobius[d] * ncr\\n\\treturn coprime;\\n\\t\\n\\ndef __starting_point():\\n\\tn = int(stdin.readline())\\n\\tarr = list(map(int, stdin.readline().split()))\\n\\tfor i in arr:\\n\\t\\tif i is 1:\\n\\t\\t\\tprint(1)\\n\\t\\t\\treturn\\n\\t\\tfre[i] += 1\\n\\tdivi[1] = n\\n\\tseive(maxn)\\n\\tcalc_mobius(maxn)\\n\\tcalc_fact()\\n\\tfor r in range(2, 8):\\n\\t\\tcoprime = count_coprime(r)\\n\\t\\tif coprime > 0:\\n\\t\\t\\tprint(r);return\\n\\tprint(-1)\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "25\n32942 232281 102984 3409 259245 57281 34629 15897 182413 163065 26397 79415 73227 299110 37051 296639 243565 229726 222334 24822 159992 89677 18592 166614 231980\n",
        "output": "-1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1043/F"
    },
    {
        "id": 1041,
        "task_id": 2133,
        "test_case_id": 1,
        "question": "Anton is growing a tree in his garden. In case you forgot, the tree is a connected acyclic undirected graph.\n\nThere are n vertices in the tree, each of them is painted black or white. Anton doesn't like multicolored trees, so he wants to change the tree such that all vertices have the same color (black or white).\n\nTo change the colors Anton can use only operations of one type. We denote it as paint(v), where v is some vertex of the tree. This operation changes the color of all vertices u such that all vertices on the shortest path from v to u have the same color (including v and u). For example, consider the tree [Image] \n\nand apply operation paint(3) to get the following: [Image] \n\nAnton is interested in the minimum number of operation he needs to perform in order to make the colors of all vertices equal.\n\n\n-----Input-----\n\nThe first line of the input contains a single integer n (1 ≤ n ≤ 200 000) — the number of vertices in the tree.\n\nThe second line contains n integers color_{i} (0 ≤ color_{i} ≤ 1) — colors of the vertices. color_{i} = 0 means that the i-th vertex is initially painted white, while color_{i} = 1 means it's initially painted black.\n\nThen follow n - 1 line, each of them contains a pair of integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — indices of vertices connected by the corresponding edge. It's guaranteed that all pairs (u_{i}, v_{i}) are distinct, i.e. there are no multiple edges.\n\n\n-----Output-----\n\nPrint one integer — the minimum number of operations Anton has to apply in order to make all vertices of the tree black or all vertices of the tree white.\n\n\n-----Examples-----\nInput\n11\n0 0 0 1 1 0 1 0 0 1 1\n1 2\n1 3\n2 4\n2 5\n5 6\n5 7\n3 8\n3 9\n3 10\n9 11\n\nOutput\n2\n\nInput\n4\n0 0 0 0\n1 2\n2 3\n3 4\n\nOutput\n0\n\n\n\n-----Note-----\n\nIn the first sample, the tree is the same as on the picture. If we first apply operation paint(3) and then apply paint(6), the tree will become completely black, so the answer is 2.\n\nIn the second sample, the tree is already white, so there is no need to apply any operations and the answer is 0.",
        "solutions": "[\"from collections import defaultdict, deque\\n\\nclass DSU:\\n    def __init__(self, n):\\n        self.parents = [i for i in range(n)]\\n        self.ranks = [0 for i in range(n)]\\n\\n    def find_parent(self, v):\\n        if self.parents[v] == v:\\n            return v\\n        self.parents[v] = self.find_parent(self.parents[v])\\n        return self.parents[v]\\n\\n    def join_sets(self, u, v):\\n        u = self.find_parent(u)\\n        v = self.find_parent(v)\\n        if u != v:\\n            if self.ranks[u] < self.ranks[v]:\\n                u, v = v, u\\n            self.parents[v] = u\\n            if self.ranks[v] == self.ranks[u]:\\n                self.ranks[u] += 1\\n\\nn = int(input())\\ndsu = DSU(n)\\ncolors = list(map(int, input().split(' ')))\\nvertices = []\\nfor i in range(n-1):\\n    u, v = [int(x)-1 for x in input().split(' ')]\\n    if colors[u] == colors[v]:\\n        dsu.join_sets(u, v)\\n    vertices.append((u,v))\\ngraph = defaultdict(list)\\nfor u, v in vertices:\\n    if colors[u] != colors[v]:\\n        u = dsu.find_parent(u)\\n        v = dsu.find_parent(v)\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n\\ndef bfs(u):\\n    d = dict()\\n    d[u] = 0\\n    q = deque()\\n    q.append(u)\\n    while q:\\n        u = q.pop()\\n        for v in graph[u]:\\n            if v not in d:\\n                d[v] = d[u] + 1\\n                q.append(v)\\n    return d\\nif graph:\\n    v = list(graph.keys())[0]\\n    d = bfs(v)\\n    u = v\\n    for i in d:\\n        if d[i] > d[u]:\\n            u = i\\n    d = bfs(u)\\n    w = u\\n    for i in d:\\n        if d[i] > d[w]:\\n            w = i\\n    print((d[w]+1)//2)\\nelse:\\n    print(0)\\n\", \"def main():\\n    n = int(input())\\n    colors = [c == '1' for c in input()[::2]]\\n    dsu = list(range(n))\\n    edges = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = u\\n                while v != a:\\n                    u = dsu[v]\\n                    dsu[v] = a\\n                    v = u\\n            else:\\n                dsu[a] = v\\n                while u != b:\\n                    v = dsu[u]\\n                    dsu[u] = b\\n                    u = v\\n        else:\\n            edges[u].append(v)\\n            edges[v].append(u)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    if not any(dsu):\\n        print(0)\\n        return\\n    d = {u: set() for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges):\\n        u = dsu[u]\\n        for v in e:\\n            v = dsu[v]\\n            d[u].add(v)\\n            d[v].add(v)\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for u in cur:\\n                visited.add(u)\\n                for v in d[u]:\\n                    if v not in visited:\\n                        nxt.append(v)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    colors = input()[::2]\\n    dsu = list(range(n))\\n    edges = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = dsu[v] = a\\n            else:\\n                dsu[a] = dsu[u] = b\\n        else:\\n            edges[u].append(v)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    d = {u: [] for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges):\\n        for v in e:\\n            d[dsu[u]].append(dsu[v])\\n            d[dsu[v]].append(dsu[u])\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for x in cur:\\n                visited.add(x)\\n                for y in d[x]:\\n                    if y not in visited:\\n                        nxt.append(y)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    colors = input()[::2]\\n    edges = [[[] for _ in range(n)] for _ in (0, 1)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        e = edges[colors[u] != colors[v]]\\n        e[u].append(v)\\n        e[v].append(u)\\n    dsu, e = [-1] * n, edges[0]\\n    for u, v in enumerate(dsu):\\n        if v == -1:\\n            nxt, c = [u], colors[u]\\n            while nxt:\\n                cur, nxt = nxt, []\\n                for v in cur:\\n                    dsu[v] = u\\n                    for v in e[v]:\\n                        if dsu[v] == -1:\\n                            nxt.append(v)\\n\\n    d = {u: [] for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges[1]):\\n        for v in e:\\n            d[dsu[u]].append(dsu[v])\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for x in cur:\\n                visited.add(x)\\n                for y in d[x]:\\n                    if y not in visited:\\n                        nxt.append(y)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from collections import defaultdict\\n    n, colors = int(input()), input()[::2]\\n    dsu, edges, d = list(range(n)), [], defaultdict(list)\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = dsu[v] = a\\n            else:\\n                dsu[a] = dsu[u] = b\\n        else:\\n            edges.append(u)\\n            edges.append(v)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    while edges:\\n        u, v = dsu[edges.pop()], dsu[edges.pop()]\\n        d[u].append(v)\\n        d[v].append(u)\\n\\n    def bfs(x):\\n        nxt, avail, t = [x], [True] * n, 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for y in cur:\\n                avail[y] = False\\n                for y in d[y]:\\n                    if avail[y]:\\n                        nxt.append(y)\\n        return t if x else cur[0]\\n\\n    print(bfs(bfs(0)) // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"f = lambda: map(int, input().split())\\ns, n = 0, int(input())\\n\\ng = [set() for i in range(n + 1)]\\nc = [0] + list(f())\\nd = [0] * (n + 1)\\n\\nfor j in range(n - 1):\\n    a, b = f()\\n    g[a].add(b)\\n    g[b].add(a)\\n\\np = [q for q, t in enumerate(g) if len(t) == 1]\\nwhile p:\\n    a = p.pop()\\n    if not g[a]: break\\n    b = g[a].pop()\\n    g[b].remove(a)\\n    if c[a] - c[b]: d[a] += 1\\n    s = max(s, d[b] + d[a])\\n    d[b] = max(d[b], d[a])\\n    if len(g[b]) == 1: p.append(b)\\n\\nprint(s + 1 >> 1)\", \"from collections import defaultdict, deque\\n \\nclass DSU:\\n    def __init__(self, n):\\n        self.parents = [i for i in range(n)]\\n        self.ranks = [0 for i in range(n)]\\n \\n    def find_parent(self, v):\\n        if self.parents[v] == v:\\n            return v\\n        self.parents[v] = self.find_parent(self.parents[v])\\n        return self.parents[v]\\n \\n    def join_sets(self, u, v):\\n        u = self.find_parent(u)\\n        v = self.find_parent(v)\\n        if u != v:\\n            if self.ranks[u] < self.ranks[v]:\\n                u, v = v, u\\n            self.parents[v] = u\\n            if self.ranks[v] == self.ranks[u]:\\n                self.ranks[u] += 1\\n \\nn = int(input())\\ndsu = DSU(n)\\ncolors = list(map(int, input().split(' ')))\\nvertices = []\\nfor i in range(n-1):\\n    u, v = map(lambda x: int(x)-1, input().split(' '))\\n    if colors[u] == colors[v]:\\n        dsu.join_sets(u, v)\\n    vertices.append((u,v))\\ngraph = defaultdict(list)\\nfor u, v in vertices:\\n    if colors[u] != colors[v]:\\n        u = dsu.find_parent(u)\\n        v = dsu.find_parent(v)\\n        graph[u].append(v)\\n        graph[v].append(u)\\n \\n \\ndef bfs(u):\\n    d = dict()\\n    d[u] = 0\\n    q = deque()\\n    q.append(u)\\n    while q:\\n        u = q.pop()\\n        for v in graph[u]:\\n            if v not in d:\\n                d[v] = d[u] + 1\\n                q.append(v)\\n    return d\\nif graph:\\n    v = list(graph.keys())[0]\\n    d = bfs(v)\\n    u = v\\n    for i in d:\\n        if d[i] > d[u]:\\n            u = i\\n    d = bfs(u)\\n    w = u\\n    for i in d:\\n        if d[i] > d[w]:\\n            w = i\\n    print((d[w]+1)//2)\\nelse:\\n    print(0)\", \"f = lambda: map(int, input().split())\\ns, n = 0, int(input())\\n\\ng = [set() for i in range(n + 1)]\\nc = [0] + list(f())\\nd = [0] * (n + 1)\\n\\nfor j in range(n - 1):\\n    a, b = f()\\n    g[a].add(b)\\n    g[b].add(a)\\n\\np = [q for q, t in enumerate(g) if len(t) == 1]\\nwhile p:\\n    a = p.pop()\\n    if not g[a]: break\\n    b = g[a].pop()\\n    g[b].remove(a)\\n    if c[a] - c[b]: d[a] += 1\\n    s = max(s, d[b] + d[a])\\n    d[b] = max(d[b], d[a])\\n    if len(g[b]) == 1: p.append(b)\\n\\nprint(s + 1 >> 1)\"]",
        "difficulty": "interview",
        "input": "11\n0 0 0 1 1 0 1 0 0 1 1\n1 2\n1 3\n2 4\n2 5\n5 6\n5 7\n3 8\n3 9\n3 10\n9 11\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/734/E"
    },
    {
        "id": 1042,
        "task_id": 2133,
        "test_case_id": 3,
        "question": "Anton is growing a tree in his garden. In case you forgot, the tree is a connected acyclic undirected graph.\n\nThere are n vertices in the tree, each of them is painted black or white. Anton doesn't like multicolored trees, so he wants to change the tree such that all vertices have the same color (black or white).\n\nTo change the colors Anton can use only operations of one type. We denote it as paint(v), where v is some vertex of the tree. This operation changes the color of all vertices u such that all vertices on the shortest path from v to u have the same color (including v and u). For example, consider the tree [Image] \n\nand apply operation paint(3) to get the following: [Image] \n\nAnton is interested in the minimum number of operation he needs to perform in order to make the colors of all vertices equal.\n\n\n-----Input-----\n\nThe first line of the input contains a single integer n (1 ≤ n ≤ 200 000) — the number of vertices in the tree.\n\nThe second line contains n integers color_{i} (0 ≤ color_{i} ≤ 1) — colors of the vertices. color_{i} = 0 means that the i-th vertex is initially painted white, while color_{i} = 1 means it's initially painted black.\n\nThen follow n - 1 line, each of them contains a pair of integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — indices of vertices connected by the corresponding edge. It's guaranteed that all pairs (u_{i}, v_{i}) are distinct, i.e. there are no multiple edges.\n\n\n-----Output-----\n\nPrint one integer — the minimum number of operations Anton has to apply in order to make all vertices of the tree black or all vertices of the tree white.\n\n\n-----Examples-----\nInput\n11\n0 0 0 1 1 0 1 0 0 1 1\n1 2\n1 3\n2 4\n2 5\n5 6\n5 7\n3 8\n3 9\n3 10\n9 11\n\nOutput\n2\n\nInput\n4\n0 0 0 0\n1 2\n2 3\n3 4\n\nOutput\n0\n\n\n\n-----Note-----\n\nIn the first sample, the tree is the same as on the picture. If we first apply operation paint(3) and then apply paint(6), the tree will become completely black, so the answer is 2.\n\nIn the second sample, the tree is already white, so there is no need to apply any operations and the answer is 0.",
        "solutions": "[\"from collections import defaultdict, deque\\n\\nclass DSU:\\n    def __init__(self, n):\\n        self.parents = [i for i in range(n)]\\n        self.ranks = [0 for i in range(n)]\\n\\n    def find_parent(self, v):\\n        if self.parents[v] == v:\\n            return v\\n        self.parents[v] = self.find_parent(self.parents[v])\\n        return self.parents[v]\\n\\n    def join_sets(self, u, v):\\n        u = self.find_parent(u)\\n        v = self.find_parent(v)\\n        if u != v:\\n            if self.ranks[u] < self.ranks[v]:\\n                u, v = v, u\\n            self.parents[v] = u\\n            if self.ranks[v] == self.ranks[u]:\\n                self.ranks[u] += 1\\n\\nn = int(input())\\ndsu = DSU(n)\\ncolors = list(map(int, input().split(' ')))\\nvertices = []\\nfor i in range(n-1):\\n    u, v = [int(x)-1 for x in input().split(' ')]\\n    if colors[u] == colors[v]:\\n        dsu.join_sets(u, v)\\n    vertices.append((u,v))\\ngraph = defaultdict(list)\\nfor u, v in vertices:\\n    if colors[u] != colors[v]:\\n        u = dsu.find_parent(u)\\n        v = dsu.find_parent(v)\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n\\ndef bfs(u):\\n    d = dict()\\n    d[u] = 0\\n    q = deque()\\n    q.append(u)\\n    while q:\\n        u = q.pop()\\n        for v in graph[u]:\\n            if v not in d:\\n                d[v] = d[u] + 1\\n                q.append(v)\\n    return d\\nif graph:\\n    v = list(graph.keys())[0]\\n    d = bfs(v)\\n    u = v\\n    for i in d:\\n        if d[i] > d[u]:\\n            u = i\\n    d = bfs(u)\\n    w = u\\n    for i in d:\\n        if d[i] > d[w]:\\n            w = i\\n    print((d[w]+1)//2)\\nelse:\\n    print(0)\\n\", \"def main():\\n    n = int(input())\\n    colors = [c == '1' for c in input()[::2]]\\n    dsu = list(range(n))\\n    edges = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = u\\n                while v != a:\\n                    u = dsu[v]\\n                    dsu[v] = a\\n                    v = u\\n            else:\\n                dsu[a] = v\\n                while u != b:\\n                    v = dsu[u]\\n                    dsu[u] = b\\n                    u = v\\n        else:\\n            edges[u].append(v)\\n            edges[v].append(u)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    if not any(dsu):\\n        print(0)\\n        return\\n    d = {u: set() for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges):\\n        u = dsu[u]\\n        for v in e:\\n            v = dsu[v]\\n            d[u].add(v)\\n            d[v].add(v)\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for u in cur:\\n                visited.add(u)\\n                for v in d[u]:\\n                    if v not in visited:\\n                        nxt.append(v)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    colors = input()[::2]\\n    dsu = list(range(n))\\n    edges = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = dsu[v] = a\\n            else:\\n                dsu[a] = dsu[u] = b\\n        else:\\n            edges[u].append(v)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    d = {u: [] for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges):\\n        for v in e:\\n            d[dsu[u]].append(dsu[v])\\n            d[dsu[v]].append(dsu[u])\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for x in cur:\\n                visited.add(x)\\n                for y in d[x]:\\n                    if y not in visited:\\n                        nxt.append(y)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    colors = input()[::2]\\n    edges = [[[] for _ in range(n)] for _ in (0, 1)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        e = edges[colors[u] != colors[v]]\\n        e[u].append(v)\\n        e[v].append(u)\\n    dsu, e = [-1] * n, edges[0]\\n    for u, v in enumerate(dsu):\\n        if v == -1:\\n            nxt, c = [u], colors[u]\\n            while nxt:\\n                cur, nxt = nxt, []\\n                for v in cur:\\n                    dsu[v] = u\\n                    for v in e[v]:\\n                        if dsu[v] == -1:\\n                            nxt.append(v)\\n\\n    d = {u: [] for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges[1]):\\n        for v in e:\\n            d[dsu[u]].append(dsu[v])\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for x in cur:\\n                visited.add(x)\\n                for y in d[x]:\\n                    if y not in visited:\\n                        nxt.append(y)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from collections import defaultdict\\n    n, colors = int(input()), input()[::2]\\n    dsu, edges, d = list(range(n)), [], defaultdict(list)\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = dsu[v] = a\\n            else:\\n                dsu[a] = dsu[u] = b\\n        else:\\n            edges.append(u)\\n            edges.append(v)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    while edges:\\n        u, v = dsu[edges.pop()], dsu[edges.pop()]\\n        d[u].append(v)\\n        d[v].append(u)\\n\\n    def bfs(x):\\n        nxt, avail, t = [x], [True] * n, 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for y in cur:\\n                avail[y] = False\\n                for y in d[y]:\\n                    if avail[y]:\\n                        nxt.append(y)\\n        return t if x else cur[0]\\n\\n    print(bfs(bfs(0)) // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"f = lambda: map(int, input().split())\\ns, n = 0, int(input())\\n\\ng = [set() for i in range(n + 1)]\\nc = [0] + list(f())\\nd = [0] * (n + 1)\\n\\nfor j in range(n - 1):\\n    a, b = f()\\n    g[a].add(b)\\n    g[b].add(a)\\n\\np = [q for q, t in enumerate(g) if len(t) == 1]\\nwhile p:\\n    a = p.pop()\\n    if not g[a]: break\\n    b = g[a].pop()\\n    g[b].remove(a)\\n    if c[a] - c[b]: d[a] += 1\\n    s = max(s, d[b] + d[a])\\n    d[b] = max(d[b], d[a])\\n    if len(g[b]) == 1: p.append(b)\\n\\nprint(s + 1 >> 1)\", \"from collections import defaultdict, deque\\n \\nclass DSU:\\n    def __init__(self, n):\\n        self.parents = [i for i in range(n)]\\n        self.ranks = [0 for i in range(n)]\\n \\n    def find_parent(self, v):\\n        if self.parents[v] == v:\\n            return v\\n        self.parents[v] = self.find_parent(self.parents[v])\\n        return self.parents[v]\\n \\n    def join_sets(self, u, v):\\n        u = self.find_parent(u)\\n        v = self.find_parent(v)\\n        if u != v:\\n            if self.ranks[u] < self.ranks[v]:\\n                u, v = v, u\\n            self.parents[v] = u\\n            if self.ranks[v] == self.ranks[u]:\\n                self.ranks[u] += 1\\n \\nn = int(input())\\ndsu = DSU(n)\\ncolors = list(map(int, input().split(' ')))\\nvertices = []\\nfor i in range(n-1):\\n    u, v = map(lambda x: int(x)-1, input().split(' '))\\n    if colors[u] == colors[v]:\\n        dsu.join_sets(u, v)\\n    vertices.append((u,v))\\ngraph = defaultdict(list)\\nfor u, v in vertices:\\n    if colors[u] != colors[v]:\\n        u = dsu.find_parent(u)\\n        v = dsu.find_parent(v)\\n        graph[u].append(v)\\n        graph[v].append(u)\\n \\n \\ndef bfs(u):\\n    d = dict()\\n    d[u] = 0\\n    q = deque()\\n    q.append(u)\\n    while q:\\n        u = q.pop()\\n        for v in graph[u]:\\n            if v not in d:\\n                d[v] = d[u] + 1\\n                q.append(v)\\n    return d\\nif graph:\\n    v = list(graph.keys())[0]\\n    d = bfs(v)\\n    u = v\\n    for i in d:\\n        if d[i] > d[u]:\\n            u = i\\n    d = bfs(u)\\n    w = u\\n    for i in d:\\n        if d[i] > d[w]:\\n            w = i\\n    print((d[w]+1)//2)\\nelse:\\n    print(0)\", \"f = lambda: map(int, input().split())\\ns, n = 0, int(input())\\n\\ng = [set() for i in range(n + 1)]\\nc = [0] + list(f())\\nd = [0] * (n + 1)\\n\\nfor j in range(n - 1):\\n    a, b = f()\\n    g[a].add(b)\\n    g[b].add(a)\\n\\np = [q for q, t in enumerate(g) if len(t) == 1]\\nwhile p:\\n    a = p.pop()\\n    if not g[a]: break\\n    b = g[a].pop()\\n    g[b].remove(a)\\n    if c[a] - c[b]: d[a] += 1\\n    s = max(s, d[b] + d[a])\\n    d[b] = max(d[b], d[a])\\n    if len(g[b]) == 1: p.append(b)\\n\\nprint(s + 1 >> 1)\"]",
        "difficulty": "interview",
        "input": "15\n0 1 0 0 1 1 0 1 1 1 1 1 0 1 0\n10 7\n10 3\n10 8\n5 7\n13 14\n8 13\n15 4\n15 13\n5 2\n9 3\n11 15\n13 6\n1 12\n9 1\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/734/E"
    },
    {
        "id": 1043,
        "task_id": 2133,
        "test_case_id": 4,
        "question": "Anton is growing a tree in his garden. In case you forgot, the tree is a connected acyclic undirected graph.\n\nThere are n vertices in the tree, each of them is painted black or white. Anton doesn't like multicolored trees, so he wants to change the tree such that all vertices have the same color (black or white).\n\nTo change the colors Anton can use only operations of one type. We denote it as paint(v), where v is some vertex of the tree. This operation changes the color of all vertices u such that all vertices on the shortest path from v to u have the same color (including v and u). For example, consider the tree [Image] \n\nand apply operation paint(3) to get the following: [Image] \n\nAnton is interested in the minimum number of operation he needs to perform in order to make the colors of all vertices equal.\n\n\n-----Input-----\n\nThe first line of the input contains a single integer n (1 ≤ n ≤ 200 000) — the number of vertices in the tree.\n\nThe second line contains n integers color_{i} (0 ≤ color_{i} ≤ 1) — colors of the vertices. color_{i} = 0 means that the i-th vertex is initially painted white, while color_{i} = 1 means it's initially painted black.\n\nThen follow n - 1 line, each of them contains a pair of integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — indices of vertices connected by the corresponding edge. It's guaranteed that all pairs (u_{i}, v_{i}) are distinct, i.e. there are no multiple edges.\n\n\n-----Output-----\n\nPrint one integer — the minimum number of operations Anton has to apply in order to make all vertices of the tree black or all vertices of the tree white.\n\n\n-----Examples-----\nInput\n11\n0 0 0 1 1 0 1 0 0 1 1\n1 2\n1 3\n2 4\n2 5\n5 6\n5 7\n3 8\n3 9\n3 10\n9 11\n\nOutput\n2\n\nInput\n4\n0 0 0 0\n1 2\n2 3\n3 4\n\nOutput\n0\n\n\n\n-----Note-----\n\nIn the first sample, the tree is the same as on the picture. If we first apply operation paint(3) and then apply paint(6), the tree will become completely black, so the answer is 2.\n\nIn the second sample, the tree is already white, so there is no need to apply any operations and the answer is 0.",
        "solutions": "[\"from collections import defaultdict, deque\\n\\nclass DSU:\\n    def __init__(self, n):\\n        self.parents = [i for i in range(n)]\\n        self.ranks = [0 for i in range(n)]\\n\\n    def find_parent(self, v):\\n        if self.parents[v] == v:\\n            return v\\n        self.parents[v] = self.find_parent(self.parents[v])\\n        return self.parents[v]\\n\\n    def join_sets(self, u, v):\\n        u = self.find_parent(u)\\n        v = self.find_parent(v)\\n        if u != v:\\n            if self.ranks[u] < self.ranks[v]:\\n                u, v = v, u\\n            self.parents[v] = u\\n            if self.ranks[v] == self.ranks[u]:\\n                self.ranks[u] += 1\\n\\nn = int(input())\\ndsu = DSU(n)\\ncolors = list(map(int, input().split(' ')))\\nvertices = []\\nfor i in range(n-1):\\n    u, v = [int(x)-1 for x in input().split(' ')]\\n    if colors[u] == colors[v]:\\n        dsu.join_sets(u, v)\\n    vertices.append((u,v))\\ngraph = defaultdict(list)\\nfor u, v in vertices:\\n    if colors[u] != colors[v]:\\n        u = dsu.find_parent(u)\\n        v = dsu.find_parent(v)\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n\\ndef bfs(u):\\n    d = dict()\\n    d[u] = 0\\n    q = deque()\\n    q.append(u)\\n    while q:\\n        u = q.pop()\\n        for v in graph[u]:\\n            if v not in d:\\n                d[v] = d[u] + 1\\n                q.append(v)\\n    return d\\nif graph:\\n    v = list(graph.keys())[0]\\n    d = bfs(v)\\n    u = v\\n    for i in d:\\n        if d[i] > d[u]:\\n            u = i\\n    d = bfs(u)\\n    w = u\\n    for i in d:\\n        if d[i] > d[w]:\\n            w = i\\n    print((d[w]+1)//2)\\nelse:\\n    print(0)\\n\", \"def main():\\n    n = int(input())\\n    colors = [c == '1' for c in input()[::2]]\\n    dsu = list(range(n))\\n    edges = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = u\\n                while v != a:\\n                    u = dsu[v]\\n                    dsu[v] = a\\n                    v = u\\n            else:\\n                dsu[a] = v\\n                while u != b:\\n                    v = dsu[u]\\n                    dsu[u] = b\\n                    u = v\\n        else:\\n            edges[u].append(v)\\n            edges[v].append(u)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    if not any(dsu):\\n        print(0)\\n        return\\n    d = {u: set() for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges):\\n        u = dsu[u]\\n        for v in e:\\n            v = dsu[v]\\n            d[u].add(v)\\n            d[v].add(v)\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for u in cur:\\n                visited.add(u)\\n                for v in d[u]:\\n                    if v not in visited:\\n                        nxt.append(v)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    colors = input()[::2]\\n    dsu = list(range(n))\\n    edges = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = dsu[v] = a\\n            else:\\n                dsu[a] = dsu[u] = b\\n        else:\\n            edges[u].append(v)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    d = {u: [] for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges):\\n        for v in e:\\n            d[dsu[u]].append(dsu[v])\\n            d[dsu[v]].append(dsu[u])\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for x in cur:\\n                visited.add(x)\\n                for y in d[x]:\\n                    if y not in visited:\\n                        nxt.append(y)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    n = int(input())\\n    colors = input()[::2]\\n    edges = [[[] for _ in range(n)] for _ in (0, 1)]\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        e = edges[colors[u] != colors[v]]\\n        e[u].append(v)\\n        e[v].append(u)\\n    dsu, e = [-1] * n, edges[0]\\n    for u, v in enumerate(dsu):\\n        if v == -1:\\n            nxt, c = [u], colors[u]\\n            while nxt:\\n                cur, nxt = nxt, []\\n                for v in cur:\\n                    dsu[v] = u\\n                    for v in e[v]:\\n                        if dsu[v] == -1:\\n                            nxt.append(v)\\n\\n    d = {u: [] for u, v in enumerate(dsu) if u == v}\\n    for u, e in enumerate(edges[1]):\\n        for v in e:\\n            d[dsu[u]].append(dsu[v])\\n\\n    def bfs(x):\\n        nxt, visited, t = [x], set(), 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for x in cur:\\n                visited.add(x)\\n                for y in d[x]:\\n                    if y not in visited:\\n                        nxt.append(y)\\n        return t, cur[0]\\n\\n    print(bfs(bfs(0)[1])[0] // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n    from collections import defaultdict\\n    n, colors = int(input()), input()[::2]\\n    dsu, edges, d = list(range(n)), [], defaultdict(list)\\n    for _ in range(n - 1):\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        if colors[u] == colors[v]:\\n            a, b = dsu[u], dsu[v]\\n            while a != dsu[a]:\\n                a = dsu[a]\\n            while b != dsu[b]:\\n                b = dsu[b]\\n            if a < b:\\n                dsu[b] = dsu[v] = a\\n            else:\\n                dsu[a] = dsu[u] = b\\n        else:\\n            edges.append(u)\\n            edges.append(v)\\n    for u, v in enumerate(dsu):\\n        dsu[u] = dsu[v]\\n    while edges:\\n        u, v = dsu[edges.pop()], dsu[edges.pop()]\\n        d[u].append(v)\\n        d[v].append(u)\\n\\n    def bfs(x):\\n        nxt, avail, t = [x], [True] * n, 0\\n        while nxt:\\n            t += 1\\n            cur, nxt = nxt, []\\n            for y in cur:\\n                avail[y] = False\\n                for y in d[y]:\\n                    if avail[y]:\\n                        nxt.append(y)\\n        return t if x else cur[0]\\n\\n    print(bfs(bfs(0)) // 2)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"f = lambda: map(int, input().split())\\ns, n = 0, int(input())\\n\\ng = [set() for i in range(n + 1)]\\nc = [0] + list(f())\\nd = [0] * (n + 1)\\n\\nfor j in range(n - 1):\\n    a, b = f()\\n    g[a].add(b)\\n    g[b].add(a)\\n\\np = [q for q, t in enumerate(g) if len(t) == 1]\\nwhile p:\\n    a = p.pop()\\n    if not g[a]: break\\n    b = g[a].pop()\\n    g[b].remove(a)\\n    if c[a] - c[b]: d[a] += 1\\n    s = max(s, d[b] + d[a])\\n    d[b] = max(d[b], d[a])\\n    if len(g[b]) == 1: p.append(b)\\n\\nprint(s + 1 >> 1)\", \"from collections import defaultdict, deque\\n \\nclass DSU:\\n    def __init__(self, n):\\n        self.parents = [i for i in range(n)]\\n        self.ranks = [0 for i in range(n)]\\n \\n    def find_parent(self, v):\\n        if self.parents[v] == v:\\n            return v\\n        self.parents[v] = self.find_parent(self.parents[v])\\n        return self.parents[v]\\n \\n    def join_sets(self, u, v):\\n        u = self.find_parent(u)\\n        v = self.find_parent(v)\\n        if u != v:\\n            if self.ranks[u] < self.ranks[v]:\\n                u, v = v, u\\n            self.parents[v] = u\\n            if self.ranks[v] == self.ranks[u]:\\n                self.ranks[u] += 1\\n \\nn = int(input())\\ndsu = DSU(n)\\ncolors = list(map(int, input().split(' ')))\\nvertices = []\\nfor i in range(n-1):\\n    u, v = map(lambda x: int(x)-1, input().split(' '))\\n    if colors[u] == colors[v]:\\n        dsu.join_sets(u, v)\\n    vertices.append((u,v))\\ngraph = defaultdict(list)\\nfor u, v in vertices:\\n    if colors[u] != colors[v]:\\n        u = dsu.find_parent(u)\\n        v = dsu.find_parent(v)\\n        graph[u].append(v)\\n        graph[v].append(u)\\n \\n \\ndef bfs(u):\\n    d = dict()\\n    d[u] = 0\\n    q = deque()\\n    q.append(u)\\n    while q:\\n        u = q.pop()\\n        for v in graph[u]:\\n            if v not in d:\\n                d[v] = d[u] + 1\\n                q.append(v)\\n    return d\\nif graph:\\n    v = list(graph.keys())[0]\\n    d = bfs(v)\\n    u = v\\n    for i in d:\\n        if d[i] > d[u]:\\n            u = i\\n    d = bfs(u)\\n    w = u\\n    for i in d:\\n        if d[i] > d[w]:\\n            w = i\\n    print((d[w]+1)//2)\\nelse:\\n    print(0)\", \"f = lambda: map(int, input().split())\\ns, n = 0, int(input())\\n\\ng = [set() for i in range(n + 1)]\\nc = [0] + list(f())\\nd = [0] * (n + 1)\\n\\nfor j in range(n - 1):\\n    a, b = f()\\n    g[a].add(b)\\n    g[b].add(a)\\n\\np = [q for q, t in enumerate(g) if len(t) == 1]\\nwhile p:\\n    a = p.pop()\\n    if not g[a]: break\\n    b = g[a].pop()\\n    g[b].remove(a)\\n    if c[a] - c[b]: d[a] += 1\\n    s = max(s, d[b] + d[a])\\n    d[b] = max(d[b], d[a])\\n    if len(g[b]) == 1: p.append(b)\\n\\nprint(s + 1 >> 1)\"]",
        "difficulty": "interview",
        "input": "42\n1 0 0 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0\n35 6\n35 39\n4 31\n31 5\n14 35\n1 2\n40 32\n35 31\n37 35\n32 38\n1 36\n3 25\n35 11\n26 35\n24 35\n3 2\n35 23\n21 1\n20 27\n16 26\n2 18\n34 39\n39 28\n3 32\n26 30\n41 7\n13 35\n1 8\n31 22\n33 21\n21 29\n28 10\n2 19\n2 17\n27 24\n9 1\n42 1\n1 15\n1 35\n12 2\n41 1\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/734/E"
    },
    {
        "id": 1044,
        "task_id": 2226,
        "test_case_id": 1,
        "question": "You are given a simple weighted connected undirected graph, consisting of $n$ vertices and $m$ edges.\n\nA path in the graph of length $k$ is a sequence of $k+1$ vertices $v_1, v_2, \\dots, v_{k+1}$ such that for each $i$ $(1 \\le i \\le k)$ the edge $(v_i, v_{i+1})$ is present in the graph. A path from some vertex $v$ also has vertex $v_1=v$. Note that edges and vertices are allowed to be included in the path multiple times.\n\nThe weight of the path is the total weight of edges in it.\n\nFor each $i$ from $1$ to $q$ consider a path from vertex $1$ of length $i$ of the maximum weight. What is the sum of weights of these $q$ paths?\n\nAnswer can be quite large, so print it modulo $10^9+7$.\n\n\n-----Input-----\n\nThe first line contains a three integers $n$, $m$, $q$ ($2 \\le n \\le 2000$; $n - 1 \\le m \\le 2000$; $m \\le q \\le 10^9$) — the number of vertices in the graph, the number of edges in the graph and the number of lengths that should be included in the answer.\n\nEach of the next $m$ lines contains a description of an edge: three integers $v$, $u$, $w$ ($1 \\le v, u \\le n$; $1 \\le w \\le 10^6$) — two vertices $v$ and $u$ are connected by an undirected edge with weight $w$. The graph contains no loops and no multiple edges. It is guaranteed that the given edges form a connected graph.\n\n\n-----Output-----\n\nPrint a single integer — the sum of the weights of the paths from vertex $1$ of maximum weights of lengths $1, 2, \\dots, q$ modulo $10^9+7$.\n\n\n-----Examples-----\nInput\n7 8 25\n1 2 1\n2 3 10\n3 4 2\n1 5 2\n5 6 7\n6 4 15\n5 3 1\n1 7 3\n\nOutput\n4361\n\nInput\n2 1 5\n1 2 4\n\nOutput\n60\n\nInput\n15 15 23\n13 10 12\n11 14 12\n2 15 5\n4 10 8\n10 2 4\n10 7 5\n3 10 1\n5 6 11\n1 13 8\n9 15 4\n4 2 9\n11 15 1\n11 12 14\n10 8 12\n3 6 11\n\nOutput\n3250\n\nInput\n5 10 10000000\n2 4 798\n1 5 824\n5 2 558\n4 1 288\n3 4 1890\n3 1 134\n2 3 1485\n4 5 284\n3 5 1025\n1 2 649\n\nOutput\n768500592\n\n\n\n-----Note-----\n\nHere is the graph for the first example: [Image] \n\nSome maximum weight paths are:   length $1$: edges $(1, 7)$ — weight $3$;  length $2$: edges $(1, 2), (2, 3)$ — weight $1+10=11$;  length $3$: edges $(1, 5), (5, 6), (6, 4)$ — weight $2+7+15=24$;  length $4$: edges $(1, 5), (5, 6), (6, 4), (6, 4)$ — weight $2+7+15+15=39$;  $\\dots$ \n\nSo the answer is the sum of $25$ terms: $3+11+24+39+\\dots$\n\nIn the second example the maximum weight paths have weights $4$, $8$, $12$, $16$ and $20$.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\nn,m,q=map(int,input().split())\\nmod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"n,m,q=map(int,input().split());mod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):neig[i]=[0]\\nfor i in range(m):a,b,w=map(int,input().split());a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w]);mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\n\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor i in range(n):\\n\\t\\tfor j, w in adj[i]:\\n\\t\\t\\tNDP[j] = max(NDP[j], DP[i] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n#print(ans)\\n#print(DP)\\n#assert all(v > 0 for v in DP)\\n\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nnedges = []\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\tnedges.append((DP[i], w))\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = nedges[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\t#assert us <= lt\\n\\t#assert until > lt, (until, lt)\\n\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor a, b, w in edges:\\n\\t\\tNDP[b] = max(NDP[b], DP[a] + w)\\n\\t\\tNDP[a] = max(NDP[a], DP[b] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = intersect[j], slope[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ndef rs(): return sys.stdin.readline().rstrip()\\ndef ri(): return int(sys.stdin.readline())\\ndef ria(): return list(map(int, sys.stdin.readline().split()))\\ndef ws(s): sys.stdout.write(s + '\\\\n')\\ndef wi(n): sys.stdout.write(str(n) + '\\\\n')\\ndef wia(a): sys.stdout.write(' '.join([str(x) for x in a]) + '\\\\n')\\n\\n\\nMOD = 10 ** 9 + 7\\nINF = 100\\n\\n\\ndef convex_hull_trick(K, M, integer=True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i, j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i, j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key=K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\ndef nn2(n):\\n    return n * (n+1) // 2\\n\\n\\ndef solve(n, m, q, edges):\\n    # k < m\\n    # dp[v][k] - max path cost ending in v and having k edges\\n    dp = [[-INF]*(m+1) for _ in range(n)]\\n    mk = [0]*(m+1)\\n\\n    dp[0][0] = 0\\n\\n    for k in range(1, m+1):\\n        for e in edges:\\n            if dp[e[0]][k-1] == -INF:\\n                continue\\n            dp[e[1]][k] = max(dp[e[1]][k], dp[e[0]][k-1] + e[2])\\n            mk[k] = max(mk[k], dp[e[1]][k])\\n\\n    ans = sum(mk) % MOD\\n    if q > m:\\n        intersect = [dp[i][m] for i in range(n)]\\n        slope = [0] * n\\n\\n        for e in edges:\\n            if e[2] > slope[e[0]]:\\n                slope[e[0]] = e[2]\\n\\n        hull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n        lt = 0\\n        for i, j in enumerate(hull_i):\\n            wt = intersect[j]\\n            w = slope[j]\\n\\n            until = min(q if i == len(hull_x) else hull_x[i], q - m)\\n            if until <= 0:\\n                continue\\n\\n            active = (until - lt)\\n\\n            ans = (ans + active * wt) % MOD\\n\\n            min_uses = lt\\n            max_uses = lt + active\\n\\n            times = nn2(max_uses) - nn2(min_uses)\\n            ans = (ans + times * w) % MOD\\n\\n            lt = until\\n            if lt == q - m: break\\n\\n\\n    return ans\\n\\ndef main():\\n    n, m, q = ria()\\n    e = []\\n    for _ in range(m):\\n        u, v, w = ria()\\n        u -= 1\\n        v -= 1\\n        e.append((u, v, w))\\n        e.append((v, u, w))\\n    wi(solve(n, m, q, e))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "7 8 25\n1 2 1\n2 3 10\n3 4 2\n1 5 2\n5 6 7\n6 4 15\n5 3 1\n1 7 3\n",
        "output": "4361\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1366/F"
    },
    {
        "id": 1045,
        "task_id": 2226,
        "test_case_id": 2,
        "question": "You are given a simple weighted connected undirected graph, consisting of $n$ vertices and $m$ edges.\n\nA path in the graph of length $k$ is a sequence of $k+1$ vertices $v_1, v_2, \\dots, v_{k+1}$ such that for each $i$ $(1 \\le i \\le k)$ the edge $(v_i, v_{i+1})$ is present in the graph. A path from some vertex $v$ also has vertex $v_1=v$. Note that edges and vertices are allowed to be included in the path multiple times.\n\nThe weight of the path is the total weight of edges in it.\n\nFor each $i$ from $1$ to $q$ consider a path from vertex $1$ of length $i$ of the maximum weight. What is the sum of weights of these $q$ paths?\n\nAnswer can be quite large, so print it modulo $10^9+7$.\n\n\n-----Input-----\n\nThe first line contains a three integers $n$, $m$, $q$ ($2 \\le n \\le 2000$; $n - 1 \\le m \\le 2000$; $m \\le q \\le 10^9$) — the number of vertices in the graph, the number of edges in the graph and the number of lengths that should be included in the answer.\n\nEach of the next $m$ lines contains a description of an edge: three integers $v$, $u$, $w$ ($1 \\le v, u \\le n$; $1 \\le w \\le 10^6$) — two vertices $v$ and $u$ are connected by an undirected edge with weight $w$. The graph contains no loops and no multiple edges. It is guaranteed that the given edges form a connected graph.\n\n\n-----Output-----\n\nPrint a single integer — the sum of the weights of the paths from vertex $1$ of maximum weights of lengths $1, 2, \\dots, q$ modulo $10^9+7$.\n\n\n-----Examples-----\nInput\n7 8 25\n1 2 1\n2 3 10\n3 4 2\n1 5 2\n5 6 7\n6 4 15\n5 3 1\n1 7 3\n\nOutput\n4361\n\nInput\n2 1 5\n1 2 4\n\nOutput\n60\n\nInput\n15 15 23\n13 10 12\n11 14 12\n2 15 5\n4 10 8\n10 2 4\n10 7 5\n3 10 1\n5 6 11\n1 13 8\n9 15 4\n4 2 9\n11 15 1\n11 12 14\n10 8 12\n3 6 11\n\nOutput\n3250\n\nInput\n5 10 10000000\n2 4 798\n1 5 824\n5 2 558\n4 1 288\n3 4 1890\n3 1 134\n2 3 1485\n4 5 284\n3 5 1025\n1 2 649\n\nOutput\n768500592\n\n\n\n-----Note-----\n\nHere is the graph for the first example: [Image] \n\nSome maximum weight paths are:   length $1$: edges $(1, 7)$ — weight $3$;  length $2$: edges $(1, 2), (2, 3)$ — weight $1+10=11$;  length $3$: edges $(1, 5), (5, 6), (6, 4)$ — weight $2+7+15=24$;  length $4$: edges $(1, 5), (5, 6), (6, 4), (6, 4)$ — weight $2+7+15+15=39$;  $\\dots$ \n\nSo the answer is the sum of $25$ terms: $3+11+24+39+\\dots$\n\nIn the second example the maximum weight paths have weights $4$, $8$, $12$, $16$ and $20$.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\nn,m,q=map(int,input().split())\\nmod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"n,m,q=map(int,input().split());mod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):neig[i]=[0]\\nfor i in range(m):a,b,w=map(int,input().split());a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w]);mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\n\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor i in range(n):\\n\\t\\tfor j, w in adj[i]:\\n\\t\\t\\tNDP[j] = max(NDP[j], DP[i] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n#print(ans)\\n#print(DP)\\n#assert all(v > 0 for v in DP)\\n\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nnedges = []\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\tnedges.append((DP[i], w))\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = nedges[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\t#assert us <= lt\\n\\t#assert until > lt, (until, lt)\\n\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor a, b, w in edges:\\n\\t\\tNDP[b] = max(NDP[b], DP[a] + w)\\n\\t\\tNDP[a] = max(NDP[a], DP[b] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = intersect[j], slope[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ndef rs(): return sys.stdin.readline().rstrip()\\ndef ri(): return int(sys.stdin.readline())\\ndef ria(): return list(map(int, sys.stdin.readline().split()))\\ndef ws(s): sys.stdout.write(s + '\\\\n')\\ndef wi(n): sys.stdout.write(str(n) + '\\\\n')\\ndef wia(a): sys.stdout.write(' '.join([str(x) for x in a]) + '\\\\n')\\n\\n\\nMOD = 10 ** 9 + 7\\nINF = 100\\n\\n\\ndef convex_hull_trick(K, M, integer=True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i, j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i, j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key=K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\ndef nn2(n):\\n    return n * (n+1) // 2\\n\\n\\ndef solve(n, m, q, edges):\\n    # k < m\\n    # dp[v][k] - max path cost ending in v and having k edges\\n    dp = [[-INF]*(m+1) for _ in range(n)]\\n    mk = [0]*(m+1)\\n\\n    dp[0][0] = 0\\n\\n    for k in range(1, m+1):\\n        for e in edges:\\n            if dp[e[0]][k-1] == -INF:\\n                continue\\n            dp[e[1]][k] = max(dp[e[1]][k], dp[e[0]][k-1] + e[2])\\n            mk[k] = max(mk[k], dp[e[1]][k])\\n\\n    ans = sum(mk) % MOD\\n    if q > m:\\n        intersect = [dp[i][m] for i in range(n)]\\n        slope = [0] * n\\n\\n        for e in edges:\\n            if e[2] > slope[e[0]]:\\n                slope[e[0]] = e[2]\\n\\n        hull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n        lt = 0\\n        for i, j in enumerate(hull_i):\\n            wt = intersect[j]\\n            w = slope[j]\\n\\n            until = min(q if i == len(hull_x) else hull_x[i], q - m)\\n            if until <= 0:\\n                continue\\n\\n            active = (until - lt)\\n\\n            ans = (ans + active * wt) % MOD\\n\\n            min_uses = lt\\n            max_uses = lt + active\\n\\n            times = nn2(max_uses) - nn2(min_uses)\\n            ans = (ans + times * w) % MOD\\n\\n            lt = until\\n            if lt == q - m: break\\n\\n\\n    return ans\\n\\ndef main():\\n    n, m, q = ria()\\n    e = []\\n    for _ in range(m):\\n        u, v, w = ria()\\n        u -= 1\\n        v -= 1\\n        e.append((u, v, w))\\n        e.append((v, u, w))\\n    wi(solve(n, m, q, e))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "2 1 5\n1 2 4\n",
        "output": "60\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1366/F"
    },
    {
        "id": 1046,
        "task_id": 2226,
        "test_case_id": 3,
        "question": "You are given a simple weighted connected undirected graph, consisting of $n$ vertices and $m$ edges.\n\nA path in the graph of length $k$ is a sequence of $k+1$ vertices $v_1, v_2, \\dots, v_{k+1}$ such that for each $i$ $(1 \\le i \\le k)$ the edge $(v_i, v_{i+1})$ is present in the graph. A path from some vertex $v$ also has vertex $v_1=v$. Note that edges and vertices are allowed to be included in the path multiple times.\n\nThe weight of the path is the total weight of edges in it.\n\nFor each $i$ from $1$ to $q$ consider a path from vertex $1$ of length $i$ of the maximum weight. What is the sum of weights of these $q$ paths?\n\nAnswer can be quite large, so print it modulo $10^9+7$.\n\n\n-----Input-----\n\nThe first line contains a three integers $n$, $m$, $q$ ($2 \\le n \\le 2000$; $n - 1 \\le m \\le 2000$; $m \\le q \\le 10^9$) — the number of vertices in the graph, the number of edges in the graph and the number of lengths that should be included in the answer.\n\nEach of the next $m$ lines contains a description of an edge: three integers $v$, $u$, $w$ ($1 \\le v, u \\le n$; $1 \\le w \\le 10^6$) — two vertices $v$ and $u$ are connected by an undirected edge with weight $w$. The graph contains no loops and no multiple edges. It is guaranteed that the given edges form a connected graph.\n\n\n-----Output-----\n\nPrint a single integer — the sum of the weights of the paths from vertex $1$ of maximum weights of lengths $1, 2, \\dots, q$ modulo $10^9+7$.\n\n\n-----Examples-----\nInput\n7 8 25\n1 2 1\n2 3 10\n3 4 2\n1 5 2\n5 6 7\n6 4 15\n5 3 1\n1 7 3\n\nOutput\n4361\n\nInput\n2 1 5\n1 2 4\n\nOutput\n60\n\nInput\n15 15 23\n13 10 12\n11 14 12\n2 15 5\n4 10 8\n10 2 4\n10 7 5\n3 10 1\n5 6 11\n1 13 8\n9 15 4\n4 2 9\n11 15 1\n11 12 14\n10 8 12\n3 6 11\n\nOutput\n3250\n\nInput\n5 10 10000000\n2 4 798\n1 5 824\n5 2 558\n4 1 288\n3 4 1890\n3 1 134\n2 3 1485\n4 5 284\n3 5 1025\n1 2 649\n\nOutput\n768500592\n\n\n\n-----Note-----\n\nHere is the graph for the first example: [Image] \n\nSome maximum weight paths are:   length $1$: edges $(1, 7)$ — weight $3$;  length $2$: edges $(1, 2), (2, 3)$ — weight $1+10=11$;  length $3$: edges $(1, 5), (5, 6), (6, 4)$ — weight $2+7+15=24$;  length $4$: edges $(1, 5), (5, 6), (6, 4), (6, 4)$ — weight $2+7+15+15=39$;  $\\dots$ \n\nSo the answer is the sum of $25$ terms: $3+11+24+39+\\dots$\n\nIn the second example the maximum weight paths have weights $4$, $8$, $12$, $16$ and $20$.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\nn,m,q=map(int,input().split())\\nmod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"n,m,q=map(int,input().split());mod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):neig[i]=[0]\\nfor i in range(m):a,b,w=map(int,input().split());a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w]);mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\n\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor i in range(n):\\n\\t\\tfor j, w in adj[i]:\\n\\t\\t\\tNDP[j] = max(NDP[j], DP[i] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n#print(ans)\\n#print(DP)\\n#assert all(v > 0 for v in DP)\\n\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nnedges = []\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\tnedges.append((DP[i], w))\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = nedges[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\t#assert us <= lt\\n\\t#assert until > lt, (until, lt)\\n\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor a, b, w in edges:\\n\\t\\tNDP[b] = max(NDP[b], DP[a] + w)\\n\\t\\tNDP[a] = max(NDP[a], DP[b] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = intersect[j], slope[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ndef rs(): return sys.stdin.readline().rstrip()\\ndef ri(): return int(sys.stdin.readline())\\ndef ria(): return list(map(int, sys.stdin.readline().split()))\\ndef ws(s): sys.stdout.write(s + '\\\\n')\\ndef wi(n): sys.stdout.write(str(n) + '\\\\n')\\ndef wia(a): sys.stdout.write(' '.join([str(x) for x in a]) + '\\\\n')\\n\\n\\nMOD = 10 ** 9 + 7\\nINF = 100\\n\\n\\ndef convex_hull_trick(K, M, integer=True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i, j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i, j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key=K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\ndef nn2(n):\\n    return n * (n+1) // 2\\n\\n\\ndef solve(n, m, q, edges):\\n    # k < m\\n    # dp[v][k] - max path cost ending in v and having k edges\\n    dp = [[-INF]*(m+1) for _ in range(n)]\\n    mk = [0]*(m+1)\\n\\n    dp[0][0] = 0\\n\\n    for k in range(1, m+1):\\n        for e in edges:\\n            if dp[e[0]][k-1] == -INF:\\n                continue\\n            dp[e[1]][k] = max(dp[e[1]][k], dp[e[0]][k-1] + e[2])\\n            mk[k] = max(mk[k], dp[e[1]][k])\\n\\n    ans = sum(mk) % MOD\\n    if q > m:\\n        intersect = [dp[i][m] for i in range(n)]\\n        slope = [0] * n\\n\\n        for e in edges:\\n            if e[2] > slope[e[0]]:\\n                slope[e[0]] = e[2]\\n\\n        hull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n        lt = 0\\n        for i, j in enumerate(hull_i):\\n            wt = intersect[j]\\n            w = slope[j]\\n\\n            until = min(q if i == len(hull_x) else hull_x[i], q - m)\\n            if until <= 0:\\n                continue\\n\\n            active = (until - lt)\\n\\n            ans = (ans + active * wt) % MOD\\n\\n            min_uses = lt\\n            max_uses = lt + active\\n\\n            times = nn2(max_uses) - nn2(min_uses)\\n            ans = (ans + times * w) % MOD\\n\\n            lt = until\\n            if lt == q - m: break\\n\\n\\n    return ans\\n\\ndef main():\\n    n, m, q = ria()\\n    e = []\\n    for _ in range(m):\\n        u, v, w = ria()\\n        u -= 1\\n        v -= 1\\n        e.append((u, v, w))\\n        e.append((v, u, w))\\n    wi(solve(n, m, q, e))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "15 15 23\n13 10 12\n11 14 12\n2 15 5\n4 10 8\n10 2 4\n10 7 5\n3 10 1\n5 6 11\n1 13 8\n9 15 4\n4 2 9\n11 15 1\n11 12 14\n10 8 12\n3 6 11\n",
        "output": "3250\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1366/F"
    },
    {
        "id": 1047,
        "task_id": 2226,
        "test_case_id": 4,
        "question": "You are given a simple weighted connected undirected graph, consisting of $n$ vertices and $m$ edges.\n\nA path in the graph of length $k$ is a sequence of $k+1$ vertices $v_1, v_2, \\dots, v_{k+1}$ such that for each $i$ $(1 \\le i \\le k)$ the edge $(v_i, v_{i+1})$ is present in the graph. A path from some vertex $v$ also has vertex $v_1=v$. Note that edges and vertices are allowed to be included in the path multiple times.\n\nThe weight of the path is the total weight of edges in it.\n\nFor each $i$ from $1$ to $q$ consider a path from vertex $1$ of length $i$ of the maximum weight. What is the sum of weights of these $q$ paths?\n\nAnswer can be quite large, so print it modulo $10^9+7$.\n\n\n-----Input-----\n\nThe first line contains a three integers $n$, $m$, $q$ ($2 \\le n \\le 2000$; $n - 1 \\le m \\le 2000$; $m \\le q \\le 10^9$) — the number of vertices in the graph, the number of edges in the graph and the number of lengths that should be included in the answer.\n\nEach of the next $m$ lines contains a description of an edge: three integers $v$, $u$, $w$ ($1 \\le v, u \\le n$; $1 \\le w \\le 10^6$) — two vertices $v$ and $u$ are connected by an undirected edge with weight $w$. The graph contains no loops and no multiple edges. It is guaranteed that the given edges form a connected graph.\n\n\n-----Output-----\n\nPrint a single integer — the sum of the weights of the paths from vertex $1$ of maximum weights of lengths $1, 2, \\dots, q$ modulo $10^9+7$.\n\n\n-----Examples-----\nInput\n7 8 25\n1 2 1\n2 3 10\n3 4 2\n1 5 2\n5 6 7\n6 4 15\n5 3 1\n1 7 3\n\nOutput\n4361\n\nInput\n2 1 5\n1 2 4\n\nOutput\n60\n\nInput\n15 15 23\n13 10 12\n11 14 12\n2 15 5\n4 10 8\n10 2 4\n10 7 5\n3 10 1\n5 6 11\n1 13 8\n9 15 4\n4 2 9\n11 15 1\n11 12 14\n10 8 12\n3 6 11\n\nOutput\n3250\n\nInput\n5 10 10000000\n2 4 798\n1 5 824\n5 2 558\n4 1 288\n3 4 1890\n3 1 134\n2 3 1485\n4 5 284\n3 5 1025\n1 2 649\n\nOutput\n768500592\n\n\n\n-----Note-----\n\nHere is the graph for the first example: [Image] \n\nSome maximum weight paths are:   length $1$: edges $(1, 7)$ — weight $3$;  length $2$: edges $(1, 2), (2, 3)$ — weight $1+10=11$;  length $3$: edges $(1, 5), (5, 6), (6, 4)$ — weight $2+7+15=24$;  length $4$: edges $(1, 5), (5, 6), (6, 4), (6, 4)$ — weight $2+7+15+15=39$;  $\\dots$ \n\nSo the answer is the sum of $25$ terms: $3+11+24+39+\\dots$\n\nIn the second example the maximum weight paths have weights $4$, $8$, $12$, $16$ and $20$.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\nn,m,q=map(int,input().split())\\nmod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"n,m,q=map(int,input().split());mod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):neig[i]=[0]\\nfor i in range(m):a,b,w=map(int,input().split());a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w]);mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\n\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor i in range(n):\\n\\t\\tfor j, w in adj[i]:\\n\\t\\t\\tNDP[j] = max(NDP[j], DP[i] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n#print(ans)\\n#print(DP)\\n#assert all(v > 0 for v in DP)\\n\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nnedges = []\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\tnedges.append((DP[i], w))\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = nedges[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\t#assert us <= lt\\n\\t#assert until > lt, (until, lt)\\n\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor a, b, w in edges:\\n\\t\\tNDP[b] = max(NDP[b], DP[a] + w)\\n\\t\\tNDP[a] = max(NDP[a], DP[b] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = intersect[j], slope[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ndef rs(): return sys.stdin.readline().rstrip()\\ndef ri(): return int(sys.stdin.readline())\\ndef ria(): return list(map(int, sys.stdin.readline().split()))\\ndef ws(s): sys.stdout.write(s + '\\\\n')\\ndef wi(n): sys.stdout.write(str(n) + '\\\\n')\\ndef wia(a): sys.stdout.write(' '.join([str(x) for x in a]) + '\\\\n')\\n\\n\\nMOD = 10 ** 9 + 7\\nINF = 100\\n\\n\\ndef convex_hull_trick(K, M, integer=True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i, j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i, j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key=K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\ndef nn2(n):\\n    return n * (n+1) // 2\\n\\n\\ndef solve(n, m, q, edges):\\n    # k < m\\n    # dp[v][k] - max path cost ending in v and having k edges\\n    dp = [[-INF]*(m+1) for _ in range(n)]\\n    mk = [0]*(m+1)\\n\\n    dp[0][0] = 0\\n\\n    for k in range(1, m+1):\\n        for e in edges:\\n            if dp[e[0]][k-1] == -INF:\\n                continue\\n            dp[e[1]][k] = max(dp[e[1]][k], dp[e[0]][k-1] + e[2])\\n            mk[k] = max(mk[k], dp[e[1]][k])\\n\\n    ans = sum(mk) % MOD\\n    if q > m:\\n        intersect = [dp[i][m] for i in range(n)]\\n        slope = [0] * n\\n\\n        for e in edges:\\n            if e[2] > slope[e[0]]:\\n                slope[e[0]] = e[2]\\n\\n        hull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n        lt = 0\\n        for i, j in enumerate(hull_i):\\n            wt = intersect[j]\\n            w = slope[j]\\n\\n            until = min(q if i == len(hull_x) else hull_x[i], q - m)\\n            if until <= 0:\\n                continue\\n\\n            active = (until - lt)\\n\\n            ans = (ans + active * wt) % MOD\\n\\n            min_uses = lt\\n            max_uses = lt + active\\n\\n            times = nn2(max_uses) - nn2(min_uses)\\n            ans = (ans + times * w) % MOD\\n\\n            lt = until\\n            if lt == q - m: break\\n\\n\\n    return ans\\n\\ndef main():\\n    n, m, q = ria()\\n    e = []\\n    for _ in range(m):\\n        u, v, w = ria()\\n        u -= 1\\n        v -= 1\\n        e.append((u, v, w))\\n        e.append((v, u, w))\\n    wi(solve(n, m, q, e))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "5 10 10000000\n2 4 798\n1 5 824\n5 2 558\n4 1 288\n3 4 1890\n3 1 134\n2 3 1485\n4 5 284\n3 5 1025\n1 2 649\n",
        "output": "768500592\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1366/F"
    },
    {
        "id": 1048,
        "task_id": 2226,
        "test_case_id": 5,
        "question": "You are given a simple weighted connected undirected graph, consisting of $n$ vertices and $m$ edges.\n\nA path in the graph of length $k$ is a sequence of $k+1$ vertices $v_1, v_2, \\dots, v_{k+1}$ such that for each $i$ $(1 \\le i \\le k)$ the edge $(v_i, v_{i+1})$ is present in the graph. A path from some vertex $v$ also has vertex $v_1=v$. Note that edges and vertices are allowed to be included in the path multiple times.\n\nThe weight of the path is the total weight of edges in it.\n\nFor each $i$ from $1$ to $q$ consider a path from vertex $1$ of length $i$ of the maximum weight. What is the sum of weights of these $q$ paths?\n\nAnswer can be quite large, so print it modulo $10^9+7$.\n\n\n-----Input-----\n\nThe first line contains a three integers $n$, $m$, $q$ ($2 \\le n \\le 2000$; $n - 1 \\le m \\le 2000$; $m \\le q \\le 10^9$) — the number of vertices in the graph, the number of edges in the graph and the number of lengths that should be included in the answer.\n\nEach of the next $m$ lines contains a description of an edge: three integers $v$, $u$, $w$ ($1 \\le v, u \\le n$; $1 \\le w \\le 10^6$) — two vertices $v$ and $u$ are connected by an undirected edge with weight $w$. The graph contains no loops and no multiple edges. It is guaranteed that the given edges form a connected graph.\n\n\n-----Output-----\n\nPrint a single integer — the sum of the weights of the paths from vertex $1$ of maximum weights of lengths $1, 2, \\dots, q$ modulo $10^9+7$.\n\n\n-----Examples-----\nInput\n7 8 25\n1 2 1\n2 3 10\n3 4 2\n1 5 2\n5 6 7\n6 4 15\n5 3 1\n1 7 3\n\nOutput\n4361\n\nInput\n2 1 5\n1 2 4\n\nOutput\n60\n\nInput\n15 15 23\n13 10 12\n11 14 12\n2 15 5\n4 10 8\n10 2 4\n10 7 5\n3 10 1\n5 6 11\n1 13 8\n9 15 4\n4 2 9\n11 15 1\n11 12 14\n10 8 12\n3 6 11\n\nOutput\n3250\n\nInput\n5 10 10000000\n2 4 798\n1 5 824\n5 2 558\n4 1 288\n3 4 1890\n3 1 134\n2 3 1485\n4 5 284\n3 5 1025\n1 2 649\n\nOutput\n768500592\n\n\n\n-----Note-----\n\nHere is the graph for the first example: [Image] \n\nSome maximum weight paths are:   length $1$: edges $(1, 7)$ — weight $3$;  length $2$: edges $(1, 2), (2, 3)$ — weight $1+10=11$;  length $3$: edges $(1, 5), (5, 6), (6, 4)$ — weight $2+7+15=24$;  length $4$: edges $(1, 5), (5, 6), (6, 4), (6, 4)$ — weight $2+7+15+15=39$;  $\\dots$ \n\nSo the answer is the sum of $25$ terms: $3+11+24+39+\\dots$\n\nIn the second example the maximum weight paths have weights $4$, $8$, $12$, $16$ and $20$.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\nn,m,q=map(int,input().split())\\nmod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"n,m,q=map(int,input().split());mod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):neig[i]=[0]\\nfor i in range(m):a,b,w=map(int,input().split());a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w]);mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\n\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor i in range(n):\\n\\t\\tfor j, w in adj[i]:\\n\\t\\t\\tNDP[j] = max(NDP[j], DP[i] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n#print(ans)\\n#print(DP)\\n#assert all(v > 0 for v in DP)\\n\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nnedges = []\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\tnedges.append((DP[i], w))\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = nedges[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\t#assert us <= lt\\n\\t#assert until > lt, (until, lt)\\n\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor a, b, w in edges:\\n\\t\\tNDP[b] = max(NDP[b], DP[a] + w)\\n\\t\\tNDP[a] = max(NDP[a], DP[b] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = intersect[j], slope[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ndef rs(): return sys.stdin.readline().rstrip()\\ndef ri(): return int(sys.stdin.readline())\\ndef ria(): return list(map(int, sys.stdin.readline().split()))\\ndef ws(s): sys.stdout.write(s + '\\\\n')\\ndef wi(n): sys.stdout.write(str(n) + '\\\\n')\\ndef wia(a): sys.stdout.write(' '.join([str(x) for x in a]) + '\\\\n')\\n\\n\\nMOD = 10 ** 9 + 7\\nINF = 100\\n\\n\\ndef convex_hull_trick(K, M, integer=True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i, j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i, j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key=K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\ndef nn2(n):\\n    return n * (n+1) // 2\\n\\n\\ndef solve(n, m, q, edges):\\n    # k < m\\n    # dp[v][k] - max path cost ending in v and having k edges\\n    dp = [[-INF]*(m+1) for _ in range(n)]\\n    mk = [0]*(m+1)\\n\\n    dp[0][0] = 0\\n\\n    for k in range(1, m+1):\\n        for e in edges:\\n            if dp[e[0]][k-1] == -INF:\\n                continue\\n            dp[e[1]][k] = max(dp[e[1]][k], dp[e[0]][k-1] + e[2])\\n            mk[k] = max(mk[k], dp[e[1]][k])\\n\\n    ans = sum(mk) % MOD\\n    if q > m:\\n        intersect = [dp[i][m] for i in range(n)]\\n        slope = [0] * n\\n\\n        for e in edges:\\n            if e[2] > slope[e[0]]:\\n                slope[e[0]] = e[2]\\n\\n        hull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n        lt = 0\\n        for i, j in enumerate(hull_i):\\n            wt = intersect[j]\\n            w = slope[j]\\n\\n            until = min(q if i == len(hull_x) else hull_x[i], q - m)\\n            if until <= 0:\\n                continue\\n\\n            active = (until - lt)\\n\\n            ans = (ans + active * wt) % MOD\\n\\n            min_uses = lt\\n            max_uses = lt + active\\n\\n            times = nn2(max_uses) - nn2(min_uses)\\n            ans = (ans + times * w) % MOD\\n\\n            lt = until\\n            if lt == q - m: break\\n\\n\\n    return ans\\n\\ndef main():\\n    n, m, q = ria()\\n    e = []\\n    for _ in range(m):\\n        u, v, w = ria()\\n        u -= 1\\n        v -= 1\\n        e.append((u, v, w))\\n        e.append((v, u, w))\\n    wi(solve(n, m, q, e))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "4 3 3\n4 2 41\n1 3 26\n4 3 24\n",
        "output": "169\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1366/F"
    },
    {
        "id": 1049,
        "task_id": 2226,
        "test_case_id": 6,
        "question": "You are given a simple weighted connected undirected graph, consisting of $n$ vertices and $m$ edges.\n\nA path in the graph of length $k$ is a sequence of $k+1$ vertices $v_1, v_2, \\dots, v_{k+1}$ such that for each $i$ $(1 \\le i \\le k)$ the edge $(v_i, v_{i+1})$ is present in the graph. A path from some vertex $v$ also has vertex $v_1=v$. Note that edges and vertices are allowed to be included in the path multiple times.\n\nThe weight of the path is the total weight of edges in it.\n\nFor each $i$ from $1$ to $q$ consider a path from vertex $1$ of length $i$ of the maximum weight. What is the sum of weights of these $q$ paths?\n\nAnswer can be quite large, so print it modulo $10^9+7$.\n\n\n-----Input-----\n\nThe first line contains a three integers $n$, $m$, $q$ ($2 \\le n \\le 2000$; $n - 1 \\le m \\le 2000$; $m \\le q \\le 10^9$) — the number of vertices in the graph, the number of edges in the graph and the number of lengths that should be included in the answer.\n\nEach of the next $m$ lines contains a description of an edge: three integers $v$, $u$, $w$ ($1 \\le v, u \\le n$; $1 \\le w \\le 10^6$) — two vertices $v$ and $u$ are connected by an undirected edge with weight $w$. The graph contains no loops and no multiple edges. It is guaranteed that the given edges form a connected graph.\n\n\n-----Output-----\n\nPrint a single integer — the sum of the weights of the paths from vertex $1$ of maximum weights of lengths $1, 2, \\dots, q$ modulo $10^9+7$.\n\n\n-----Examples-----\nInput\n7 8 25\n1 2 1\n2 3 10\n3 4 2\n1 5 2\n5 6 7\n6 4 15\n5 3 1\n1 7 3\n\nOutput\n4361\n\nInput\n2 1 5\n1 2 4\n\nOutput\n60\n\nInput\n15 15 23\n13 10 12\n11 14 12\n2 15 5\n4 10 8\n10 2 4\n10 7 5\n3 10 1\n5 6 11\n1 13 8\n9 15 4\n4 2 9\n11 15 1\n11 12 14\n10 8 12\n3 6 11\n\nOutput\n3250\n\nInput\n5 10 10000000\n2 4 798\n1 5 824\n5 2 558\n4 1 288\n3 4 1890\n3 1 134\n2 3 1485\n4 5 284\n3 5 1025\n1 2 649\n\nOutput\n768500592\n\n\n\n-----Note-----\n\nHere is the graph for the first example: [Image] \n\nSome maximum weight paths are:   length $1$: edges $(1, 7)$ — weight $3$;  length $2$: edges $(1, 2), (2, 3)$ — weight $1+10=11$;  length $3$: edges $(1, 5), (5, 6), (6, 4)$ — weight $2+7+15=24$;  length $4$: edges $(1, 5), (5, 6), (6, 4), (6, 4)$ — weight $2+7+15+15=39$;  $\\dots$ \n\nSo the answer is the sum of $25$ terms: $3+11+24+39+\\dots$\n\nIn the second example the maximum weight paths have weights $4$, $8$, $12$, $16$ and $20$.",
        "solutions": "[\"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\n#lev contains height from root,lower neighbour, higher neighbours\\n#lev[0] contains 0 (because it is the root), higher neighbours (=neighbours)\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1\\n    b-=1\\n    neig[a][0]+=1\\n    neig[b][0]+=1\\n    neig[a].append([b,w])\\n    neig[b].append([a,w])\\n    mxw=max(mxw,w)\\n    wgts[a]=max(wgts[a],w)\\n    wgts[b]=max(wgts[b],w)\\nposs=[-1]*n\\nposs[0]=0\\nsol=0\\ncurw=0\\nhasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:\\n        hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:\\n            curmx=poss[i]\\n            ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:\\n                ov=True\\n    curw=curmx\\n    sol+=curw\\n    sol%=mod\\n    l+=1\\nif l==q:\\n    print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:\\n                    mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:\\n                sol+=rem*curw\\n                sol+=mx*((rem*(rem+1))//2)\\n                sol%=mod\\n                ov=True\\n            else:\\n                sol+=(gd-1)*curw\\n                sol+=mx*((gd*(gd-1))//2)\\n                sol%=mod\\n                for i in range(n):\\n                    poss[i]+=gd*wgts[i]\\n                    curw=max(curw,poss[i])\\n                sol+=curw\\n                rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn,m,q=map(int,input().split())\\nmod=1000000007\\nmxw=0\\nwgts=[0]*n\\nneig=[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False\\nl=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):\\n                newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])\\n    curmx=0\\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:\\n        rem=q-l\\n        sol+=rem*curw\\n        sol%=mod\\n        sol+=mxw*((rem*(rem+1))//2)\\n        sol%=mod\\n        print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])\\n            gd=-1\\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx\\n                    loss=curw-poss[i]\\n                    loss+=diff-1\\n                    att=loss//diff\\n                    if gd==-1:\\n                        gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\nn,m,q=map(int,input().split())\\nmod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):\\n    neig[i]=[0]\\n\\nfor i in range(m):\\n    a,b,w=map(int,input().split())\\n    a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w])\\n    mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"n,m,q=map(int,input().split());mod,mxw = 1000000007,0;wgts,neig=[0]*n,[0]*n\\nfor i in range(n):neig[i]=[0]\\nfor i in range(m):a,b,w=map(int,input().split());a-=1;b-=1;neig[a][0]+=1;neig[b][0]+=1;neig[a].append([b,w]);neig[b].append([a,w]);mxw=max(mxw,w);wgts[a]=max(wgts[a],w);wgts[b]=max(wgts[b],w)\\nposs=[-1]*n;poss[0]=0;sol=0;curw=0;hasmxw=[False]*n\\nfor i in range(n):\\n    if wgts[i]==mxw:hasmxw[i]=True\\nov=False;l=0\\nwhile l<q and not ov and l<3000:\\n    newposs=[-1]*n;curmx=0\\n    for i in range(n):\\n        if poss[i]>=0:\\n            for j in range(1,neig[i][0]+1):newposs[neig[i][j][0]]=max(newposs[neig[i][j][0]],poss[i]+neig[i][j][1])    \\n    for i in range(n):\\n        poss[i]=newposs[i]\\n        if poss[i]>curmx:curmx=poss[i];ov=hasmxw[i]\\n        else:\\n            if poss[i]==curmx and hasmxw[i]:ov=True\\n    curw=curmx;sol+=curw;sol%=mod;l+=1\\nif l==q:print(sol)\\nelse:\\n    if ov:rem=q-l;sol+=rem*curw;sol%=mod;sol+=mxw*((rem*(rem+1))//2);sol%=mod;print(sol)\\n    else:\\n        rem=q-l\\n        while not ov:\\n            mx=0;gd=-1\\n            for i in range(n):\\n                if poss[i]==curw:mx=max(mx,wgts[i])            \\n            for i in range(n):\\n                if wgts[i]>mx and poss[i]>=0:\\n                    diff=wgts[i]-mx;loss=curw-poss[i];loss+=diff-1;att=loss//diff\\n                    if gd==-1:gd=att\\n                    gd=min(att,gd)\\n            if gd==-1 or gd>rem:sol+=rem*curw;sol+=mx*((rem*(rem+1))//2);sol%=mod;ov=True\\n            else:\\n                sol+=(gd-1)*curw;sol+=mx*((gd*(gd-1))//2);sol%=mod\\n                for i in range(n):poss[i]+=gd*wgts[i];curw=max(curw,poss[i])\\n                sol+=curw;rem-=gd\\n        print(sol)\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\n\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor i in range(n):\\n\\t\\tfor j, w in adj[i]:\\n\\t\\t\\tNDP[j] = max(NDP[j], DP[i] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n#print(ans)\\n#print(DP)\\n#assert all(v > 0 for v in DP)\\n\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nnedges = []\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\tnedges.append((DP[i], w))\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = nedges[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\t#assert us <= lt\\n\\t#assert until > lt, (until, lt)\\n\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn, m, q = list(map(int, input().split()))\\n\\nMOD = 10 ** 9 + 7\\n\\nedges = []\\nadj = [[] for _ in range(n)]\\n\\nfor i in range(m):\\n\\ta, b, w = list(map(int, input().split()))\\n\\ta -= 1\\n\\tb -= 1\\n\\tedges.append((a, b, w))\\n\\tadj[a].append((b, w))\\n\\tadj[b].append((a, w))\\n\\n\\nINF = 10 ** 18\\nans = 0\\n\\nDP = [-INF] * n\\nDP[0] = 0\\n\\n# Paths are at most m long\\nfor plen in range(m):\\n\\tNDP = [-INF] * n\\n\\n\\tfor a, b, w in edges:\\n\\t\\tNDP[b] = max(NDP[b], DP[a] + w)\\n\\t\\tNDP[a] = max(NDP[a], DP[b] + w)\\n\\n\\tDP = NDP\\n\\tans = (ans + max(DP)) % MOD\\n\\n\\\"\\\"\\\" From PyRival \\\"\\\"\\\"\\ndef convex_hull_trick(K, M, integer = True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i,j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i,j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key = K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\nslope, intersect = [], []\\nfor a, b, w in edges:\\n\\ti = max(a, b, key=lambda i: DP[i])\\n\\tassert DP[i] > 0\\n\\n\\t#print(f'edge ({a+1}, {b+1}, {w}) usable from {usable_from} with distance {w_at_time}', file=sys.stderr)\\n\\tslope.append(w)\\n\\tintersect.append(DP[i])\\n\\n# For each edge, figure out the interval in which it is the best option\\nhull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n#print(hull_i)\\n#print(hull_x)\\n\\ndef tri(x):\\n\\treturn (x * (x + 1)) // 2\\n\\nlt = 0\\nfor i, j in enumerate(hull_i):\\n\\twt, w = intersect[j], slope[j]\\n\\n\\tuntil = min(q if i == len(hull_x) else hull_x[i], q - m)\\n\\n\\tif until <= 0: continue\\n\\n\\tactive = (until - lt)\\n\\tans = (ans + active * wt) % MOD\\n\\n\\tmin_uses = lt\\n\\tmax_uses = lt + active\\n\\n\\ttimes = tri(max_uses) - tri(min_uses)\\n\\tans = (ans + times * w) % MOD\\n\\n\\t#print(f'since {lt} to {until} use {(wt, w)} from {min_uses} to {max_uses} ({times}) times')\\n\\t#print(ans)\\n\\n\\tlt = until\\n\\tif lt == q - m: break\\n\\nprint(ans)\\n\\n\", \"import sys\\ndef rs(): return sys.stdin.readline().rstrip()\\ndef ri(): return int(sys.stdin.readline())\\ndef ria(): return list(map(int, sys.stdin.readline().split()))\\ndef ws(s): sys.stdout.write(s + '\\\\n')\\ndef wi(n): sys.stdout.write(str(n) + '\\\\n')\\ndef wia(a): sys.stdout.write(' '.join([str(x) for x in a]) + '\\\\n')\\n\\n\\nMOD = 10 ** 9 + 7\\nINF = 100\\n\\n\\ndef convex_hull_trick(K, M, integer=True):\\n    \\\"\\\"\\\"\\n    Given lines on the form y = K[i] * x + M[i] this function returns intervals,\\n    such that on each interval the convex hull is made up of a single line.\\n    Input:\\n        K: list of the slopes\\n        M: list of the constants (value at x = 0)\\n        interger: boolean for turning on / off integer mode. Integer mode is exact, it\\n                  works by effectively flooring the seperators of the intervals.\\n    Return:\\n        hull_i: on interval j, line i = hull_i[j] is >= all other lines\\n        hull_x: interval j and j + 1 is separated by x = hull_x[j], (hull_x[j] is the last x in interval j)\\n    \\\"\\\"\\\"\\n    if integer:\\n        intersect = lambda i, j: (M[j] - M[i]) // (K[i] - K[j])\\n    else:\\n        intersect = lambda i, j: (M[j] - M[i]) / (K[i] - K[j])\\n\\n    assert len(K) == len(M)\\n\\n    hull_i = []\\n    hull_x = []\\n    order = sorted(list(range(len(K))), key=K.__getitem__)\\n    for i in order:\\n        while True:\\n            if not hull_i:\\n                hull_i.append(i)\\n                break\\n            elif K[hull_i[-1]] == K[i]:\\n                if M[hull_i[-1]] >= M[i]:\\n                    break\\n                hull_i.pop()\\n                if hull_x: hull_x.pop()\\n            else:\\n                x = intersect(i, hull_i[-1])\\n                if hull_x and x <= hull_x[-1]:\\n                    hull_i.pop()\\n                    hull_x.pop()\\n                else:\\n                    hull_i.append(i)\\n                    hull_x.append(x)\\n                    break\\n    return hull_i, hull_x\\n\\n\\ndef nn2(n):\\n    return n * (n+1) // 2\\n\\n\\ndef solve(n, m, q, edges):\\n    # k < m\\n    # dp[v][k] - max path cost ending in v and having k edges\\n    dp = [[-INF]*(m+1) for _ in range(n)]\\n    mk = [0]*(m+1)\\n\\n    dp[0][0] = 0\\n\\n    for k in range(1, m+1):\\n        for e in edges:\\n            if dp[e[0]][k-1] == -INF:\\n                continue\\n            dp[e[1]][k] = max(dp[e[1]][k], dp[e[0]][k-1] + e[2])\\n            mk[k] = max(mk[k], dp[e[1]][k])\\n\\n    ans = sum(mk) % MOD\\n    if q > m:\\n        intersect = [dp[i][m] for i in range(n)]\\n        slope = [0] * n\\n\\n        for e in edges:\\n            if e[2] > slope[e[0]]:\\n                slope[e[0]] = e[2]\\n\\n        hull_i, hull_x = convex_hull_trick(slope, intersect)\\n\\n        lt = 0\\n        for i, j in enumerate(hull_i):\\n            wt = intersect[j]\\n            w = slope[j]\\n\\n            until = min(q if i == len(hull_x) else hull_x[i], q - m)\\n            if until <= 0:\\n                continue\\n\\n            active = (until - lt)\\n\\n            ans = (ans + active * wt) % MOD\\n\\n            min_uses = lt\\n            max_uses = lt + active\\n\\n            times = nn2(max_uses) - nn2(min_uses)\\n            ans = (ans + times * w) % MOD\\n\\n            lt = until\\n            if lt == q - m: break\\n\\n\\n    return ans\\n\\ndef main():\\n    n, m, q = ria()\\n    e = []\\n    for _ in range(m):\\n        u, v, w = ria()\\n        u -= 1\\n        v -= 1\\n        e.append((u, v, w))\\n        e.append((v, u, w))\\n    wi(solve(n, m, q, e))\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\"]",
        "difficulty": "interview",
        "input": "5 4 6\n2 4 1\n1 5 2\n5 2 5\n3 4 1\n",
        "output": "87\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1366/F"
    },
    {
        "id": 1050,
        "task_id": 2246,
        "test_case_id": 5,
        "question": "There are n cities and n - 1 roads in the Seven Kingdoms, each road connects two cities and we can reach any city from any other by the roads.\n\nTheon and Yara Greyjoy are on a horse in the first city, they are starting traveling through the roads. But the weather is foggy, so they can’t see where the horse brings them. When the horse reaches a city (including the first one), it goes to one of the cities connected to the current city. But it is a strange horse, it only goes to cities in which they weren't before. In each such city, the horse goes with equal probabilities and it stops when there are no such cities. \n\nLet the length of each road be 1. The journey starts in the city 1. What is the expected length (expected value of length) of their journey? You can read about expected (average) value by the link https://en.wikipedia.org/wiki/Expected_value.\n\n\n-----Input-----\n\nThe first line contains a single integer n (1 ≤ n ≤ 100000) — number of cities.\n\nThen n - 1 lines follow. The i-th line of these lines contains two integers u_{i} and v_{i} (1 ≤ u_{i}, v_{i} ≤ n, u_{i} ≠ v_{i}) — the cities connected by the i-th road.\n\nIt is guaranteed that one can reach any city from any other by the roads.\n\n\n-----Output-----\n\nPrint a number — the expected length of their journey. The journey starts in the city 1.\n\nYour answer will be considered correct if its absolute or relative error does not exceed 10^{ - 6}.\n\nNamely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if $\\frac{|a - b|}{\\operatorname{max}(1, b)} \\leq 10^{-6}$.\n\n\n-----Examples-----\nInput\n4\n1 2\n1 3\n2 4\n\nOutput\n1.500000000000000\n\nInput\n5\n1 2\n1 3\n3 4\n2 5\n\nOutput\n2.000000000000000\n\n\n\n-----Note-----\n\nIn the first sample, their journey may end in cities 3 or 4 with equal probability. The distance to city 3 is 1 and to city 4 is 2, so the expected length is 1.5.\n\nIn the second sample, their journey may end in city 4 or 5. The distance to the both cities is 2, so the expected length is 2.",
        "solutions": "[\"from sys import stdin\\nfrom decimal import Decimal as D\\ninput = stdin.readline\\nn = int(input())\\nadj = [[] for i in range(n+1)]\\ntree = [[] for i in range(n+1)]\\nvisit = [-1]*(n+1)\\nlength = [-1]*(n+1)\\nfor i in range(n-1):\\n    a, b = map(int,input().split())\\n    adj[a].append(b)\\n    adj[b].append(a)\\nbfsord = []\\n\\nfrom collections import deque\\nQ = deque()\\nQ.append(1)\\nvisit[1] = 0\\nwhile len(Q):\\n    p = Q.popleft()\\n    bfsord.append(p)\\n    for q in adj[p]:\\n        if visit[q] != -1: continue\\n        visit[q] = visit[p]+1\\n        Q.append(q)\\n        tree[p].append(q)\\n\\nfor p in reversed(bfsord):\\n    if not tree[p]: length[p] = D(0)\\n    else: length[p] = D(1) + sum(length[q] for q in tree[p])/len(tree[p])\\nprint(length[1])\", \"import sys\\n\\ndef r():\\n    return list(map(int, input().split()))\\n\\nn = int(input())\\nedge = [r() for i in range(n-1)]\\n\\nadj = [[] for i in range(n+1)]\\nfor e in edge:\\n    adj[e[0]].append(e[1])\\n    adj[e[1]].append(e[0])\\n\\nprob = [0.0 for i in range(n+1)]\\nd = [0 for i in range(n+1)]\\nvisited = [False for i in range(n+1)]\\n\\nprob[1] = 1\\nans = 0.0\\n\\nst = [1]\\nvisited[1] = True\\nwhile st:\\n    u = st.pop()\\n    cnt = len(adj[u])\\n    if u != 1:\\n        cnt -= 1\\n    if cnt == 0:\\n        ans = ans + prob[u]*d[u]\\n    else:\\n        for v in adj[u]:\\n            if not visited[v]:\\n                visited[v] = True\\n                prob[v] = prob[u]*(1.0/cnt)\\n                d[v] = d[u]+1\\n                st.append(v)\\n                \\nprint(ans)\\n\\n\", \"from queue import *\\n\\nn = int(input())\\ng = [[] for x in range(n)]\\n\\nfor i in range(n-1):\\n  a, b = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  g[a].append(b)\\n  g[b].append(a)\\n\\nused = [0]*n\\n\\nanw = 0\\n\\ndef solve(v, d, r):\\n  nonlocal anw\\n  q = Queue()\\n  q.put((v, d, r))\\n  while not q.empty():\\n    v, d, r = q.get(v)\\n    used[v] = True\\n    for u in g[v]:\\n      if not used[u]:\\n        q.put((u, d+1, r*(len(g[v])-(v!=0))))\\n        #print(\\\"put\\\", u)\\n    #print(\\\"so at\\\", v, \\\"len\\\", len(g[v]))\\n    if v != 0 and len(g[v]) == 1:\\n      #print(\\\"At \\\", v, \\\"is\\\", d, r)\\n      anw += d/r\\n  \\nsolve(0, 0, 1)\\nprint(anw)\", \"def f():\\n    n = int(input())\\n\\n    if n == 1:\\n        print(0)\\n        return\\n\\n    road = {}\\n    visited = {}\\n    for i in range(n-1):\\n        u, v = input().split()\\n        if u in road:\\n            road[u].append(v)\\n        else:\\n            road[u] = [v]\\n        if v in road:\\n            road[v].append(u)\\n        else:\\n            road[v] = [u]\\n        visited[u] = False\\n        visited[v] = False\\n\\n    prob = {}\\n    length = {}\\n    res = []\\n\\n    queue = []\\n    for k in road['1']:\\n        prob[k] = 1.0 / len(road['1'])\\n        length[k] = 1\\n        queue.append(k)\\n        visited['1'] = True\\n\\n    while len(queue) > 0:\\n        cur = queue[0]\\n        del queue[0]\\n        dest = []\\n        for k in road[cur]:\\n            if not visited[k]:\\n                dest.append(k)\\n        if len(dest) == 0:\\n            res.append((cur, prob[cur], length[cur]))\\n            continue\\n        for k in dest:\\n            prob[k] = prob[cur] / len(dest)\\n            length[k] = length[cur] + 1\\n            queue.append(k)\\n            visited[cur] = True\\n\\n    val = 0.0\\n    for item in res:\\n        val += item[1] * item[2]\\n\\n    print(val)\\n\\nf()\\n\", \"from collections import deque\\nimport sys\\nfrom decimal import Decimal\\n\\nreadline = sys.stdin.readline\\nn = int(input())\\nedges = [[]*n for _ in [None]*n]\\n\\nfor _ in [None]*(n-1):\\n    a, b = map(int, readline().split())\\n    edges[a-1].append(b-1)\\n    edges[b-1].append(a-1)\\n\\nvisited = [False]*n\\nvisited[0] = True\\ncnt = 0\\ntotal = 0\\ndq = deque()\\nappend, pop = dq.append, dq.popleft\\nappend((0, 0, Decimal(1)))\\n\\nwhile dq:\\n    pos, l, multiple = pop()\\n    flag = True\\n    to = [x for x in edges[pos] if visited[x] == False]\\n\\n    if to:\\n        x = len(to)\\n        multiple /= Decimal(x)\\n        for v in to:\\n            visited[v] = True\\n            append((v, l+1, multiple))\\n    else:\\n        total += Decimal(l) * multiple\\n\\nprint(total)\", \"import sys\\nfrom collections import deque\\nread=lambda:sys.stdin.readline().rstrip()\\nreadi=lambda:int(sys.stdin.readline())\\nwriteln=lambda x:sys.stdout.write(str(x)+\\\"\\\\n\\\")\\nwrite=lambda x:sys.stdout.write(x)\\nN = readi()\\nif N == 1:\\n    writeln(0)\\n    return\\n\\nG = [set() for _ in range(N)]\\nfor _ in range(N-1):\\n    u, v = list(map(int, read().split()))\\n    G[u-1].add(v-1)\\n    G[v-1].add(u-1)\\n\\nvisited = [0]*N\\nprev = [0]*N\\nvisited[0] = 1\\nprev[0] = 1\\nq = deque([0])\\nlengths = []\\nev = 0\\nwhile q:\\n    cur = q.popleft()\\n    n_neighbor = len(G[cur])\\n    if cur == 0:\\n        divs = n_neighbor\\n    else:\\n        divs = n_neighbor - 1\\n    if n_neighbor == 1 and cur != 0:\\n        ev += (visited[cur] - 1)*prev[cur]\\n        continue\\n     \\n    for nxt in G[cur]:\\n        if not visited[nxt]:\\n            visited[nxt] = visited[cur] + 1\\n            prev[nxt] = (1 / divs)*prev[cur]\\n            q.append(nxt)\\n\\nprint('%.12f' % ev)\\n\", \"#!/usr/bin/env python\\n\\nimport sys\\n\\ndef explore(src, prob, length, adj_mat, gains, visited):\\n    nexts = set(adj_mat[src]) - visited\\n    if nexts:\\n        go_prob = 1 / len(nexts)\\n        length += 1\\n        for dst in nexts:\\n            visited.add(dst)\\n            explore(dst, prob * go_prob, length, adj_mat, gains, visited)\\n    else:\\n        gains[src] += prob * length\\n\\ndef main():\\n  n = int(sys.stdin.readline())\\n  adj_mat = [[] for __ in range(n)]\\n  node_exp = [0 for __ in range(n)]\\n  for __ in range(n-1):\\n      u, v = list(map(int, sys.stdin.readline().split()))\\n      adj_mat[u-1].append(v-1)\\n      adj_mat[v-1].append(u-1)\\n  stk = [0]\\n  visited= set([0])\\n  prob = [1] * n\\n  length = [0] * n\\n  while stk:\\n      nxt = stk.pop()\\n      nexts = set(adj_mat[nxt]) - visited\\n      if nexts:\\n          go_prob = 1 / len(nexts)\\n          for dst in nexts:\\n              visited.add(dst)\\n              stk.append(dst)\\n              prob[dst] *= prob[nxt] * go_prob\\n              length[dst] = length[nxt] + 1\\n      else:\\n        node_exp[nxt] += prob[nxt] * length[nxt]\\n\\n  #explore(0, 1, 0, adj_mat,node_exp, set([0]))\\n  print(\\\"{:.13f}\\\".format(sum(node_exp)))\\n\\nmain()\\n\", \"import sys\\n\\n\\ndef main():\\n    n = int(input())\\n    if n == 1:\\n        print(0)\\n        return\\n    x = [[] for i in range(n)]\\n    for i in range(n - 1):\\n        a, b = list(map(int, sys.stdin.readline().split()))\\n        x[a - 1].append(b - 1)\\n        x[b - 1].append(a - 1)\\n\\n    isl = [False] * n\\n    for i in range(1, n):\\n        if len(x[i]) == 1:\\n            isl[i] = True\\n\\n    u = [False] * n\\n    u[0] = True\\n    q = [(0, 0, 1 / len(x[0]))]\\n    a = 0\\n    while len(q) != 0:\\n        c, s, t = q.pop()\\n        for i in x[c]:\\n            if not u[i]:\\n                if isl[i]:\\n                    a += (s + 1) * t\\n                else:\\n                    q.append((i, s + 1, t / (len(x[i])-1)))\\n                u[i] = True\\n    print(a)\\n\\n\\nmain()\\n\", \"n = int(input())\\nr = [[]for _ in range(n+1)]\\n\\nfor _ in range(n-1):\\n\\tu,v = list(map(int,input().split()))\\n\\tu -= 1\\n\\tv -= 1\\n\\tr[u].append(v)\\n\\tr[v].append(u)\\n\\nif n == 1:\\n\\tprint(0)\\nelse:\\n\\tst = [(0,0,1/len(r[0]))]\\n\\tans = 0\\n\\trsum = 0\\n\\tvisit = [False] * n\\n\\tvisit[0] = True\\n\\twhile st:\\n\\t\\tv,t,m = st.pop()\\n\\t\\tfor l in r[v]:\\n\\t\\t\\tif not visit[l]:\\n\\t\\t\\t\\tif l != 0 and len(r[l]) == 1:\\n\\t\\t\\t\\t\\tans += (t+1)*m\\n\\t\\t\\t\\t\\trsum += 1\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\tst.append((l,t+1,m/(len(r[l])-1)))\\n\\t\\t\\t\\tvisit[l] = True\\n\\tprint(ans)\\n\", \"from collections import deque, Counter\\n\\ndef mean_of_ways(graph, n):\\n    leafs = []\\n    distances = [None] * n\\n    probabilities = [0] * n\\n    probabilities[0] = 1\\n    distances[0] = 0\\n    used = [False] * n\\n    used[0] = True\\n\\n    active_vertices = deque([0])\\n\\n    while active_vertices:\\n        where = active_vertices.popleft()\\n\\n        # remember what happens if we change place of used assignment\\n        is_leaf = True\\n\\n        leafs_count = ([not used[v] for v in graph[where]]).count(True)\\n        if leafs_count > 1:\\n            probabilities[where] /= leafs_count\\n\\n        for to in graph[where]:\\n            if not used[to]:\\n                active_vertices.append(to)\\n                distances[to] = distances[where] + 1\\n                probabilities[to] = probabilities[where]\\n                used[to] = True\\n                is_leaf = False\\n\\n        if is_leaf:\\n            leafs.append(where)\\n\\n    Mx = 0\\n    for i in leafs:\\n        Mx += probabilities[i] * distances[i]\\n    return Mx\\n\\ndef main():\\n    n = int(input())\\n\\n    graph = [list() for i in range(n)]\\n    for i in range(n-1):\\n        # It there are vertices in descending order we can don't check\\n        # used it or not by making edges one directional\\n        u, v = list(map(int, input().split()))\\n        u -= 1\\n        v -= 1\\n        graph[u].append(v)\\n        graph[v].append(u)\\n\\n    print(mean_of_ways(graph, n))\\n\\nmain()\\n\", \"\\\"\\\"\\\"\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout\\n\\n\\ndef main():\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n\\n    # iterative DFS\\n    stack = [(1, 0, 1)]\\n    while len(stack):\\n        u, plen, prb = stack.pop()\\n\\n        visited[u] = True\\n        nBranch = 0\\n        for v in g[u]:\\n            if not visited[v]: nBranch += 1\\n        \\n        if nBranch > 0: \\n            probability = prb * (1 / nBranch)\\n            for v in g[u]:\\n                if not visited[v]:\\n                    stack.append((v, plen+1, probability))\\n        \\n        if nBranch == 0:\\n            paths.append(plen * prb)\\n\\n        \\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\n# node, probablity, path\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"from collections import deque\\ns = 0\\nn = int(input())\\nls = [[] for i in range(n + 1)]\\nfor i in range(1, n):\\n    a, b = list(map(int, input().split()))\\n    ls[a].append(b)\\n    ls[b].append(a)\\narr = [0 for i in range(n + 1)]\\nq = deque()\\nq.append([1, 1, 0])\\nwhile q:\\n    x = q.pop()\\n    node = x[0]\\n    arr[node] = 1\\n    to_explore = []\\n    for i in ls[node]:\\n        if arr[i] == 0:\\n            to_explore.append(i)\\n\\n    if len(to_explore) == 0:\\n        # leaf\\n        s = s + x[1] * x[2]\\n        continue\\n    n = len(to_explore)\\n    for i in to_explore:\\n        q.append([i, x[1] / n, x[2] + 1])\\nprint(s)\\n\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\n# print(L)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"\\\"\\\"\\\"\\n    # recursive DFS gives RTE, implement iterative DFS\\n    Author      : Arif Ahmad\\n    Date        : \\n    Algo        : \\n    Difficulty  : \\n\\\"\\\"\\\"\\nfrom sys import stdin, stdout, setrecursionlimit\\nimport threading\\n\\n\\ng = None\\nvisited = None\\npaths = None\\n\\ndef dfs(u, plen, prb):\\n    nonlocal visited, paths\\n\\n    visited[u] = True\\n    \\n    nBranch = 0\\n    for v in g[u]:\\n        if not visited[v]: nBranch += 1\\n    \\n    if nBranch > 0: \\n        probability = prb * (1 / nBranch)\\n        for v in g[u]:\\n            #print(prb, nBranch)\\n            if not visited[v]:\\n                dfs(v, plen+1, probability)\\n\\n    if nBranch == 0:\\n        paths.append(plen * prb)\\n\\n\\ndef main():\\n    nonlocal g, visited, paths\\n\\n    n = int(stdin.readline().strip())\\n    g = [[] for i in range(n+1)]\\n    for _ in range(n-1):\\n        u, v = [int(_) for _ in stdin.readline().strip().split()]\\n        g[u].append(v)\\n        g[v].append(u)\\n\\n    visited = [False for i in range(n+1)]\\n    paths = []\\n    #setrecursionlimit(n+10)\\n    dfs(1, 0, 1)\\n\\n    ans = sum(paths) \\n    stdout.write(str(ans) + '\\\\n')\\n\\ndef __starting_point():\\n    setrecursionlimit(10**6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"def dfs(start):\\n    visited = [0, True] + [False]*(n-1)\\n    stack = [(x, 1, 1/len(tree[start])) for x in tree[start]]\\n    while stack:\\n        v, l, p = stack.pop()\\n        if leafs[v]:\\n            L.append(l*p)\\n        if not visited[v]:\\n            visited[v] = True\\n            for u in tree[v]:\\n                if not visited[u]:\\n                    stack.append((u, l+1, p/(len(tree[v])-1)))\\n\\n\\nn = int(input())\\ntree = {x: [] for x in range(1, n+1)}\\nL = []\\nfor _ in range(n-1):\\n    u, v = list(map(int, input().split()))\\n    tree[u].append(v)\\n    tree[v].append(u)\\nleafs = [False, True] + [len(tree[x]) == 1 for x in range(2, n+1)]\\ndfs(1)\\nprint(sum(L))\\n\\n\\\"\\\"\\\"\\n7\\n1 2\\n1 3\\n1 4\\n4 7\\n2 6\\n2 5\\n\\\"\\\"\\\"\\n\", \"n = int(input())\\n\\nd = { x : [] for x in range(1, n + 1)}#defaultdict(list)\\nfor x in range(n - 1):\\n    s,de = map(int, input().split())\\n    d[s].append(de)\\n    d[de].append(s)\\n\\ndef dfs():\\n    lst = [(0, 1.0, 1)]\\n    visited = set({1})\\n    ans = 0\\n\\n    while lst:\\n        depth,prob,source = lst.pop()\\n        \\n        for neigh in d[source]:\\n            if neigh not in visited:\\n                visited.add(neigh)\\n                lst.append((depth + 1, prob*(1/(len(d[source]) - (1 if depth != 0 else 0))), neigh))\\n        \\n        if depth != 0 and len(d[source]) == 1:\\n            ans += prob*depth\\n\\n    return ans\\n\\nprint(\\\"{:.8f}\\\".format(dfs()))\", \"n = int(input())\\n\\ntree = {x: [] for x in range(1, n+1)}\\n\\nfor _ in range(n-1):\\n    k,l = list(map(int, input().split()))\\n    tree[k].append(l)\\n    tree[l].append(k)\\n\\n#for i in range(99999):\\n#    tree[i+1].append(i+2)\\n#    tree[i+2].append(i+1)\\n\\nvisited = set()\\ns = 0\\n\\na = [(1,1,0)]\\n\\nwhile a:\\n    v,p,l = a.pop()\\n    visited.add(v) \\n    k = 0\\n    for vv in tree[v]:\\n        if vv not in visited:\\n            k += 1\\n    if k <= 0:\\n        s += p*l\\n    else:\\n        for vv in tree[v]:\\n            if vv not in visited:\\n                a.append((vv,p*1.0/k,l+1))\\n\\nprint(s)\", \"from collections import defaultdict\\nfrom collections import deque\\nfrom sys import stdin\\nlines = deque(line.strip() for line in stdin.readlines())\\n\\ndef nextline():\\n    return lines.popleft()\\n\\ndef types(cast):\\n    return tuple(int(x) for x in nextline().split())\\n\\ndef ints():\\n    return types(int)\\n\\ndef strs():\\n    return nextline().split()\\n\\ndef main():\\n    # lines will now contain all of the input's lines in a list\\n    n = int(nextline())\\n    graph = defaultdict(list)\\n    for _ in range(1, n):\\n        a, b = ints()\\n        graph[a].append(b)\\n        graph[b].append(a)\\n    stack = [(1, 1, 0)]\\n    visited = set()\\n    expected = 0\\n    while stack:\\n        city, denominator, length = stack.pop()\\n        visited.add(city)\\n        choices = [choice for choice in graph[city] if choice not in visited]\\n        if choices:\\n            for choice in choices:\\n                stack.append((choice, denominator * len(choices), length + 1))\\n        else:\\n            expected += length / denominator\\n    print(expected)\\n\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import queue\\nn = int(input())\\ndic = {}\\n\\nfor i in range(n+1):\\n    dic[i] = []\\nfor i in range(n-1):\\n    u,v = [ int(x) for x in input().split() ]\\n    dic[u].append(v)\\n    dic[v].append(u)\\n\\nweight = {1:(0,1)} # No.City:(length,weight)\\nq = queue.Queue()\\nq.put(1)\\nwhile(q.empty() == False):\\n    city = q.get()\\n    choice = len(dic[city])\\n    if city > 1:\\n        choice = choice - 1\\n    for to_city in dic[city]:\\n        if to_city not in weight:\\n            weight[to_city] = (weight[city][0]+1, weight[city][1]/choice)\\n            q.put(to_city)\\n    if(choice > 0):\\n        weight[city] = (weight[city][0], 0)\\n\\nsum = 0\\nfor city in weight:\\n    sum = sum + weight[city][0] * weight[city][1]\\n\\nprint(\\\"%.15f\\\" % sum)\\n\\n\", \"def __starting_point():\\n    n = int(input())\\n    \\n    edges = [[] for i in range(n+1)]\\n    edges[0] = None\\n    for i in range(n-1):\\n        n1,n2 = list(map(int, input().split()))\\n        edges[n1].append(n2)\\n        edges[n2].append(n1)\\n\\n    # stack = [(node, height, prob),...]\\n    stack = [(1,0,1.0)]\\n    sev = 0\\n    visited = [False for i in range(n+1)]\\n    \\n    while len(stack) > 0:\\n        (node, height, prob) = stack.pop()\\n        visited[node] = True\\n        children = len(edges[node]) - 1\\n        if node == 1:\\n            # since root has no parent\\n            children += 1\\n            \\n        for child in edges[node]:\\n            if visited[child]:\\n                continue\\n            stack.append((child, height+1, prob/children))\\n        \\n        if children == 0:\\n            sev += height * prob\\n    print(sev)\\n            \\n    \\n    \\n    \\n    \\n\\n\\n\\n__starting_point()\", \"class Node:\\n\\tdef __init__(self, index):\\n\\t\\tself.neighbours = []\\n\\t\\tself.index = index\\n\\t\\tself.prob = 1.0\\n\\t\\tself.vis = False\\n\\t\\tself.length = 0\\n\\n\\tdef addNode(self, city):\\n\\t\\tself.neighbours.append(city)\\n\\t\\tif self.index == 1:\\n\\t\\t\\tself.prob = 1.0 / len(self.neighbours)\\n\\t\\telse:\\n\\t\\t\\tl = len(self.neighbours)\\n\\t\\t\\tself.prob = 1.0  if l < 2 else (1.0 / (l - 1))\\n\\nn = int(input())\\nif n == 1:\\n\\tprint(0)\\n\\treturn\\n\\t\\ncities = {}\\n\\nfor i in range(n-1):\\n\\ta, b = [int(k) for k in input().split()]\\n\\t#print (\\\"test \\\", a, \\\" to \\\", b)\\n\\tif a not in cities:\\n\\t\\tcities[a] = Node(a)\\n\\tcities[a].addNode(b)\\n\\tif b not in cities:\\n\\t\\tcities[b] = Node(b)\\n\\tcities[b].addNode(a)\\n\\nif len(cities) == 2:\\n\\tprint(1)\\n\\treturn\\n\\n\\n# for i in range(1, n + 1, 1):\\n# \\tprint (\\\"city.index \\\", cities[i].index, ' roads ', cities[i].neighbours, ' prob ', cities[i].prob)\\n\\n\\n# deadends = []\\n# deadendsProb = []\\n\\n\\n# def Parse(city, prob, length, oldCity):\\n# \\t#print ('parse ', city.index)\\n# \\tnewprob = 1.0 if oldCity == 0  else cities[oldCity].prob\\n# \\t#print ('nnewProb ', newprob)\\n# \\tprob *= newprob\\n# \\t#print (city.index, ' len ', len(city.neighbours))\\n# \\tif len(city.neighbours) == 1 and oldCity != 0:\\n# \\t\\tdeadends.append(length)\\n# \\t\\tdeadendsProb.append(prob)\\n# \\telse:\\n# \\t\\t#print (city.neighbours)\\n# \\t\\tlength += 1\\n# \\t\\tfor c in city.neighbours:\\n# \\t\\t\\tif c != oldCity:\\n# \\t\\t\\t\\tParse(cities[c], prob, length, city.index )\\n\\n# Parse(cities[1], 1.0, 0, 0)\\n# #for i in range(len(deadends)):\\n# #\\tprint('len ', deadends[i], ' prob ', deadendsProb[i])\\n\\n# ans = sum(map(lambda l, p: l * p, deadends, deadendsProb))\\n# #print('ans', ans)\\n# print(ans)\\n\\n\\ndef inorder(city):\\n\\ts = []\\n\\ts.append(city)\\n\\t#print ('index ', city.index)\\n\\t#print ('neighbours ', city.neighbours)\\n\\twhile s:\\n\\t\\tcity = s.pop()\\n\\t\\tcity.vis = True\\n\\t\\tif city.neighbours:\\n\\t\\t\\tif city.index == 1 or len(city.neighbours) > 1:\\n\\t\\t\\t\\tfor c in city.neighbours:\\n\\t\\t\\t\\t\\tif not cities[c].vis:\\n\\t\\t\\t\\t\\t\\tcities[c].length = city.length + 1\\n\\t\\t\\t\\t\\t\\tcities[c].prob *= city.prob\\n\\t\\t\\t\\t\\t\\ts.append(cities[c])\\n\\t\\t\\telse:\\n\\t\\t\\t\\tyield (city.index, city.prob, city.length)\\n\\n\\n\\ntest = sum([city[1] * city[2] for city in inorder(cities[1])])\\nprint(test)\\n\\n\\n\\n# this is a tree\\n\", \"from sys import setrecursionlimit\\nimport threading\\n\\n\\nlength = 0\\nvisited = None\\ng = None\\n\\n\\ndef dfs(v, path_length, chance):\\n    nonlocal length\\n    visited.add(v)\\n    edges = 0\\n    for n in g[v]:\\n        if n not in visited:\\n            edges += 1\\n    for n in g[v]:\\n        if n not in visited:\\n            dfs(n, path_length + 1, chance * (1 / edges))\\n    if edges == 0:\\n        length += path_length * chance\\n\\n\\ndef main():\\n    nonlocal visited, g\\n    vert = int(input())\\n    g = [[] for _ in range(vert + 1)]\\n    visited = set()\\n    for _ in range(vert - 1):\\n        a, b = list(map(int, input().split()))\\n        g[a].append(b)\\n        g[b].append(a)\\n    dfs(1, 0, 1)\\n    print(length)\\n\\n\\ndef __starting_point():\\n    setrecursionlimit(10 ** 6)\\n    threading.stack_size(134217728)\\n    thread = threading.Thread(target=main)\\n    thread.start()\\n\\n__starting_point()\", \"n = int(input())\\ng = [[] for i in range(n+1)]\\nfor i in range(n-1):\\n    a,b = [int(x) for x in input().split()]\\n    g[a].append(b)\\n    g[b].append(a)\\n\\nd = [0 for i in range(n+1)]\\nv = [0 for i in range(n+1)]\\ne = [0 for i in range(n+1)]\\np = [0.0 for i in range(n+1)]\\np[1] = 1.0\\nen = 0\\nq = [1]\\nwhile len(q):\\n    c = q.pop(0)\\n    v[c] = 1\\n    a = 0\\n    for ne in g[c]:\\n        if not v[ne]:\\n            q.append(ne)\\n            d[ne] = d[c] + 1\\n            p[ne] = p[c]/len(g[c]) if c == 1 else p[c]/(len(g[c])-1)\\n            a += 1\\n    if a == 0:\\n        e[c] = 1\\n        en += 1\\nave = 0.0\\nfor i in range(n+1):\\n    if e[i]:\\n        ave += d[i]*p[i]\\nprint(ave)\\n\", \"n=int(input())\\nif n==1:\\n\\tprint(0)\\n\\treturn\\ngraph=[None]*n\\nfor i in range(n-1):\\n\\tu,v=map(int,input().split())\\n\\tu-=1\\n\\tv-=1\\n\\tif graph[u]==None:\\n\\t\\tgraph[u]=[v]\\n\\telse:\\n\\t\\tgraph[u].append(v)\\n\\tif graph[v]==None:\\n\\t\\tgraph[v]=[u]\\n\\telse:\\n\\t\\tgraph[v].append(u)\\n\\nis_checked=[False]*n\\nlens=[None]*n\\nlens[0]=0\\ndivs=[None]*n\\ndivs[0]=1\\nstack=[0]\\nans=0\\nwhile stack:\\n\\ttop=stack.pop()\\n\\tis_checked[top]=True\\n\\tvals=graph[top]\\n\\tfor i in range(len(vals)):\\n\\t\\tcur=vals[i]\\n\\t\\tif not is_checked[cur]:\\n\\t\\t\\tlens[cur]=lens[top]+1\\n\\t\\t\\tdivs[cur]=divs[top]*(len(vals)-(1 if top!=0 else 0))\\n\\t\\t\\tif len(graph[cur])==1:\\n\\t\\t\\t\\tans+=lens[cur]/divs[cur]\\n\\t\\t\\t\\tis_checked[cur]=None\\n\\t\\t\\t\\tlens[cur]=None\\n\\t\\t\\t\\tdivs[cur]=None\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\tstack.append(cur)\\nprint('{:.8f}'.format(ans))\"]",
        "difficulty": "interview",
        "input": "1\n",
        "output": "0.000000000000000\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/839/C"
    },
    {
        "id": 1051,
        "task_id": 2305,
        "test_case_id": 2,
        "question": "We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.\nAdditionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.\nFor each k=1, 2, ..., N, solve the following problem:\n - Find the number of simple paths that visit a vertex painted in the color k one or more times.\nNote: The simple paths from Vertex u to v and from v to u are not distinguished.\n\n-----Constraints-----\n - 1 \\leq N \\leq 2 \\times 10^5\n - 1 \\leq c_i \\leq N\n - 1 \\leq a_i,b_i \\leq N\n - The given graph is a tree.\n - All values in input are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN\nc_1 c_2 ... c_N\na_1 b_1\n:\na_{N-1} b_{N-1}\n\n-----Output-----\nPrint the answers for k = 1, 2, ..., N in order, each in its own line.\n\n-----Sample Input-----\n3\n1 2 1\n1 2\n2 3\n\n-----Sample Output-----\n5\n4\n0\n\nLet P_{i,j} denote the simple path connecting Vertex i and j.\nThere are 5 simple paths that visit a vertex painted in the color 1 one or more times:\nP_{1,1}\\,,\\,P_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,3}\\,,\\,P_{3,3} \nThere are 4 simple paths that visit a vertex painted in the color 2 one or more times:\nP_{1,2}\\,,\\,P_{1,3}\\,,\\,P_{2,2}\\,,\\,P_{2,3} \nThere are no simple paths that visit a vertex painted in the color 3 one or more times.",
        "solutions": "[\"import sys\\nfrom collections import defaultdict\\n\\n# \\u518d\\u5e30\\u5236\\u9650\\u3092\\u7de9\\u548c\\u3059\\u308b\\u304a\\u307e\\u3058\\u306a\\u3044\\nsys.setrecursionlimit(10**6)\\n\\ndef sum(n):return n*(n+1)//2\\n\\n# \\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3068\\u3001\\u300c\\u8272i\\u3092\\u5c01\\u9396\\u3057\\u305f\\u3068\\u304d\\u306b\\u5230\\u9054\\u3067\\u304d\\u306a\\u3044\\u9802\\u70b9\\u306e\\u500b\\u6570\\u300d\\u3092\\u6301\\u3064\\u8f9e\\u66f8\\u3092\\u8fd4\\u3059\\ndef dfs(v,p):\\n    ret=defaultdict(int)\\n    size=1\\n    for vv in g[v]:\\n        if vv==p:\\n            continue\\n        ss,d=dfs(vv,v)\\n        size+=ss\\n        ans[c[v]]+=sum(ss-d[c[v]])\\n        \\n        # \\u30de\\u30fc\\u30b8\\u30c6\\u30af\\n        if len(ret)<len(d):\\n            ret,d=d,ret\\n        for vvv in d:\\n            ret[vvv]+=d[vvv]\\n    ret[c[v]]=size\\n    return size,ret\\n\\nn=int(input())\\nc=list(map(int,input().split()))\\ng=[[] for _ in range(n+1)]\\nans=[0]*(n+1)\\nfor _ in range(n-1):\\n    s,t=list(map(int, input().split()))\\n    s-=1\\n    t-=1\\n    g[s].append(t)\\n    g[t].append(s)\\n_,ret=dfs(0,-1)\\nfor i in range(1,n+1):\\n    print((sum(n)-ans[i]-sum(n-ret[i])))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef input():\\n\\treturn sys.stdin.readline()[:-1]\\n\\nn = int(input())\\nc = list(map(int, input().split()))\\nans = [n*(n+1)//2 for _ in range(n)]\\nfor i in range(n):\\n\\tc[i] -= 1\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n\\ta, b = map(int, input().split())\\n\\tadj[a-1].append(b-1)\\n\\tadj[b-1].append(a-1)\\n\\ns = 0\\ndp = [0 for _ in range(n)]\\n\\ndef dfs(x, p):\\n\\tnonlocal s\\n\\tpres = s + dp[c[x]]\\n\\tfor v in adj[x]:\\n\\t\\tif v == p:\\n\\t\\t\\tcontinue\\n\\t\\tdp[c[x]] = -s\\n\\t\\tdfs(v, x)\\n\\t\\tans[c[x]] -= (s+dp[c[x]]) * (s+dp[c[x]]+1) // 2\\n\\ts += 1\\n\\tdp[c[x]] = pres - s\\n\\treturn\\n\\ndfs(0, -1)\\nfor i in range(n):\\n\\tans[i] -= (s+dp[i]) * (s+dp[i]+1) // 2\\n\\nprint(*ans, sep=\\\"\\\\n\\\")\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append(t)\\n        self.E[t].append(f)\\n\\n\\ndef make_order(g, v):\\n    seen = [False]*g.N\\n    last_order = [-1]*g.N\\n    parent = [-1]*g.N\\n    size = [1]*g.N\\n    counter = {\\\"last\\\": 0}\\n\\n    def recur(v):\\n        seen[v] = True\\n        for to in g.E[v]:\\n            if seen[to] == True:\\n                continue\\n            parent[to] = v\\n            recur(to)\\n            size[v] += size[to]\\n\\n        last_order[counter[\\\"last\\\"]] = v\\n        counter[\\\"last\\\"] += 1\\n\\n    recur(v)\\n    return last_order, parent, size\\n\\n\\ndef merge(d: dict, e: dict):\\n    if len(d) < len(e):\\n        d, e = e, d\\n    for k in e:\\n        d[k] += e[k]\\n    return d\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\nans = [0]*N\\n\\nret = {}\\nlast_order, parent, size = make_order(g, 0)\\nfor curr in last_order:\\n    cn = c[curr]\\n    rrr = defaultdict(int)\\n    for dest in g.E[curr]:\\n        if dest == parent[curr]:\\n            continue\\n        child = ret.pop(dest)\\n\\n        n = size[dest]-child[cn]\\n        ans[cn] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        rrr = merge(rrr, child)\\n\\n    rrr[cn] = size[curr]\\n    ret[curr] = rrr\\n\\n\\ntot = N*(N+1)//2\\nfor color in range(N):\\n    if color != c[0]:\\n        n = N-ret[0][color]\\n        ans[color] += n*(n+1)//2\\n    print((tot-ans[color]))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\n \\nN = int(input())  # \\u9802\\u70b9\\u6570\\nC = [int(x) - 1  for x in input().split()]  # \\u9802\\u70b9\\u306e\\u8272\\u756a\\u53f7 (1~N) -> (0~N-1)\\n \\nE = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    E[a-1].append(b-1)\\n    E[b-1].append(a-1)\\n    \\ndef dfs(u, p = -1):\\n \\n    color = C[u]  # \\u8272\\u756a\\u53f7\\n    VC[color].append(u)  # \\u8272\\u3054\\u3068\\u306b stack \\u3057\\u3066\\u304a\\u304f\\n    \\n    s = 1  # \\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\uff09\\n    for v in E[u]:\\n        if v == p:\\n            continue\\n            \\n        child[u] = 0  # v \\u306e\\u65b9\\u5411\\u306b\\u3042\\u308b\\u5b50\\u5b6b\\u306e\\u3046\\u3061\\u8fbf\\u308c\\u308b\\u9802\\u70b9\\u306e\\u6570 = s - \\u5b50\\u5b6b\\u306b\\u3042\\u308b\\u3001\\u540c\\u8272\\u3092\\u6839\\u3068\\u3059\\u308b\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u306e\\u5408\\u8a08\\n        ret = dfs(v, u)\\n \\n        s += ret  # \\u7d14\\u7c8b\\u306a\\u90e8\\u5206\\u6728\\u306e\\u30b5\\u30a4\\u30ba\\u3092\\u8a08\\u7b97\\n        child[u] += ret  # \\u540c\\u8272\\u306e\\u9802\\u70b9\\u304c\\u5207\\u308a\\u96e2\\u3055\\u308c\\u305f\\u3042\\u3068\\u306e\\u90e8\\u5206\\u6728\\u30b5\\u30a4\\u30ba\\n        \\n        ans[color] -= child[u] * (child[u] + 1) // 2        \\n \\n    VC[color].pop()\\n    if VC[color]: # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u3042\\u308b\\u5834\\u5408\\n        child[VC[color][-1]] -= s  # \\u4e00\\u756a\\u8fd1\\u3044\\u540c\\u8272\\u304b\\u3089\\u3001\\u81ea\\u5206\\u306e\\u90e8\\u5206\\u6728\\u306e\\u5927\\u304d\\u3055\\u3092\\u3072\\u304f\\n    else:  # \\u4e0a\\u4f4d\\u306b\\u540c\\u3058\\u8272\\u304c\\u306a\\u3044\\u5834\\u5408\\n        root_size[color] -= s  # \\u4e0a\\u4f4d\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\n        \\n    return s\\n\\nchild = [0] * N  # \\u8fbf\\u3063\\u3066\\u3044\\u308b\\u5b50\\u306e\\u4e2d\\u3067\\u306e\\u306e\\u6b8b\\u308a\\u6570\\u3000\\u3068\\u308a\\u3042\\u3048\\u305a 0\\u3067\\u521d\\u671f\\u5316\\nroot_size = [N] * N  # \\u6839\\u65b9\\u5411\\u306b\\u6b8b\\u3063\\u3066\\u3044\\u308b\\u9802\\u70b9\\u306e\\u6570\\uff08\\u3092\\u683c\\u7d0d\\u3059\\u308b\\u3002\\u64cd\\u4f5c\\u524d\\u306f N \\u5168\\u90e8\\uff09\\nVC = [ [] for _ in range(N)]  # \\u8272\\u756a\\u53f7\\u3054\\u3068\\nans = [N*(N+1)//2] * N  # \\u8272\\u3054\\u3068\\u306e\\uff08\\u6b8b\\u308a\\u306e\\uff09\\u7d44\\u307f\\u5408\\u308f\\u305b\\u6570\\u3000\\u521d\\u671f\\u5024\\u306f \\u5168\\u70b9\\u306e\\u7d44\\u307f\\u5408\\u308f\\u305b\\n \\ndfs(0)\\n \\nfor i in range(N):\\n    print(ans[i] - root_size[i] * (root_size[i] + 1) // 2)\", \"import sys\\n\\ninput=sys.stdin.readline\\nsys.setrecursionlimit(10**7)\\n\\nN=int(input())\\nc=list(map(int,input().split()))\\nfor i in range(N):\\n    c[i]-=1\\nedge=[[] for i in range(N)]\\nfor i in range(N-1):\\n    a,b=map(int,input().split())\\n    edge[a-1].append(b-1)\\n    edge[b-1].append(a-1)\\n\\nsubtree=[1]*N\\ndef size(v,pv):\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            size(nv,v)\\n            subtree[v]+=subtree[nv]\\n\\nsize(0,-1)\\n\\ndata=[[] for i in range(N)]\\nParent=[[0] for i in range(N)]\\nparent=[0]*N\\ndef dfs(v,pv,nop):\\n    pp=parent[nop]\\n    parent[nop]=v\\n    Parent[nop].append(v)\\n    data[c[v]].append((parent[c[v]],v))\\n    for nv in edge[v]:\\n        if nv!=pv:\\n            dfs(nv,v,c[v])\\n    parent[nop]=pp\\n\\ndfs(0,-1,0)\\n\\nfor i in range(N):\\n    dic={v:subtree[v] for v in Parent[i]}\\n    for p,ch in data[i]:\\n        dic[p]-=subtree[ch]\\n\\n    res=N*(N+1)//2\\n    for p in dic:\\n        res-=dic[p]*(dic[p]+1)//2\\n    print(res)\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nhead = [-1] * (N + 1)\\nto = [0] * (N - 1 << 1)\\nnxt = [0] * (N - 1 << 1)\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    x, y = x-1, y-1\\n    nxt[i] = head[x]\\n    to[i] = y\\n    head[x] = i\\n    j = i + N - 1\\n    nxt[j] = head[y]\\n    to[j] = x\\n    head[y] = j\\n\\ndef EulerTour(n, i=0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    ANS = [f(n)] * n\\n    \\n    P = [-1] * n\\n    ct = 0\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    while i >= 0:\\n        e = head[i]\\n        if e < 0:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = to[e]\\n        if P[i] == j:\\n            head[i] = nxt[e]\\n            continue\\n        \\n        P[j] = i\\n        head[i] = nxt[e]\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    done = [0] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            continue\\n        done[i] = 1\\n        if i: ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n        for a in X[i]:\\n            if done[a]: continue\\n            P[a] = i\\n            Q.append(~a)\\n            Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\nn = int(input())\\nc = list(map(lambda x: int(x) - 1, input().split()))\\n\\ngraph = [[] for _ in range(n)]\\nfor _ in range(n - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    graph[a].append(b)\\n    graph[b].append(a)\\n\\nans = [n * (n + 1) // 2] * n\\nn_subtree = [1] * n  # n_st[i] = (\\u9802\\u70b9i\\u306e\\u90e8\\u5206\\u6728\\u306e\\u9802\\u70b9\\u6570)\\ncnt = [0] * n  # cnt[i] = (\\u8a2a\\u554f\\u6e08\\u307f\\u9802\\u70b9\\u306e\\u3046\\u3061\\u3001\\u8272i\\u3092\\u901a\\u904e\\u3057\\u306a\\u3044\\u3068\\u305f\\u3069\\u308a\\u7740\\u3051\\u306a\\u3044\\u9802\\u70b9\\u6570)\\nn_visited = 0\\n\\n\\ndef dfs(v, v_p):\\n    nonlocal n_visited\\n    cnt_v_before = cnt[c[v]]\\n    for v_next in graph[v]:\\n        if v_next == v_p:\\n            continue\\n\\n        m_prev = n_visited - cnt[c[v]]\\n        dfs(v_next, v)\\n        n_subtree[v] += n_subtree[v_next]\\n        m_next = n_visited - cnt[c[v]]\\n\\n        m = m_next - m_prev\\n        ans[c[v]] -= m * (m + 1) // 2\\n\\n    cnt[c[v]] = cnt_v_before + n_subtree[v]\\n    n_visited += 1\\n\\n\\ndfs(0, -1)\\n\\nfor i in range(n):\\n    m = n - cnt[i]\\n    ans[i] -= m * (m + 1) // 2\\n\\nprint(*ans, sep='\\\\n')\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\n# ci = [set() for _ in range(n)]\\n# for i,c in enumerate(C):\\n#     ci[c-1].add(i)\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = list(map(int,input().split()))\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n\\nans = [n*(n+1)//2]*n\\ncparent = [[] for _ in range(n)]\\nroot_size = [n]*n\\n# index:color\\nreached = [False]*n\\nsize = [0]*n\\n# index:vertex\\n\\ndef dfs(p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        if reached[nxt]:continue\\n        reached[nxt] = True\\n        size[p] = 0\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] += ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    # size[p] += s\\n    if cparent[c]:\\n        size[cparent[c][-1]] -= s\\n    else:\\n        root_size[c] -= s\\n    # ans[c] -= size[p] * (size[p] - 1)//2\\n    return s\\n\\nreached = [False]*n\\nreached[0] = True\\n\\ndfs(0)\\nfor i in range(n):\\n    print((ans[i] - root_size[i]*(root_size[i]+1)//2))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if ind == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if ind >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind]\\n        if P[i] == j:\\n            IND[i] += 1\\n            continue\\n        P[j] = i\\n        IND[i] += 1\\n        i = j\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nC = list(map(int,input().split()))\\n\\npath = [set() for _ in range(n)]\\n\\nfor i in range(n-1):\\n    a,b = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    path[a].add(b)\\n    path[b].add(a)\\n    \\nans = [n*(n+1)//2]*n\\nsize = [0]*n\\ncparent = [[] for _ in range(n)]\\nreached = [False]*n\\nroot_size = [n]*n\\n\\ndef dfs (p):\\n    c = C[p] - 1\\n    cparent[c].append(p)\\n    s = 1\\n    for nxt in path[p]:\\n        #print (nxt)\\n        \\n        if reached[nxt]:\\n            continue\\n        size[p] = 0\\n        reached[nxt] = True\\n        ret = dfs(nxt)\\n        s += ret\\n        size[p] +=ret\\n        ans[c] -= size[p] * (size[p] + 1)//2\\n    cparent[c].pop()\\n    if cparent[c]:\\n        size[cparent[c][-1]] -=s\\n    else:\\n        root_size[c] -=s    \\n    return s\\n\\nreached[0] = True\\ndfs (0)\\n\\nfor i in range(n):\\n    print (ans[i]-root_size[i]*(root_size[i]+1)//2)\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef main():\\n    n=int(input())\\n    color=list(map(int,input().split()))\\n    nodes=[[]for i in range(n)]\\n    cut=[[0]for i in range(n+1)]\\n    ans=[0]*(n+1)\\n    for i in range(n-1):\\n        a,b=map(int,input().split())\\n        nodes[a-1].append(b-1)\\n        nodes[b-1].append(a-1)\\n    def num(p,parent):\\n        s=0\\n        for child in nodes[p]:\\n            if child==parent:\\n                continue\\n            cut[color[p]].append(0)\\n            nc=num(child,p)\\n            group=nc-cut[color[p]].pop()\\n            ans[color[p]]+=group*(group+1)//2\\n            s+=nc\\n        s+=1\\n        cut[color[p]][-1]+=s\\n        return s\\n    num(0,-1)\\n    for a,c in zip(ans[1:],cut[1:]):\\n        group=n-c[0]\\n        c=group*(group+1)//2\\n        print(n*(n+1)//2-c-a)\\nmain()\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    Q = [~i0, i0]\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    while Q:\\n        i = Q.pop()\\n        if i < 0:\\n            i = ~i\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                p = P[i]\\n                k = ET2[i] - ET1[i] + 1 - USED[C[p]] + ORG[i]\\n                ANS[C[p]] -= f(k)\\n                TMP[p] += k\\n            continue\\n        if i >= 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            if ET1[i] == 0: ET1[i] = ct\\n        for a in X[i][::-1]:\\n            if a != P[i]:\\n                P[a] = i\\n                for k in range(len(X[a])):\\n                    if X[a][k] == i:\\n                        del X[a][k]\\n                        break\\n                Q.append(~a)\\n                Q.append(a)\\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10**9)\\n\\ndef solve():\\n    N = int(input())\\n    Cs = [A-1 for A in map(int, input().split())]\\n    adjL = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = list(map(int, input().split()))\\n        a, b = a-1, b-1\\n        adjL[a].append(b)\\n        adjL[b].append(a)\\n\\n    sizeSubtrees = [1] * N\\n    numVs = [0] * N\\n    anss = [0] * N\\n\\n    def dfs(vNow, vPar, clrPar):\\n        clrNow = Cs[vNow]\\n        numVNow = numVs[clrNow]\\n        numVPar = numVs[clrPar]\\n        for v2 in adjL[vNow]:\\n            if v2 == vPar: continue\\n            dfs(v2, vNow, clrNow)\\n            sizeSubtrees[vNow] += sizeSubtrees[v2]\\n        numVs[clrNow] = numVNow + sizeSubtrees[vNow]\\n        if clrPar != -1:\\n            k = sizeSubtrees[vNow] - (numVs[clrPar]-numVPar)\\n            anss[clrPar] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for clr in range(N):\\n        if clr == Cs[0]: continue\\n        k = N - numVs[clr]\\n        anss[clr] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    anss = [numAll-ans for ans in anss]\\n\\n    print(('\\\\n'.join(map(str, anss))))\\n\\n\\nsolve()\\n\", \"import sys\\nimport itertools\\n\\nsys.setrecursionlimit(3 * 10**5)\\n\\n\\nclass Node:\\n  def __init__(self, color):\\n    self.color = int(color) - 1\\n    self.children = []\\n\\n  def add_child(self, node):\\n    self.children.append(node)\\n\\n  @staticmethod\\n  def prepare_nodes(n):\\n    nodes = tuple(map(Node, input().split()))\\n    for _ in range(n - 1):\\n      a, b = [int(x) - 1 for x in input().split()]\\n      nodes[a].add_child(nodes[b])\\n      nodes[b].add_child(nodes[a])\\n\\n    return nodes\\n\\n\\ndef main():\\n  n = int(input())\\n  nodes = Node.prepare_nodes(n)\\n\\n  dp = [0 for _ in range(n)]\\n  ans_list = [n*(n + 1) // 2 for _ in range(n)]\\n\\n  def update_ans(s, color):\\n    ans_list[color] -= (s + dp[color]) * (s + dp[color] + 1) // 2\\n\\n  def dfs(s, node, parent):\\n    color = node.color\\n    pre_s = s + dp[color]\\n    for child in node.children:\\n      if child == parent: continue\\n\\n      dp[color] = -s\\n      s = dfs(s, child, node)\\n      update_ans(s, color)\\n\\n    s += 1\\n    dp[color] = pre_s - s\\n\\n    return s\\n\\n  s = dfs(0, nodes[0], Node(-1))\\n\\n  for i in range(n):\\n    update_ans(s, i)\\n    print((ans_list[i]))\\n\\n\\nmain()\\n\", \"def main():\\n    import sys\\n    sys.setrecursionlimit(10**9)\\n    input = sys.stdin.readline\\n\\n    N = int(input())\\n    C = [c-1 for c in map(int, input().split())]\\n    tree = [[] for _ in range(N)]\\n    for _ in range(N-1):\\n        a, b = map(int, input().split())\\n        tree[a-1].append(b-1)\\n        tree[b-1].append(a-1)\\n\\n    # 1\\u56de\\u4ee5\\u4e0a > \\u5168\\u4f53 - 1\\u56de\\u3082\\u901a\\u3089\\u306a\\u3044\\n    # \\u9078\\u3079\\u308b\\u6570 = n(n+1)//2\\n    size_subtree = [1] * N\\n    num_v = [0] * N\\n    ans = [0] * N\\n\\n    def dfs(v_now, v_parent, color_parent):\\n        color_now = C[v_now]\\n        num_v_now = num_v[color_now]\\n        num_v_parent = num_v[color_parent]\\n        for v2 in tree[v_now]:\\n            if v2 == v_parent: continue\\n            dfs(v2, v_now, color_now)\\n            size_subtree[v_now] += size_subtree[v2]\\n        num_v[color_now] = num_v_now + size_subtree[v_now]\\n        if color_parent != -1:\\n            k = size_subtree[v_now] - (num_v[color_parent]-num_v_parent)\\n            ans[color_parent] += k*(k+1)//2\\n\\n    dfs(0, -1, -1)\\n\\n    for color in range(N):\\n        if color == C[0]: continue\\n        k = N - num_v[color]\\n        ans[color] += k*(k+1)//2\\n\\n    numAll = N*(N+1)//2\\n    ans = [numAll-ans for ans in ans]\\n\\n    print(*ans, sep='\\\\n')\\n\\nmain()\\n\", \"#!/usr/bin/env python3\\nimport sys\\nsys.setrecursionlimit(10**8)\\nfrom collections import defaultdict\\nINF = float(\\\"inf\\\")\\nMOD = 10**9+7\\n\\n\\nclass Graph(object):\\n    def __init__(self, N):\\n        self.N = N\\n        self.E = defaultdict(list)\\n\\n    def add_edge(self, f, t, w=1):\\n        self.E[f].append((t, w))\\n        self.E[t].append((f, w))\\n\\n\\nN = int(input())\\nc = [x-1 for x in map(int, input().split())]\\nA = [None]*(N-1)\\nB = [None]*(N-1)\\nfor i in range(N-1):\\n    A[i], B[i] = list(map(int, input().split()))\\n\\ng = Graph(N)\\nfor a, b in zip(A, B):\\n    g.add_edge(a-1, b-1)\\n\\n# k=1, 2, ..., N\\u306b\\u5bfe\\u3057\\u3066\\n# \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u4ee5\\u4e0a\\u901a\\u308b\\u3088\\u3046\\u306a\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\u3092\\u6c42\\u3081\\u308b\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570 - \\u8272k\\u304c\\u5857\\u3089\\u308c\\u3066\\u3044\\u308b\\u9802\\u70b9\\u3092\\u4e00\\u5ea6\\u3082\\u901a\\u3089\\u306a\\u3044\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u6570\\n# \\u5168\\u30d1\\u30b9\\u306e\\u500b\\u6570\\u306fN*(N+1)/2\\n\\n# \\u30b0\\u30e9\\u30d5\\u3092\\u8272k\\u306e\\u30ce\\u30fc\\u30c9\\u3067\\u5206\\u5272\\u3057\\u3066\\u3001\\u90e8\\u5206\\u30b0\\u30e9\\u30d5\\u5185\\u3067\\u306e\\u5358\\u7d14\\u30d1\\u30b9\\u306e\\u7dcf\\u6570\\u3092\\u6c42\\u3081\\u308c\\u3070\\u826f\\u3044\\n# \\u5404\\u30ce\\u30fc\\u30c9\\u306b\\u72b6\\u614b\\u3068\\u3057\\u3066\\u8f9e\\u66f8\\u3092\\u3082\\u305f\\u305b\\u308b\\u3002\\n# x\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n# o\\u8f9e\\u66f8\\u306f\\u8272i\\u3092\\u901a\\u3089\\u305a\\u306b\\u5230\\u9054\\u4e0d\\u53ef\\u80fd\\u306a\\u9802\\u70b9\\u306e\\u6570\\u3092\\u6301\\u3064\\u3002\\n\\n# \\u56de\\u7b54\\u7528\\nans = [0]*N\\n\\n\\ndef f(curr, par=-1):\\n    # \\u518d\\u5e30\\u95a2\\u6570\\n    # curr: \\u73fe\\u5728\\u306e\\u7bc0\\u70b9\\n    # par : \\u89aa\\u7bc0\\u70b9\\u306e\\u756a\\u53f7\\n    ret = defaultdict(int)\\n    size = 1\\n    for dest, w in g.E[curr]:\\n        if dest == par:\\n            continue\\n        sz, child = f(dest, curr)\\n        size += sz\\n\\n        # \\u81ea\\u8eab\\u306e\\u8272\\u3068\\u540c\\u3058\\u5834\\u5408\\u3001\\u5b50\\u306e\\u9802\\u70b9\\u306e\\u6570\\u304b\\u3089\\u52a0\\u7b97\\n        n = sz-child[c[curr]]\\n        ans[c[curr]] += n*(n+1)//2\\n\\n        # \\u30de\\u30fc\\u30b8\\n        if len(ret) < len(child):\\n            child, ret = ret, child\\n        for key in child:\\n            ret[key] += child[key]\\n\\n    ret[c[curr]] = size\\n    return size, ret\\n\\n\\nsz, ret = f(0)\\nfor color in range(N):\\n    if color != c[0]:\\n        n = sz-ret[color]\\n        ans[color] += n*(n+1)//2\\n\\ntot = N*(N+1)//2\\nfor a in ans:\\n    print((tot-a))\\n\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [0] * n\\n    i = i0\\n    while i >= 0:\\n        if IND[i] == 0:\\n            if i: ORG[i] = USED[C[P[i]]]\\n            ct += 1\\n            ET1[i] = ct\\n        \\n        if IND[i] >= len(X[i]):\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        if P[i] == X[i][IND[i]]:\\n            IND[i] += 1\\n            continue\\n        P[X[i][IND[i]]] = i\\n        IND[i], i = IND[i] + 1, X[i][IND[i]]\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\ninput = lambda: sys.stdin.readline().rstrip()\\nN = int(input())\\nC = [int(a) - 1 for a in input().split()]\\nX = [[] for i in range(N)]\\nfor i in range(N-1):\\n    x, y = map(int, input().split())\\n    X[x-1].append(y-1)\\n    X[y-1].append(x-1)\\n\\ndef EulerTour(n, X, i0):\\n    f = lambda k: k * (k + 1) // 2\\n    USED = [0] * n\\n    ORG = [0] * n\\n    TMP = [0] * n\\n    P = [-1] * n\\n    ct = -1\\n    ET1 = [0] * n\\n    ET2 = [0] * n\\n    ANS = [f(n)] * n\\n    IND = [len(x) for x in X]\\n    i = i0\\n    while i >= 0:\\n        ind = IND[i]\\n        if not ind:\\n            ET2[i] = ct\\n            USED[C[i]] += 1 + TMP[i]\\n            if i:\\n                k = ET2[i] - ET1[i] + 1 - USED[C[P[i]]] + ORG[i]\\n                ANS[C[P[i]]] -= f(k)\\n                TMP[P[i]] += k\\n            i = P[i]\\n            continue\\n        \\n        j = X[i][ind - 1]\\n        if P[i] == j:\\n            IND[i] -= 1\\n            continue\\n        \\n        P[j] = i\\n        IND[i] -= 1\\n        i = j\\n        ORG[i] = USED[C[P[i]]]\\n        ct += 1\\n        ET1[i] = ct\\n    \\n    for i in range(n):\\n        ANS[i] -= f(n - USED[i])\\n    return ANS\\n\\nprint(*EulerTour(N, X, 0), sep = \\\"\\\\n\\\")\", \"import sys\\nsys.setrecursionlimit(10**6)\\nstdin = sys.stdin\\n\\nns = lambda: stdin.readline().rstrip()\\nni = lambda: int(stdin.readline().rstrip())\\nnm = lambda: map(int, stdin.readline().split())\\nnl = lambda: list(map(int, stdin.readline().split()))\\n\\nn = ni()\\ncolor = nl()\\ng = [list() for _ in range(n)]\\nfor _ in range(n-1):\\n    a, b = nm()\\n    g[a-1].append(b-1)\\n    g[b-1].append(a-1)\\n\\nsize = [1]*n\\npar = [-1]*n\\nh = [list() for _ in range(n+1)]\\nidx = [0]*n\\ndef dfs(v, p):\\n    h[color[v]].append(v)\\n    for x in g[v]:\\n        if x == p: continue\\n        par[x] = v\\n        size[v] += dfs(x, v)\\n        h[color[v]].append(v)\\n    h[color[v]].append(v)\\n    return size[v]\\ndfs(0, -1)\\n# print(size)\\nfor v in range(n):\\n    if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n        idx[v] += 1\\nacc = [0]*(n+10)\\nans = [n*(n+1)//2]*(n+1)\\nfor col in range(1, n+1):\\n    # print(h[col])\\n    acc[-1] = 0\\n    p = [-1]\\n    for v in h[col]:\\n        if p[-1] == v:\\n            p.pop()\\n            if idx[v] >= len(g[v]):\\n                acc[p[-1]] += size[v]\\n                continue\\n            q = size[g[v][idx[v]]] - acc[v]\\n            idx[v] += 1\\n            if idx[v] < len(g[v]) and g[v][idx[v]] == par[v]:\\n              idx[v] += 1\\n            ans[col] -= q*(q+1)//2\\n            acc[v] = 0\\n            # print(v, q)\\n            p.append(v)\\n        else:\\n            p.append(v)\\n    q = n - acc[-1]\\n    # print(-1, q)\\n    ans[col] -= q*(q+1)//2\\nprint(*ans[1:], sep='\\\\n')\"]",
        "difficulty": "interview",
        "input": "1\n1\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc163/tasks/abc163_f"
    },
    {
        "id": 1052,
        "task_id": 2346,
        "test_case_id": 1,
        "question": "You are given a rooted tree with vertices numerated from $1$ to $n$. A tree is a connected graph without cycles. A rooted tree has a special vertex named root.\n\nAncestors of the vertex $i$ are all vertices on the path from the root to the vertex $i$, except the vertex $i$ itself. The parent of the vertex $i$ is the nearest to the vertex $i$ ancestor of $i$. Each vertex is a child of its parent. In the given tree the parent of the vertex $i$ is the vertex $p_i$. For the root, the value $p_i$ is $-1$.\n\n [Image] An example of a tree with $n=8$, the root is vertex $5$. The parent of the vertex $2$ is vertex $3$, the parent of the vertex $1$ is vertex $5$. The ancestors of the vertex $6$ are vertices $4$ and $5$, the ancestors of the vertex $7$ are vertices $8$, $3$ and $5$ \n\nYou noticed that some vertices do not respect others. In particular, if $c_i = 1$, then the vertex $i$ does not respect any of its ancestors, and if $c_i = 0$, it respects all of them.\n\nYou decided to delete vertices from the tree one by one. On each step you select such a non-root vertex that it does not respect its parent and none of its children respects it. If there are several such vertices, you select the one with the smallest number. When you delete this vertex $v$, all children of $v$ become connected with the parent of $v$.\n\n [Image] An example of deletion of the vertex $7$. \n\nOnce there are no vertices matching the criteria for deletion, you stop the process. Print the order in which you will delete the vertices. Note that this order is unique.\n\n\n-----Input-----\n\nThe first line contains a single integer $n$ ($1 \\le n \\le 10^5$) — the number of vertices in the tree.\n\nThe next $n$ lines describe the tree: the $i$-th line contains two integers $p_i$ and $c_i$ ($1 \\le p_i \\le n$, $0 \\le c_i \\le 1$), where $p_i$ is the parent of the vertex $i$, and $c_i = 0$, if the vertex $i$ respects its parents, and $c_i = 1$, if the vertex $i$ does not respect any of its parents. The root of the tree has $-1$ instead of the parent index, also, $c_i=0$ for the root. It is guaranteed that the values $p_i$ define a rooted tree with $n$ vertices.\n\n\n-----Output-----\n\nIn case there is at least one vertex to delete, print the only line containing the indices of the vertices you will delete in the order you delete them. Otherwise print a single integer $-1$.\n\n\n-----Examples-----\nInput\n5\n3 1\n1 1\n-1 0\n2 1\n3 0\n\nOutput\n1 2 4 \n\nInput\n5\n-1 0\n1 1\n1 1\n2 0\n3 0\n\nOutput\n-1\n\nInput\n8\n2 1\n-1 0\n1 0\n1 1\n1 1\n4 0\n5 1\n7 0\n\nOutput\n5 \n\n\n\n-----Note-----\n\nThe deletion process in the first example is as follows (see the picture below, the vertices with $c_i=1$ are in yellow):\n\n  first you will delete the vertex $1$, because it does not respect ancestors and all its children (the vertex $2$) do not respect it, and $1$ is the smallest index among such vertices;  the vertex $2$ will be connected with the vertex $3$ after deletion;  then you will delete the vertex $2$, because it does not respect ancestors and all its children (the only vertex $4$) do not respect it;  the vertex $4$ will be connected with the vertex $3$;  then you will delete the vertex $4$, because it does not respect ancestors and all its children (there are none) do not respect it (vacuous truth);  you will just delete the vertex $4$;  there are no more vertices to delete. \n\n [Image] \n\nIn the second example you don't need to delete any vertex:\n\n  vertices $2$ and $3$ have children that respect them;  vertices $4$ and $5$ respect ancestors. \n\n [Image] \n\nIn the third example the tree will change this way:\n\n [Image]",
        "solutions": "[\"n = int(input())\\nparent = [-1]*n\\nisresp = [1]*n\\nfor i in range(n):\\n\\tp, r = list(map(int, input().split()))\\n\\tp -= 1\\n\\tif r == 0:\\n\\t\\tisresp[i] = 0\\n\\t\\tif p>=0:\\n\\t\\t\\tisresp[p] = 0\\nnore = []\\nfor i in range(n):\\n\\tif isresp[i] == 1:\\n\\t\\tnore.append(i+1)\\nif not nore:\\n\\tprint(-1)\\nelse:\\n\\tprint(\\\" \\\".join(map(str, nore)))\\n\", \"n = int(input())\\n\\ntree  = [[] for _ in range(n)]\\ninf = [0] * n\\n\\nfor i in range(n):\\n    ind, f = map(int, input().split())\\n    inf[i] = f\\n    if ind != -1:\\n        tree[ind - 1].append(i)\\n\\nflag = False\\n\\nfor i in range(n):\\n    if inf[i] == 1:\\n        for x in tree[i]:\\n            if inf[x] != 1:\\n                break\\n        else:\\n            flag = True\\n            print(i + 1, end=' ')\\n\\nif not flag:\\n    print(-1)\", \"import sys\\n\\nn = int(sys.stdin.readline().strip())\\nR = [0] * n\\n\\nfor i in range (0, n):\\n    line = sys.stdin.readline().strip().split()\\n    p = int(line[0])\\n    c = int(line[1])\\n    if c == 0:\\n        R[i] = 1\\n        if p != -1:\\n            R[p - 1] = 1\\n\\nans = [0] * (n - sum(R))\\nj = 0\\nfor i in range (0, n):\\n    if R[i] == 0:\\n        ans[j] = str(i + 1)\\n        j = j + 1\\nans = \\\" \\\".join(ans)\\n\\nif ans == \\\"\\\":\\n    print(-1)\\nelse:\\n    print(ans)\", \"a=int(input())\\nA = [0]*(a+2)\\nB = [0]*a\\nfor i in range(a):\\n    q,w = map(int,input().split())\\n    A[i]+=1\\n    A[q-1]+=1\\n    if w == 1:\\n        B[q-1]+=1\\n        B[i]+=1\\nC = []\\nfor i in range(a):\\n    if A[i] == B[i]:\\n        C.append(i+1)\\nif len(C)==0:\\n    print(-1)\\nelse:\\n    print(*C)\", \"n = int(input())\\nlis = []\\nfor _ in range(n):\\n    lis.append(list(map(int, input().split())))\\n    \\nx = list([0]*n)\\np = list([0]*n)\\n\\nres = []\\nfor i in lis:\\n    if i[0] != -1:\\n        p[i[0]-1] += (i[-1])\\n        x[i[0]-1] += 1    \\nfor i in range(n):\\n    if lis[i][-1] == 1:\\n        if x[i] == p[i]:\\n            res.append(i+1)\\n\\nif len(res) == 0:\\n    print(-1)\\nelse:\\n    print(\\\" \\\".join(map(str, sorted(res))))\\n\", \"from collections import defaultdict\\n\\n\\nn = int(input())\\ng = defaultdict(list)\\ncs = [None] * n\\n\\nfor v, _ in enumerate(range(n)):\\n    p, c = map(int, input().split())\\n    if p != -1:\\n        g[p - 1].append(v)\\n    cs[v] = c\\n\\nr = []\\nused = [False] * n\\n\\ndef dfs(g, stack):\\n    nonlocal r, used\\n    while stack:\\n        v = stack.pop()\\n        used[v] = True\\n\\n        flag = True if cs[v] == 1 else False\\n        for u in g[v]:\\n            if not used[u]:\\n                stack.append(u)\\n            if cs[u] == 0:\\n                flag = False\\n\\n        if flag:\\n            r.append(v)\\n\\n\\nfor v in range(n):\\n    if not used[v]:\\n        dfs(g, [v])\\n\\nif r:\\n    for el in sorted(r):\\n        print(el + 1, end=\\\" \\\")\\nelse:\\n    print(-1)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn=int(input())\\nEDGE=[list(map(int,input().split())) for i in range(n)]\\n\\nCHILDLIST=[[] for i in range(n+1)]\\n\\n\\nfor i in range(n):\\n    if EDGE[i][0]==-1:\\n        continue\\n    CHILDLIST[EDGE[i][0]].append(i+1)\\n\\nANS=[]\\n\\nfor i in range(1,n+1):\\n    if EDGE[i-1][1]==1:\\n\\n        \\n        for chi in CHILDLIST[i]:\\n            if EDGE[chi-1][1]==0:\\n                break\\n\\n        else:\\n            ANS.append(i)\\n\\nANS.sort()\\nif ANS==[]:\\n    print(-1)\\nprint(*ANS)\\n\", \"import sys\\nN = int(input())\\nEr = set(range(1, N+1))\\nfor i in range(1, N + 1):\\n    p, c = map(int, sys.stdin.readline().split())\\n    if c == 0:\\n        if i in Er:\\n            Er.remove(i)\\n        if p in Er:\\n            Er.remove(p)\\nEr = list(Er)\\nif not Er:\\n    print(-1)\\nelse:\\n    Er.sort()\\n    print(*Er)\", \"from collections import defaultdict as dd\\nimport math\\nimport heapq\\ndef nn():\\n\\treturn int(input())\\n\\ndef li():\\n\\treturn list(input())\\n\\ndef mi():\\n\\treturn list(map(int, input().split()))\\n\\ndef lm():\\n\\treturn list(map(int, input().split()))\\n\\nn=nn()\\n\\n\\nparentlist=[]\\n\\n\\nfor i in range(n):\\n\\tparentlist.append(lm())\\n\\n\\n\\n\\nchildlist=dd(list)\\n\\nfor i in range(1,n+1):\\n\\tchildlist[i]=[]\\n\\n\\n\\nfor vertex, pair in enumerate(parentlist):\\n\\t#print(vertex,pair)\\n\\t#print(childlist)\\n\\tparent=pair[0]\\n\\t#print(parent)\\n\\tif not parent==-1:\\n\\t\\t#print(childlist[parent])\\n\\t\\t(childlist[parent]).append((vertex+1,pair[1]))\\n\\nblist=[]\\n\\nfor vertex in childlist:\\n\\tif childlist[vertex]==[] or all([child[1]==1 for child in childlist[vertex]]):\\n\\t\\tif parentlist[vertex-1][1]==1:\\n\\t\\t\\tblist.append(vertex)\\n\\ndeleted=sorted(blist)\\n\\n\\nif len(deleted)==0:\\n\\tprint(-1)\\n\\nelse:\\n\\tprint(*deleted)\\n\\t\\n\\n\\n\\n\\n\\n\\n\\n\\n\", \"#! /usr/bin/env python\\n# -*- coding: utf-8 -*-\\n# vim:fenc=utf-8\\n#\\n\\n\\\"\\\"\\\"\\n549 C\\n\\\"\\\"\\\"\\n\\nn = int(input())\\nd = [1 for i in range(n+1)]\\n\\nfor i in range(1, n+1):\\n    pi, ci = list(map(int, input().split()))\\n    if ci == 0:\\n        d[i] = 0\\n        if pi != -1:\\n            d[pi] = 0\\n\\ne = []\\nfor i in range(1, n+1):\\n    if d[i] == 1:\\n        e.append(i)\\n\\nif len(e) == 0:\\n    print(-1)\\n    quit()\\nelse:\\n    print(\\\" \\\".join(map(str, e)))\\n\", \"n = int(input().strip())\\n\\nparent = [None for i in range(n+1)]\\nchildren = [[] for i in range(n+1)]\\ncs = [None for i in range(n+1)]\\n\\nfor i in range(1, n+1):\\n\\tp, c = list(map(int, input().split()))\\n\\tparent[i] = p\\n\\tif p != -1:\\n\\t\\tchildren[p].append(i)\\n\\tcs[i] = c\\n\\n# print(parent)\\n# print(children)\\n# print(cs)\\n\\nto_delete = []\\nfor i in range(1, n+1):\\n\\tif parent[i] > 0 and cs[i] == 1 and sum([cs[item] for item in children[i]]) == len(children[i]):\\n\\t\\tto_delete.append(i)\\n\\nif len(to_delete) > 0:\\n\\tprint(' '.join([str(item) for item in to_delete]))\\nelse:\\n\\tprint(-1)\\n\", \"3\\n\\ndef ir(): return int(input())\\ndef ia(): return [int(i) for i in input().split()]\\n\\nn = ir()\\n\\np = [None] * n\\nc = [None] * n\\nadj =  [ [ ] for i in range(n) ]\\n\\ngood = [False] * n\\n\\nfor i in range(n):\\n\\tp0, c0 = ia()\\n\\tc[i] = c0\\n\\tif p0 == -1:\\n\\t\\tp[i] = p0\\n\\t\\tgood[i] = True\\n\\telse:\\n\\t\\tp0 -= 1\\n\\t\\tp[i] = p0\\n\\t\\tadj[p0].append(i)\\n\\t\\tif c0 == 0:\\n\\t\\t\\tgood[i] = True\\n\\t\\t\\tgood[p0] = True\\n\\nans = [ ]\\nfor i, e in enumerate(good):\\n\\tif not e:\\n\\t\\tans.append(i)\\n\\nans.sort()\\n\\nif ans == []:\\n\\tprint(-1)\\nelse:\\n\\tans = [str(e + 1) for e in ans]\\n\\tprint( \\\" \\\".join(ans) )\", \"n = int(input())\\nmass = [0 for i in range(n + 10)]\\narr = [0 for i in range(n + 10)]\\nnnn = [0 for i in range(n + 10)]\\nfor i in range(n):\\n    a, b = map(int, input().split())\\n    if a != -1:\\n        mass[a] += 1\\n        arr[a] += b\\n    nnn[i + 1] = b\\nans = []\\nfor i in range(n):\\n    if mass[i + 1] == arr[i + 1] and nnn[i + 1] == 1:\\n        ans.append(i + 1)\\nif len(ans) == 0:\\n    ans.append(-1)\\nfor i in ans:\\n    print(i, end = ' ')\", \"'''input\\n5\\n-1 0\\n1 1\\n1 1\\n2 0\\n3 0\\n'''\\nfrom sys import stdin, stdout\\nimport sys\\nfrom collections import defaultdict, deque\\n\\nsys.setrecursionlimit(15000)\\n\\n\\ndef bfs(graph, parent, n):\\n\\tmyq = deque()\\n\\tmyq.append([parent, 0])\\n\\tans = []\\n\\twhile len(myq) != 0:\\n\\t\\tnode = myq.popleft()\\n\\t\\tflag = 0\\n\\t\\tfor i in graph[node[0]]:\\n\\t\\t\\tmyq.append(i)\\n\\t\\t\\tif i[1] == 1:\\n\\t\\t\\t\\tcontinue\\n\\t\\t\\telse:\\n\\t\\t\\t\\tflag = 1\\n\\t\\tif flag == 0 and node[1] == 1:\\n\\t\\t\\tans.append(node[0])\\n\\treturn ans\\n\\n# main starts\\nn  = int(stdin.readline().strip())\\ngraph = defaultdict(list)\\nparent = -1\\nfor i in range(1, n + 1):\\n\\tnode, key = list(map(int, stdin.readline().split()))\\n\\tif node == -1:\\n\\t\\tparent = i\\n\\t\\tcontinue\\t\\t\\n\\tgraph[node].append([i, key])\\n#print(graph)\\nans = bfs(graph, parent, n)\\nans.sort()\\nif len(ans) > 0:\\n\\tprint(*ans)\\nelse:\\n\\tprint(-1)\", \"class node:\\n    def __init__(self, value, parent, respect):\\n        self.value = value\\n        self.children = []\\n        self.parent = parent\\n        self.respect = respect\\n\\nall_nodes = {}\\nroot = -1\\n\\nnot_respectors = []\\n\\nimport sys\\n\\nn = int(sys.stdin.readline())\\nfor i in range(n):\\n    line = sys.stdin.readline()[:-1]\\n    node_parent, node_res = line.split(\\\" \\\")\\n    if i+1 in all_nodes:\\n        all_nodes[i+1].parent = int(node_parent)\\n        all_nodes[i+1].respect = int(node_res)\\n    else:\\n        all_nodes[i+1] = node(i+1, int(node_parent), int(node_res))\\n\\n    if int(node_parent) == -1:\\n        root = all_nodes[i+1]\\n    else:\\n        int_node_parent = int(node_parent)\\n        if int_node_parent not in all_nodes:\\n            all_nodes[int_node_parent] = node(int_node_parent, -1, -1)\\n        all_nodes[int_node_parent].children.append(all_nodes[i+1])\\n\\n    if int(node_res) == 1:\\n        not_respectors.append(all_nodes[i+1])\\n\\ndef get_val(n):\\n    return n.value\\n\\n\\nnot_respecters = sorted(not_respectors, key=get_val)\\ntot = []\\nfor n in not_respectors:\\n    b = True\\n    for c in n.children:\\n        if c.respect == 0:\\n            b = False\\n            break\\n    if b:\\n        tot.append(n)\\n\\n\\nif len(tot) == 0:\\n    print(\\\"-1\\\")\\nelse:\\n    print(\\\" \\\".join(map(str, list(map(get_val, tot)))))\\n\", \"n = int(input())\\np = [0]\\nc = [0]\\na = []\\nm = dict()\\nfor i in range(n+1):\\n\\tm[i] = []\\nfor i in range(n):\\n\\tx,y = list(map(int,input().split()))\\n\\tp.append(x)\\n\\tc.append(y)\\n\\tif x!=-1:\\n\\t\\tm[x].append(i+1)\\nfor i in range(1,n+1):\\n\\tif c[i]==1:\\n\\t\\tif len(m[i])==0:\\n\\t\\t\\ta.append(i)\\n\\t\\telse:\\n\\t\\t\\tfor j in m[i]:\\n\\t\\t\\t\\tif c[j]==0:\\n\\t\\t\\t\\t\\tbreak\\n\\t\\t\\telse:\\n\\t\\t\\t\\ta.append(i)\\n\\nif len(a)==0:\\n\\tprint(-1)\\nelse:\\n\\tprint(*sorted(a))\\n\", \"def bs (arr, l, r, x): \\n  \\n    # Check base case \\n    if r >= l: \\n  \\n        mid = l + (r - l)//2\\n  \\n        # If element is present at the middle itself \\n        if arr[mid] == x: \\n            return 1 \\n          \\n        # If element is smaller than mid, then it can only \\n        # be present in left subarray \\n        elif arr[mid] > x: \\n            return bs(arr, l, mid-1, x) \\n  \\n        # Else the element can only be present in right subarray \\n        else: \\n            return bs(arr, mid+1, r, x) \\n  \\n    else: \\n        # Element is not present in the array \\n        return 0\\n\\nn=int(input())\\nnrp=[]\\nrc=[]\\nfor i in range(n):\\n    p,c=map(int,input().split())\\n    if c:nrp.append(i+1)\\n    if not c:rc.append(p)\\ns=[]\\nrc.sort()\\nfor i in nrp:\\n    if not bs(rc,0,len(rc)-1,i):s.append(i)\\nif not s:print(\\\"-1\\\")\\nelse:print(*sorted(s))\", \"n = int(input())\\nparent_respect = [0] * (n + 1)\\ngraph = [[] for i in range(n + 1)]\\nroot = None\\nfor i in range(1, n + 1):\\n    p, r = [int(j) for j in input().split()]\\n    if p == -1:\\n        root = i\\n        continue\\n    graph[p].append(i)\\n    parent_respect[i] = r\\n\\n# print(graph)\\nresult = []\\n\\nfor i in range(1, n + 1):\\n    if parent_respect[i] == 1:\\n        all_c_r = False\\n        for c in graph[i]:\\n            if parent_respect[c] == 0:\\n                all_c_r = True\\n\\n        if not all_c_r:\\n            result.append(i)\\n\\nif result:\\n    for i in result:\\n        print(i, end=' ')\\nelse:\\n    print(-1)\\n\", \"v = int(input())\\n\\nneed_check = {}\\ntree = {}\\n\\nkids = {}\\n\\ncount = 1\\nwhile count <= v:\\n    parent, to_check = list(map(int, input().split()))\\n    node = count\\n\\n    tree[node] = parent\\n\\n    if to_check:\\n        need_check[node] = 1\\n\\n    if parent not in kids:\\n        kids[parent] = []\\n\\n    kids[parent].append(node)\\n\\n    count += 1\\n\\ndeletion = []\\nfor node in sorted(need_check.keys()):\\n    # determine if we'll delete him\\n    if node in kids and kids[node]:\\n        to_delete = True\\n\\n        for kid in kids[node]:\\n            if kid not in need_check:\\n                to_delete = False\\n                break\\n\\n        if not to_delete:\\n            continue\\n\\n        next_parent = tree[node]\\n        for kid in kids[node]:\\n            tree[kid] = next_parent\\n\\n        del kids[node]\\n    deletion.append(node)\\n\\nif not deletion:\\n    print('-1')\\nelse:\\n    print(' '.join(map(str, deletion)))\\n\", \"import sys\\nsys.setrecursionlimit(2000)\\nfrom collections import Counter\\nfrom functools import reduce\\n# sys.stdin.readline()\\nfrom copy import deepcopy\\n\\n\\ndef __starting_point():\\n\\n    # single variables\\n    n = [int(val) for val in sys.stdin.readline().split()][0]\\n\\n    # On each step you select such a non-root vertex that it does not \\n    # respect its parent and none of its children respects it\\n\\n    tree = {i+1 : [[], []] for i in range(n)}\\n    for i in range(n):\\n        p, c = [int(val) for val in sys.stdin.readline().split()]\\n            \\n        tree[i+1][0].append((p, c))\\n        if(p != -1):\\n            tree[p][1].append((i, c))\\n\\n    d = []\\n    for i in range(1, n+1):\\n        p, c = tree[i]\\n        if(p[0][1] == 1 and sum([x[1] for x in c]) == len(c)):\\n            d.append(i)\\n\\n   \\n    if(len(d) == 0):\\n        print(-1)\\n    else: \\n        for val in d:\\n            print(val, end=' ')\\n        print('')\\n\\n\\n\\n\\n__starting_point()\", \"n = int(input())\\n\\nrespectingChildren = [0 for i in range(n)]\\ntotalChildren = [0 for i in range(n)]\\nCi = []\\nPi = []\\nroot = 0\\n\\nfor i in range(n):\\n    pi, ci = list(map(int, input().split()))\\n    Ci.append(ci)\\n    if pi==-1:\\n        Pi.append(pi)\\n        root = i\\n        continue\\n\\n    Pi.append(pi-1)\\n    totalChildren[pi-1] += 1\\n    if ci==0:\\n        respectingChildren[pi-1] += 1\\n\\nans = []\\nfor i in range(n):\\n    if i==root:\\n        continue\\n\\n    if Ci[i]==1:\\n        if respectingChildren[i]==0:\\n            ans.append(i+1)\\n            parentOfI = Pi[i]\\n            totalChildren[parentOfI] += totalChildren[i]\\n\\nif ans==[]:\\n    print(-1)\\nelse:\\n    print(*ans)\\n        \\n    \\n\"]",
        "difficulty": "interview",
        "input": "5\n3 1\n1 1\n-1 0\n2 1\n3 0\n",
        "output": "1 2 4 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1143/C"
    },
    {
        "id": 1053,
        "task_id": 2394,
        "test_case_id": 1,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n",
        "output": "20\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1054,
        "task_id": 2394,
        "test_case_id": 2,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n",
        "output": "114\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1055,
        "task_id": 2394,
        "test_case_id": 3,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "3 5\n2 1\n3 1\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1056,
        "task_id": 2394,
        "test_case_id": 4,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "2 1\n1 2\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1057,
        "task_id": 2394,
        "test_case_id": 5,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "2 5\n2 1\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1058,
        "task_id": 2394,
        "test_case_id": 6,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "15 1\n12 9\n13 7\n1 3\n10 4\n9 2\n2 15\n11 4\n2 14\n10 8\n6 7\n12 5\n8 7\n3 10\n10 2\n",
        "output": "346\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1059,
        "task_id": 2394,
        "test_case_id": 7,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "4 2\n3 4\n2 4\n3 1\n",
        "output": "7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1060,
        "task_id": 2394,
        "test_case_id": 8,
        "question": "A tree is an undirected connected graph without cycles. The distance between two vertices is the number of edges in a simple path between them.\n\nLimak is a little polar bear. He lives in a tree that consists of n vertices, numbered 1 through n.\n\nLimak recently learned how to jump. He can jump from a vertex to any vertex within distance at most k.\n\nFor a pair of vertices (s, t) we define f(s, t) as the minimum number of jumps Limak needs to get from s to t. Your task is to find the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Input-----\n\nThe first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ 5) — the number of vertices in the tree and the maximum allowed jump distance respectively.\n\nThe next n - 1 lines describe edges in the tree. The i-th of those lines contains two integers a_{i} and b_{i} (1 ≤ a_{i}, b_{i} ≤ n) — the indices on vertices connected with i-th edge.\n\nIt's guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer, denoting the sum of f(s, t) over all pairs of vertices (s, t) such that s < t.\n\n\n-----Examples-----\nInput\n6 2\n1 2\n1 3\n2 4\n2 5\n4 6\n\nOutput\n20\n\nInput\n13 3\n1 2\n3 2\n4 2\n5 2\n3 6\n10 6\n6 7\n6 13\n5 8\n5 9\n9 11\n11 12\n\nOutput\n114\n\nInput\n3 5\n2 1\n3 1\n\nOutput\n3\n\n\n\n-----Note-----\n\nIn the first sample, the given tree has 6 vertices and it's displayed on the drawing below. Limak can jump to any vertex within distance at most 2. For example, from the vertex 5 he can jump to any of vertices: 1, 2 and 4 (well, he can also jump to the vertex 5 itself). [Image] \n\nThere are $\\frac{n \\cdot(n - 1)}{2} = 15$ pairs of vertices (s, t) such that s < t. For 5 of those pairs Limak would need two jumps: (1, 6), (3, 4), (3, 5), (3, 6), (5, 6). For other 10 pairs one jump is enough. So, the answer is 5·2 + 10·1 = 20.\n\nIn the third sample, Limak can jump between every two vertices directly. There are 3 pairs of vertices (s < t), so the answer is 3·1 = 3.",
        "solutions": "[\"\\\"\\\"\\\"\\n#If FastIO not needed, used this and don't forget to strip\\n#import sys, math\\n#input = sys.stdin.readline\\n\\\"\\\"\\\"\\n\\nimport os\\nimport sys\\nfrom io import BytesIO, IOBase\\nimport heapq as h \\nfrom bisect import bisect_left, bisect_right\\n\\nfrom types import GeneratorType\\nBUFSIZE = 8192\\nclass FastIO(IOBase):\\n    newlines = 0\\n \\n    def __init__(self, file):\\n        import os\\n        self.os = os\\n        self._fd = file.fileno()\\n        self.buffer = BytesIO()\\n        self.writable = \\\"x\\\" in file.mode or \\\"r\\\" not in file.mode\\n        self.write = self.buffer.write if self.writable else None\\n \\n    def read(self):\\n        while True:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            if not b:\\n                break\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines = 0\\n        return self.buffer.read()\\n \\n    def readline(self):\\n        while self.newlines == 0:\\n            b = self.os.read(self._fd, max(self.os.fstat(self._fd).st_size, BUFSIZE))\\n            self.newlines = b.count(b\\\"\\\\n\\\") + (not b)\\n            ptr = self.buffer.tell()\\n            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)\\n        self.newlines -= 1\\n        return self.buffer.readline()\\n \\n    def flush(self):\\n        if self.writable:\\n            self.os.write(self._fd, self.buffer.getvalue())\\n            self.buffer.truncate(0), self.buffer.seek(0)\\n \\n \\nclass IOWrapper(IOBase):\\n    def __init__(self, file):\\n        self.buffer = FastIO(file)\\n        self.flush = self.buffer.flush\\n        self.writable = self.buffer.writable\\n        self.write = lambda s: self.buffer.write(s.encode(\\\"ascii\\\"))\\n        self.read = lambda: self.buffer.read().decode(\\\"ascii\\\")\\n        self.readline = lambda: self.buffer.readline().decode(\\\"ascii\\\")\\n \\n \\nsys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)\\ninput = lambda: sys.stdin.readline().rstrip(\\\"\\\\r\\\\n\\\")\\n\\nfrom collections import defaultdict as dd, deque as dq, Counter as dc\\nimport math, string\\n\\n\\ndef getInts():\\n    return [int(s) for s in input().split()]\\n\\ndef getInt():\\n    return int(input())\\n\\ndef getStrs():\\n    return [s for s in input().split()]\\n\\ndef getStr():\\n    return input()\\n\\ndef listStr():\\n    return list(input())\\n\\ndef getMat(n):\\n    return [getInts() for _ in range(n)]\\n\\nMOD = 10**9+7\\n\\n\\n\\\"\\\"\\\"\\nEach edge goes from parent U to child V\\nEdge appears on S_V * (N - S_V) paths\\n\\nFor each path of length L, (L + (-L)%K)/K\\n\\n\\nL%K 0, 1, 2, 3, 4\\n(K - L%K)%K K K-1 K-2 ...\\n0 K-1 K-2 ...\\n\\n\\\"\\\"\\\"\\ndef bootstrap(f, stack=[]):\\n    def wrappedfunc(*args, **kwargs):\\n        if stack:\\n            return f(*args, **kwargs)\\n        else:\\n            to = f(*args, **kwargs)\\n            while True:\\n                if type(to) is GeneratorType:\\n                    stack.append(to)\\n                    to = next(to)\\n                else:\\n                    stack.pop()\\n                    if not stack:\\n                        break\\n                    to = stack[-1].send(to)\\n            return to\\n    return wrappedfunc\\n\\ndef solve():\\n    N, K = getInts()\\n    graph = dd(set)\\n    for i in range(N-1):\\n        A, B = getInts()\\n        graph[A].add(B)\\n        graph[B].add(A)\\n    dp_count = [[0 for j in range(5)] for i in range(N+1)]\\n    dp_total = [0 for j in range(N+1)]\\n    nonlocal ans\\n    ans = 0\\n    @bootstrap\\n    def dfs(node,parent,depth):\\n        nonlocal ans\\n        dp_count[node][depth % K] = 1\\n        dp_total[node] = 1\\n        for neigh in graph[node]:\\n            if neigh != parent:\\n                yield dfs(neigh,node,depth+1)\\n                for i in range(K):\\n                    for j in range(K):\\n                        diff = (i+j-2*depth)%K\\n                        req = (-diff)%K\\n                        ans += req * dp_count[node][i] * dp_count[neigh][j]\\n                for i in range(K):\\n                    dp_count[node][i] += dp_count[neigh][i]\\n                dp_total[node] += dp_total[neigh]\\n        ans += dp_total[node] * (N - dp_total[node])\\n        yield\\n    dfs(1,-1,0)\\n    return ans//K\\n    \\n    \\nprint(solve())\\n\"]",
        "difficulty": "interview",
        "input": "12 3\n5 11\n10 11\n6 4\n8 9\n4 12\n10 7\n4 1\n3 1\n2 12\n9 4\n9 10\n",
        "output": "88\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/771/C"
    },
    {
        "id": 1061,
        "task_id": 2461,
        "test_case_id": 1,
        "question": "Ilya is very fond of graphs, especially trees. During his last trip to the forest Ilya found a very interesting tree rooted at vertex 1. There is an integer number written on each vertex of the tree; the number written on vertex i is equal to a_{i}.\n\nIlya believes that the beauty of the vertex x is the greatest common divisor of all numbers written on the vertices on the path from the root to x, including this vertex itself. In addition, Ilya can change the number in one arbitrary vertex to 0 or leave all vertices unchanged. Now for each vertex Ilya wants to know the maximum possible beauty it can have.\n\nFor each vertex the answer must be considered independently.\n\nThe beauty of the root equals to number written on it.\n\n\n-----Input-----\n\nFirst line contains one integer number n — the number of vertices in tree (1 ≤ n ≤ 2·10^5).\n\nNext line contains n integer numbers a_{i} (1 ≤ i ≤ n, 1 ≤ a_{i} ≤ 2·10^5).\n\nEach of next n - 1 lines contains two integer numbers x and y (1 ≤ x, y ≤ n, x ≠ y), which means that there is an edge (x, y) in the tree.\n\n\n-----Output-----\n\nOutput n numbers separated by spaces, where i-th number equals to maximum possible beauty of vertex i.\n\n\n-----Examples-----\nInput\n2\n6 2\n1 2\n\nOutput\n6 6 \n\nInput\n3\n6 2 3\n1 2\n1 3\n\nOutput\n6 6 6 \n\nInput\n1\n10\n\nOutput\n10",
        "solutions": "[\"from sys import stdin\\nfrom fractions import gcd\\n\\nn = int(stdin.readline().strip())\\nv = list(map(int, stdin.readline().strip().split()))\\n\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    x, y = list(map(int, stdin.readline().strip().split()))\\n    adj[x-1].append(y-1)\\n    adj[y-1].append(x-1)\\n\\nroot_divisors = []\\ncnt = [0]*200001\\nd = 1\\nwhile d*d <= v[0]:\\n    if v[0] % d == 0:\\n        root_divisors.append(d)\\n        cnt[d] += 1\\n        if v[0]//d != d:\\n            root_divisors.append(v[0]//d)\\n            cnt[v[0]//d] += 1\\n    d += 1    \\ns = [0]\\nvisited = [False]*n\\nvisited[0] = True\\nlevel = [1]*n\\nres1 = [0]*n\\nres2 = [0]*n\\nres1[0] = v[0]\\nd = 1\\nwhile s:\\n    x = s[-1]\\n    any_more = False\\n    while adj[x]:\\n        y = adj[x].pop()\\n        if not visited[y]:\\n            visited[y] = True\\n            any_more = True\\n            s.append(y)\\n            level[y] = level[x]+1\\n            res2[y] = gcd(res2[x], v[y])\\n            for d in root_divisors:\\n                if v[y] % d == 0:\\n                    cnt[d] += 1\\n                if cnt[d] == level[y] or cnt[d] == level[y]-1:\\n                    res1[y] = max(res1[y], res2[y], d)\\n            break\\n    if not any_more:\\n        s.pop()\\n        for d in root_divisors:\\n            if v[x] % d == 0:\\n                cnt[d] -= 1\\n        \\nprint(' '.join(list(map(str, res1))))\"]",
        "difficulty": "interview",
        "input": "2\n6 2\n1 2",
        "output": "6 6",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/842/C"
    },
    {
        "id": 1062,
        "task_id": 2461,
        "test_case_id": 2,
        "question": "Ilya is very fond of graphs, especially trees. During his last trip to the forest Ilya found a very interesting tree rooted at vertex 1. There is an integer number written on each vertex of the tree; the number written on vertex i is equal to a_{i}.\n\nIlya believes that the beauty of the vertex x is the greatest common divisor of all numbers written on the vertices on the path from the root to x, including this vertex itself. In addition, Ilya can change the number in one arbitrary vertex to 0 or leave all vertices unchanged. Now for each vertex Ilya wants to know the maximum possible beauty it can have.\n\nFor each vertex the answer must be considered independently.\n\nThe beauty of the root equals to number written on it.\n\n\n-----Input-----\n\nFirst line contains one integer number n — the number of vertices in tree (1 ≤ n ≤ 2·10^5).\n\nNext line contains n integer numbers a_{i} (1 ≤ i ≤ n, 1 ≤ a_{i} ≤ 2·10^5).\n\nEach of next n - 1 lines contains two integer numbers x and y (1 ≤ x, y ≤ n, x ≠ y), which means that there is an edge (x, y) in the tree.\n\n\n-----Output-----\n\nOutput n numbers separated by spaces, where i-th number equals to maximum possible beauty of vertex i.\n\n\n-----Examples-----\nInput\n2\n6 2\n1 2\n\nOutput\n6 6 \n\nInput\n3\n6 2 3\n1 2\n1 3\n\nOutput\n6 6 6 \n\nInput\n1\n10\n\nOutput\n10",
        "solutions": "[\"from sys import stdin\\nfrom fractions import gcd\\n\\nn = int(stdin.readline().strip())\\nv = list(map(int, stdin.readline().strip().split()))\\n\\nadj = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    x, y = list(map(int, stdin.readline().strip().split()))\\n    adj[x-1].append(y-1)\\n    adj[y-1].append(x-1)\\n\\nroot_divisors = []\\ncnt = [0]*200001\\nd = 1\\nwhile d*d <= v[0]:\\n    if v[0] % d == 0:\\n        root_divisors.append(d)\\n        cnt[d] += 1\\n        if v[0]//d != d:\\n            root_divisors.append(v[0]//d)\\n            cnt[v[0]//d] += 1\\n    d += 1    \\ns = [0]\\nvisited = [False]*n\\nvisited[0] = True\\nlevel = [1]*n\\nres1 = [0]*n\\nres2 = [0]*n\\nres1[0] = v[0]\\nd = 1\\nwhile s:\\n    x = s[-1]\\n    any_more = False\\n    while adj[x]:\\n        y = adj[x].pop()\\n        if not visited[y]:\\n            visited[y] = True\\n            any_more = True\\n            s.append(y)\\n            level[y] = level[x]+1\\n            res2[y] = gcd(res2[x], v[y])\\n            for d in root_divisors:\\n                if v[y] % d == 0:\\n                    cnt[d] += 1\\n                if cnt[d] == level[y] or cnt[d] == level[y]-1:\\n                    res1[y] = max(res1[y], res2[y], d)\\n            break\\n    if not any_more:\\n        s.pop()\\n        for d in root_divisors:\\n            if v[x] % d == 0:\\n                cnt[d] -= 1\\n        \\nprint(' '.join(list(map(str, res1))))\"]",
        "difficulty": "interview",
        "input": "3\n6 2 3\n1 2\n1 3",
        "output": "6 6 6",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/842/C"
    },
    {
        "id": 1063,
        "task_id": 2464,
        "test_case_id": 1,
        "question": "You are given a tree (an undirected connected acyclic graph) consisting of $n$ vertices and $n - 1$ edges. A number is written on each edge, each number is either $0$ (let's call such edges $0$-edges) or $1$ (those are $1$-edges).\n\nLet's call an ordered pair of vertices $(x, y)$ ($x \\ne y$) valid if, while traversing the simple path from $x$ to $y$, we never go through a $0$-edge after going through a $1$-edge. Your task is to calculate the number of valid pairs in the tree.\n\n\n-----Input-----\n\nThe first line contains one integer $n$ ($2 \\le n \\le 200000$) — the number of vertices in the tree.\n\nThen $n - 1$ lines follow, each denoting an edge of the tree. Each edge is represented by three integers $x_i$, $y_i$ and $c_i$ ($1 \\le x_i, y_i \\le n$, $0 \\le c_i \\le 1$, $x_i \\ne y_i$) — the vertices connected by this edge and the number written on it, respectively.\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the number of valid pairs of vertices.\n\n\n-----Example-----\nInput\n7\n2 1 1\n3 2 0\n4 2 1\n5 2 0\n6 7 1\n7 2 1\n\nOutput\n34\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example:\n\n[Image]",
        "solutions": "[\"class UnionFind:\\n    def __init__(self, N):\\n        self.par = [i for i in range(N)]\\n        self.rank = [1 for i in range(N)]\\n        self.rank[0] = 0\\n    def union(self, x, y):\\n        if not self.is_same_set(x, y):\\n            par_x = self.find_par(x)\\n            par_y = self.find_par(y)\\n\\n            if self.rank[par_x] > self.rank[par_y]:\\n                self.rank[par_x] += self.rank[par_y]\\n                self.rank[par_y] = 0\\n                self.par[par_y] = par_x\\n            else:\\n                self.rank[par_y] += self.rank[par_x]\\n                self.rank[par_x] = 0\\n                self.par[par_x] = par_y\\n\\n    def find_par(self, x):\\n        if self.par[x] == x: return x\\n        self.par[x] = self.find_par(self.par[x])\\n        return self.par[x]\\n\\n    def is_same_set(self, x, y):\\n        return self.find_par(x) == self.find_par(y)\\n\\n    def size(self, x):\\n        return self.rank[self.find_par(x)]\\n# 2 unionfind, para 0 e para 1 formando 2 florestas\\n# lista de adj\\n# verificar todos os componentes existentes e adicionar na resposta n * (n-1)\\n\\nn = int(input())\\n\\nadj = [[] for i in range(n+1)]\\n\\nuf0 = UnionFind(n+1)\\nuf1 = UnionFind(n+1)\\n\\nfor i in range(n-1):\\n    x, y, c = list(map(int, input().split()))\\n\\n    if c == 0:\\n        uf0.union(x, y)\\n    else:\\n        uf1.union(x, y)\\n    adj[x].append(y)\\n    adj[y].append(x)\\nfor i in range(n+1):\\n    uf0.find_par(i)\\n    uf1.find_par(i)\\n\\nresp = 0\\n\\nfor i in set(uf0.par):\\n    resp += uf0.rank[i] * (uf0.rank[i] - 1)\\nfor i in set(uf1.par):\\n    resp += uf1.rank[i] * (uf1.rank[i] - 1)\\n\\n# pra cada componente do 0-uf verificar se existe esse vertice na 1-uf e ele for conectado com algu\\u00e9m, se sim, multiplicar (n-1)*(m-1) sendo n o componente da 0-uf e m o componente da 1-f e adicionar na resposta\\n\\nfor i in range(len(uf0.par)):\\n    if uf0.rank[uf0.find_par(i)] > 1:\\n        if uf1.rank[uf1.find_par(i)] > 1:\\n            resp += (uf0.rank[uf0.find_par(i)] - 1) * (uf1.rank[uf1.find_par(i)] - 1)\\nprint(resp)\\n\", \"class UnionFind:\\n    def __init__(self, N):\\n        self.par = [i for i in range(N)]\\n        self.rank = [1 for i in range(N)]\\n        self.rank[0] = 0\\n    def union(self, x, y):\\n        if not self.is_same_set(x, y):\\n            par_x = self.find_par(x)\\n            par_y = self.find_par(y)\\n\\n            if self.rank[par_x] > self.rank[par_y]:\\n                self.rank[par_x] += self.rank[par_y]\\n                self.rank[par_y] = 0\\n                self.par[par_y] = par_x\\n            else:\\n                self.rank[par_y] += self.rank[par_x]\\n                self.rank[par_x] = 0\\n                self.par[par_x] = par_y\\n\\n    def find_par(self, x):\\n        if self.par[x] == x: return x\\n        self.par[x] = self.find_par(self.par[x])\\n        return self.par[x]\\n\\n    def is_same_set(self, x, y):\\n        return self.find_par(x) == self.find_par(y)\\n\\n    def size(self, x):\\n        return self.rank[self.find_par(x)]\\n# 2 unionfind, para 0 e para 1 formando 2 florestas\\n# lista de adj\\n# verificar todos os componentes existentes e adicionar na resposta n * (n-1)\\n\\nn = int(input())\\n\\nadj = [[] for i in range(n+1)]\\n\\nuf0 = UnionFind(n+1)\\nuf1 = UnionFind(n+1)\\n\\nfor i in range(n-1):\\n    x, y, c = list(map(int, input().split()))\\n\\n    if c == 0:\\n        uf0.union(x, y)\\n    else:\\n        uf1.union(x, y)\\n    adj[x].append(y)\\n    adj[y].append(x)\\n\\n    uf0.find_par(x)\\n    uf1.find_par(y)\\n\\nresp = 0\\n\\nresp += sum([uf0.rank[i] * (uf0.rank[i] - 1) for i in set(uf0.par)])\\nresp += sum([uf1.rank[i] * (uf1.rank[i] - 1) for i in set(uf1.par)])\\n# pra cada componente do 0-uf verificar se existe esse vertice na 1-uf e ele for conectado com algu\\u00e9m, se sim, multiplicar (n-1)*(m-1) sendo n o componente da 0-uf e m o componente da 1-f e adicionar na resposta\\n#ja_visto = set()\\nfor i in range(len(uf0.par)):\\n    if uf0.rank[uf0.find_par(i)] > 1: #and not uf0.find_par(i) in ja_visto:\\n        #ja_visto.add(uf0.find_par(i))\\n        if uf1.rank[uf1.find_par(i)] > 1:\\n            resp += (uf0.rank[uf0.find_par(i)] - 1) * (uf1.rank[uf1.find_par(i)] - 1)\\nprint(resp)\\n\", \"from collections import deque\\n\\ndef bfs(source, graph, mark, num, fcount):\\n\\tvisited = [source]\\n\\tq = deque()\\n\\tmark[source] = True\\n\\tq.append(source)\\n\\twhile q:\\n\\t\\tu = q.popleft()\\n\\t\\tfor v, c in g[u]:\\n\\t\\t\\tif c == num and not mark[v]:\\n\\t\\t\\t\\tmark[v] = True\\n\\t\\t\\t\\tvisited.append(v)\\n\\t\\t\\t\\tq.append(v)\\n\\tif len(visited) > 1:\\n\\t\\tfor u in visited:\\n\\t\\t\\tfcount[u] = len(visited)\\n\\nn = int(input())\\nedges = [tuple(map(int, input().split())) for _ in range(0, n - 1)]\\ng = [[] for _ in range(0, n)]\\ncnt = [[0 for _ in range(0, n)] for _ in range(0, 2)]\\nfor u, v, c in edges:\\n\\tg[u - 1].append((v - 1, c))\\n\\tg[v - 1].append((u - 1, c))\\n\\nres = 0\\nfor link in range(0, 2):\\n\\tmark = [False] * n\\n\\tfor u in range(0, n):\\n\\t\\tif not mark[u]:\\n\\t\\t\\tbfs(u, g, mark, link, cnt[link])\\n\\t\\t\\tres += cnt[link][u] * (cnt[link][u] - 1)\\n\\nfor i in range(0, n):\\n\\tif cnt[0][i] > 0 and cnt[1][i] > 1:\\n\\t\\tres += (cnt[0][i] - 1) * (cnt[1][i] - 1)\\n\\nprint(int(res))\", \"class Deque:\\n\\tdef __init__(self):\\n\\t\\tself.buff = [0] * 400000\\n\\t\\tself.l = 0\\n\\t\\tself.r = 0\\n\\tdef append(self, x):\\n\\t\\tself.buff[self.r] = x\\n\\t\\tself.r = self.r + 1\\n\\tdef popleft(self):\\n\\t\\told_left = self.l\\n\\t\\tself.l = self.l + 1\\n\\t\\treturn self.buff[old_left]\\n\\tdef __bool__(self):\\n\\t\\treturn self.l != self.r\\n\\nq = Deque()\\n\\ndef bfs(source, graph, mark, num, fcount):\\n\\tvisited = [source]\\n\\tmark[source] = True\\n\\tq.append(source)\\n\\twhile q:\\n\\t\\tu = q.popleft()\\n\\t\\tfor v, c in g[u]:\\n\\t\\t\\tif c == num and not mark[v]:\\n\\t\\t\\t\\tmark[v] = True\\n\\t\\t\\t\\tvisited.append(v)\\n\\t\\t\\t\\tq.append(v)\\n\\tif len(visited) > 1:\\n\\t\\tfor u in visited:\\n\\t\\t\\tfcount[u] = len(visited)\\n\\nn = int(input())\\nedges = [tuple(map(int, input().split())) for _ in range(0, n - 1)]\\ng = [[] for _ in range(0, n)]\\ncnt = [[0 for _ in range(0, n)] for _ in range(0, 2)]\\nfor u, v, c in edges:\\n\\tg[u - 1].append((v - 1, c))\\n\\tg[v - 1].append((u - 1, c))\\n\\nres = 0\\nfor link in range(0, 2):\\n\\tmark = [False] * n\\n\\tfor u in range(0, n):\\n\\t\\tif not mark[u]:\\n\\t\\t\\tbfs(u, g, mark, link, cnt[link])\\n\\t\\t\\tres += cnt[link][u] * (cnt[link][u] - 1)\\n\\nfor i in range(0, n):\\n\\tif cnt[0][i] > 0 and cnt[1][i] > 1:\\n\\t\\tres += (cnt[0][i] - 1) * (cnt[1][i] - 1)\\n\\nprint(int(res))\", \"from collections import defaultdict\\n\\ndef colour(a, graph, cur, i):\\n    top = set()\\n    top.add(i)\\n    while len(top):\\n        x = top.pop()\\n        a[x] = cur\\n        for y in graph[x]:\\n            if a[y] == 0:\\n                top.add(y)\\n\\ndef colour_graph(a, graph, n):\\n    cur = 0\\n    for i in range(1, n + 1):\\n        if a[i] or i not in graph:\\n            continue\\n        else:\\n            cur += 1\\n            colour(a, graph, cur, i)\\n\\ndef count(col):\\n    ans = 0\\n    for el in col:\\n        if col[el] > 1:\\n            ans += col[el]*(col[el] - 1)\\n    return ans\\n\\nn = int(input())\\n\\n\\ngraph0 = defaultdict(set)\\ngraph1 = defaultdict(set)\\n\\n\\nvertex0 = set()\\n\\na0 = [0]*(n + 1)  \\na1 = [0]*(n + 1)\\n\\nfor i in range(n - 1):\\n    x, y, c = list(map(int, input().split()))\\n    if c == 0:\\n        graph0[x].add(y)\\n        graph0[y].add(x)\\n        vertex0.add(x)\\n        vertex0.add(y)\\n    else:\\n        graph1[x].add(y)\\n        graph1[y].add(x)\\n\\n\\ncolour_graph(a0, graph0, n)\\ncolour_graph(a1, graph1, n)\\n\\nanswer = 0\\ncol0 = defaultdict(int)\\ncol1 = defaultdict(int)\\n\\nfor i in range(n + 1):\\n    if a0[i]:\\n        col0[a0[i]] += 1\\n    if a1[i]:\\n        col1[a1[i]] += 1\\n\\nanswer += count(col0) + count(col1)\\n\\ncol = defaultdict(int)\\nfor v in vertex0:\\n    col[a1[v]] += col0[a0[v]] - 1\\nfor el in col:\\n    if el:\\n        answer += col[el]*(col1[el] - 1)\\n        \\nprint(answer)\\n\", \"class dsu:\\n\\tdef __init__(self,n):\\n\\t\\tself.arr = [i for i in range(n)]\\n\\t\\tself.size = [1 for i in range(n)]\\n\\t\\tself.s = n\\n\\tdef find(self,i):\\n\\t\\twhile(i!=self.arr[i]):\\n\\t\\t\\ti = self.arr[i]\\n\\t\\treturn i\\n\\t\\tif(self.arr[i]!=i):\\n\\t\\t\\tself.arr[i] = self.find(self.arr[i])\\n\\t\\treturn self.arr[i]\\n\\tdef union(self,i,j):\\n\\t\\tfa = self.find(i)\\n\\t\\tfb = self.find(j)\\n\\t\\tif(fa == fb):\\n\\t\\t\\treturn\\n\\t\\ts1 = self.size[fa]\\n\\t\\ts2 = self.size[fb]\\n\\t\\tif(s1<s2):\\n\\t\\t\\tself.arr[fa] = fb\\n\\t\\t\\tself.size[fb] += self.size[fa]\\n\\t\\telse:\\n\\t\\t\\tself.arr[fb] = fa\\n\\t\\t\\tself.size[fa] += self.size[fb]\\nn = int(input())\\nzero = dsu(n)\\none = dsu(n)\\nfor i in range(n-1):\\n\\ta = [int(i) for i in input().split(' ')]\\n\\tx = a[0]-1\\n\\ty = a[1]-1\\n\\tc = a[2]\\n\\tif(c==0):\\n\\t\\tzero.union(x,y)\\n\\telse:\\n\\t\\tone.union(x,y)\\ncount = 0\\nfor i in range(n):\\n\\tif(zero.arr[i] == i):\\n\\t\\tcount+=zero.size[i]*(zero.size[i]-1)\\n\\tif(one.arr[i] == i):\\n\\t\\tcount+=one.size[i]*(one.size[i]-1)\\n\\tcount += (zero.size[zero.find(i)]-1)*(one.size[one.find(i)]-1)\\nprint(count)\", \"import sys\\n\\nclass Disjoint:\\n    def __init__(self, n):\\n        self.n = n\\n        self.size = [1] * self.n\\n        self.parent = [i for i in range(self.n)]\\n    \\n    def root(self, node):\\n        self.parent[node] = node if self.parent[node] == node else self.root(self.parent[node])\\n        return self.parent[node]\\n    \\n    def join(self, a, b):\\n        a = self.root(a)\\n        b = self.root(b)\\n        if a == b:\\n            return False\\n        if self.size[a] > self.size[b]:\\n            a, b = b, a\\n            \\n        self.size[b] += self.size[a]\\n        self.parent[a] = b\\n        return True\\n    \\n    \\n\\ninp = [int(x) for x in sys.stdin.read().split()]\\nn = inp[0]\\n\\nwhite = Disjoint(n)\\nblack = Disjoint(n)\\n\\ninp_idx = 1\\nfor i in range(n - 1):\\n    x, y, c = inp[inp_idx], inp[inp_idx + 1], inp[inp_idx + 2]\\n    x -= 1\\n    y -= 1\\n    inp_idx += 3\\n    \\n    if c == 0:\\n        white.join(x, y)\\n    else:\\n        black.join(x, y)\\n    \\nans = 0\\nfor i in range(n):\\n    rootW = white.root(i)\\n    rootB = black.root(i)\\n    ans += white.size[rootW] - 1\\n    ans += black.size[rootB] - 1\\n    \\n    ans += (white.size[rootW] - 1) * (black.size[rootB] - 1)\\n    \\nprint(ans)\\n\", \"class UnionFind():\\n\\n    def __init__(self, n):\\n        self.n = n\\n\\n        self.root = [-1]*(n+1)\\n\\n        self.rnk = [0]*(n+1)\\n\\n\\n    def Find_Root(self, x):\\n        if(self.root[x] < 0):\\n            return x\\n        else:\\n\\n            self.root[x] = self.Find_Root(self.root[x])\\n            return self.root[x]\\n\\n    def Unite(self, x, y):\\n\\n        x = self.Find_Root(x)\\n        y = self.Find_Root(y)\\n\\n        if(x == y):\\n            return \\n\\n        elif(self.rnk[x] > self.rnk[y]):\\n            self.root[x] += self.root[y]\\n            self.root[y] = x\\n\\n        else:\\n            self.root[y] += self.root[x]\\n            self.root[x] = y\\n\\n            if(self.rnk[x] == self.rnk[y]):\\n                self.rnk[y] += 1\\n\\n    def isSameGroup(self, x, y):\\n        return self.Find_Root(x) == self.Find_Root(y)\\n\\n\\n    def Count(self, x):\\n        return -self.root[self.Find_Root(x)]\\n\\nimport sys\\ninput = sys.stdin.readline\\n\\nN = int(input())\\nuni0 = UnionFind(N)\\nuni1 = UnionFind(N)\\nfor _ in range(N-1):\\n    a, b, c = map(int, input().split())\\n    if c == 0:\\n        uni0.Unite(a-1, b-1)\\n    else:\\n        uni1.Unite(a-1, b-1)\\n\\ng0 = {}\\ng1 = {}\\nfor i in range(N):\\n    if uni0.Count(i) != 1:\\n        r = uni0.Find_Root(i)\\n        if not r in g0.keys():\\n            g0[r] = [i]\\n        else:\\n            g0[r].append(i)\\n    if uni1.Count(i) != 1:\\n        r = uni1.Find_Root(i)\\n        if not r in g1.keys():\\n            g1[r] = [i]\\n        else:\\n            g1[r].append(i)\\n\\nans = 0\\nfor v_list in g1.values():\\n    c = 0\\n    for n in v_list:\\n        if uni0.Count(n) == 1:\\n            c += 1\\n    l = len(v_list)\\n    ans += c*(l-1)\\n\\nfor v_list in g0.values():\\n    c = 0\\n    for n in v_list:\\n        if uni1.Count(n) != 1:\\n            r = uni1.Find_Root(n)\\n            c += len(g1[r])-1\\n    c += len(v_list)-1\\n    ans += len(v_list)*c\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.parent = [-1] * n\\n        self.cnt = n\\n\\n    def root(self, x):\\n        if self.parent[x] < 0:\\n            return x\\n        else:\\n            self.parent[x] = self.root(self.parent[x])\\n            return self.parent[x]\\n\\n    def merge(self, x, y):\\n        x = self.root(x)\\n        y = self.root(y)\\n        if x == y:\\n            return\\n        if self.parent[x] > self.parent[y]:\\n            x, y = y, x\\n        self.parent[x] += self.parent[y]\\n        self.parent[y] = x\\n        self.cnt -= 1\\n        \\n    def is_same(self, x, y):\\n        return self.root(x) == self.root(y)\\n\\n    def get_size(self, x):\\n        return -self.parent[self.root(x)]\\n    \\n    def get_cnt(self):\\n        return self.cnt\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\nuf0 = UnionFind(n)\\nuf1 = UnionFind(n)\\ntree0 = [[] for i in range(n)]\\ntree1 = [[] for i in range(n)]\\nfor i in range(n - 1):\\n    a, b, cost = info[i]\\n    a -= 1\\n    b -= 1\\n    if cost == 0:\\n        tree0[a].append(b)\\n        tree0[b].append(a)\\n        uf0.merge(a, b)\\n    else:\\n        tree1[a].append(b)\\n        tree1[b].append(a)\\n        uf1.merge(a, b)\\n\\nans0 = [0] * n\\nans1 = [0] * n\\nfor i in range(n):\\n    ans0[i] = uf0.get_size(i) - 1\\n    ans1[i] = uf1.get_size(i) - 1\\n\\nans = 0\\nfor i in range(n):\\n    ans += ans0[i]\\n    ans += ans1[i]\\n    ans += ans0[i] * ans1[i]\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\nn = int(input())\\nabd = [list(map(int,input().split())) for i in range(n-1)]\\ngraph = [[] for i in range(n+1)]\\ndeg = [0]*(n+1)\\nif n == 2:\\n  print(2)\\n  return\\nfor a,b,d in abd:\\n  graph[a].append((b,d))\\n  graph[b].append((a,d))\\n  deg[a] += 1\\n  deg[b] += 1\\ndp = [[0]*3 for i in range(n+1)]\\nstack = []\\nroot = 0\\nfor i in range(1,n+1):\\n  if deg[i] == 1:\\n    stack.append(i)\\n  elif not root:\\n    root = i\\n    deg[root] += 1\\nwhile stack:\\n  x = stack.pop()\\n  if dp[x][0] == dp[x][1] == 0:\\n    for y,d in graph[x]:\\n      dp[y][d] += 1\\n      deg[y] -= 1\\n      if deg[y] == 1:\\n        stack.append(y)\\n  else:\\n    for y,d in graph[x]:\\n      if deg[y] > 1:\\n        dp[y][d] += dp[x][d]+1\\n        if d == 1:\\n          dp[y][2] += dp[x][0]+dp[x][2]\\n        deg[y] -= 1\\n        if deg[y] == 1:\\n          stack.append(y)\\nstack = [root]\\ndeg[root] -= 1\\nwhile stack:\\n  x = stack.pop()\\n  for y,d in graph[x]:\\n    if deg[y] == 1:\\n      deg[y] -= 1\\n      dp[y][d] += dp[x][d]-dp[y][d]\\n      if d == 1:\\n        dp[y][2] += dp[x][2]+dp[x][0]-dp[y][0]-dp[y][2]\\n      stack.append(y)\\nans = 0\\nfor i,j,k in dp:\\n  ans += i+j+k\\nprint(ans)\", \"def addEdge(arr: [], x, y):\\n    x, y = getRoot(arr, x), getRoot(arr, y)\\n    if arr[x] <= arr[y]:\\n        arr[x] += arr[y]\\n        arr[y] = x\\n    else:\\n        arr[y] += arr[x]\\n        arr[x] = y\\n\\n\\ndef getRoot(arr: [], id):\\n    while arr[id] > 0:\\n        id = arr[id]\\n    return id\\n\\n\\ndef solve(zeroArr: [], oneArr: []):\\n    Ans = 0\\n    for i in range(1, len(zeroArr)):\\n        if zeroArr[i] < 0:\\n            Ans += zeroArr[i] * (-1) * (zeroArr[i] * (-1) - 1)\\n        if oneArr[i] < 0:\\n            Ans += oneArr[i] * (-1) * (oneArr[i] * (-1) - 1)\\n        if (zeroArr[i] != -1 and oneArr[i] != -1):\\n            one, zero = getRoot(oneArr, i), getRoot(zeroArr, i)\\n            Ans += (oneArr[one] * (-1) - 1) * (zeroArr[zero] * (-1) - 1)\\n    return Ans\\n\\nn = int(input())\\nzeroArr, oneArr = [-1] * (n + 1), [-1] * (n + 1)\\nwhile n > 1:\\n    a, b, w = [int(x) for x in input().split()]\\n    if w == 0:\\n        addEdge(zeroArr, a, b)\\n    else:\\n        addEdge(oneArr, a, b)\\n    n -= 1\\nprint(solve(zeroArr, oneArr))\\n\", \"mod = 1000000007\\neps = 10**-9\\n\\n\\ndef main():\\n    import sys\\n    from collections import deque\\n    input = sys.stdin.readline\\n\\n    class UnionFind():\\n        def __init__(self, n):\\n            self.n = n\\n            self.root = [-1] * (n + 1)\\n            self.rnk = [0] * (n + 1)\\n\\n        def find_root(self, x):\\n            while self.root[x] >= 0:\\n                x = self.root[x]\\n            return x\\n\\n        def unite(self, x, y):\\n            x = self.find_root(x)\\n            y = self.find_root(y)\\n            if x == y:\\n                return\\n            elif self.rnk[x] > self.rnk[y]:\\n                self.root[x] += self.root[y]\\n                self.root[y] = x\\n            else:\\n                self.root[y] += self.root[x]\\n                self.root[x] = y\\n                if self.rnk[x] == self.rnk[y]:\\n                    self.rnk[y] += 1\\n\\n        def isSameGroup(self, x, y):\\n            return self.find_root(x) == self.find_root(y)\\n\\n        def size(self, x):\\n            return -self.root[self.find_root(x)]\\n\\n    N = int(input())\\n    adj = [[] for _ in range(N+1)]\\n    UF0 = UnionFind(N+1)\\n    UF1 = UnionFind(N+1)\\n    for _ in range(N-1):\\n        a, b, c = list(map(int, input().split()))\\n        adj[a].append((b, c))\\n        adj[b].append((a, c))\\n        if c == 0:\\n            UF0.unite(a, b)\\n        else:\\n            UF1.unite(a, b)\\n\\n    ans = 0\\n    roots = set()\\n    for v in range(1, N+1):\\n        r = UF1.find_root(v)\\n        if r not in roots:\\n            roots.add(r)\\n            s = -UF1.root[r]\\n            ans += s * (s-1)\\n    #print(ans)\\n\\n    roots = set()\\n    for v in range(1, N+1):\\n        r = UF0.find_root(v)\\n        if r not in roots:\\n            roots.add(r)\\n            s = -UF0.root[r]\\n            ans += s * (s-1)\\n    #print(ans)\\n\\n    for v in range(1, N+1):\\n        W = 0\\n        B = 0\\n        flg0 = 0\\n        flg1 = 0\\n        for u, c in adj[v]:\\n            if flg0 and flg1:\\n                break\\n            if c == 0:\\n                if flg0:\\n                    continue\\n                W += UF0.size(u) - 1\\n                flg0 = 1\\n            else:\\n                if flg1:\\n                    continue\\n                B += UF1.size(u) - 1\\n                flg1 = 1\\n        ans += W * B\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(2*10**5)\\nI = lambda : list(map(int,input().split()))\\n\\nn,=I()\\nr=[[1 for i in range(n)],[1 for i in range(n)]]\\np=[[i for i in range(n)],[i for i in range(n)]]\\ndef find(x,c):\\n\\tif x!=p[c][x]:\\n\\t\\tp[c][x]=find(p[c][x],c)\\n\\treturn p[c][x]\\n\\ndef union(a,b,c):\\n\\tx=find(a,c)\\n\\ty=find(b,c)\\n\\tmm=min(x,y)\\n\\tif x!=y:\\n\\t\\tp[c][y]=p[c][x]=mm\\n\\t\\tr[c][mm]+=r[c][max(x,y)]\\nan=0\\nfor i in range(n-1):\\n\\ta,b,c=I()\\n\\tunion(a-1,b-1,c)\\nvis=[0]*n\\ncc=[]\\nfor i in range(n):\\n\\ts0=r[0][i]\\n\\ts1=r[1][i]\\n\\tif p[0][i]==i:\\n\\t\\tan+=(s0-1)*s0\\n\\tif p[1][i]==i:\\n\\t\\tan+=(s1-1)*s1\\n\\tan+=(r[1][find(i,1)]-1)*(r[0][find(i,0)]-1)\\nprint(an)\", \"import sys;sys.setrecursionlimit(10**9)\\nclass UnionFind:\\n  def __init__(self,n):\\n    self.n=[-1]*n\\n    self.r=[0]*n\\n    self.siz=n\\n  def find_root(self,x):\\n    if self.n[x]<0:\\n      return x\\n    else:\\n      self.n[x]=self.find_root(self.n[x])\\n      return self.n[x]\\n  def unite(self,x,y):\\n    x=self.find_root(x)\\n    y=self.find_root(y)\\n    if x==y:return\\n    elif self.r[x]>self.r[y]:\\n      self.n[x]+=self.n[y]\\n      self.n[y]=x\\n    else:\\n      self.n[y]+=self.n[x]\\n      self.n[x]=y\\n      if self.r[x]==self.r[y]:\\n        self.r[y]+=1\\n    self.siz-=1\\n  def root_same(self,x,y):\\n    return self.find_root(x)==self.find_root(y)\\n  def count(self,x):\\n    return -self.n[self.find_root(x)]\\n  def size(self):\\n    return self.siz\\n\\nn=int(input())\\nouf=UnionFind(n)\\nzuf=UnionFind(n)\\nfor _ in range(n-1):\\n  a,b,c=map(int,input().split())\\n  a-=1\\n  b-=1\\n  if c==0:\\n    zuf.unite(a,b)\\n  else:\\n    ouf.unite(a,b)\\nans=0\\nfor i in range(n):\\n  m=zuf.count(i)\\n  if zuf.find_root(i)==i:ans+=m*(m-1)\\n  mm=ouf.count(i)\\n  if ouf.find_root(i)==i:ans+=mm*(mm-1)\\n  ans+=(m-1)*(mm-1)\\nprint(ans)\", \"class UnionFindVerSize():\\n    def __init__(self, N):\\n        self._parent = [n for n in range(0, N)]\\n        self._size = [1] * N\\n\\n    def find_root(self, x):\\n        if self._parent[x] == x: return x\\n        self._parent[x] = self.find_root(self._parent[x])\\n        return self._parent[x]\\n\\n    def unite(self, x, y):\\n        gx = self.find_root(x)\\n        gy = self.find_root(y)\\n        if gx == gy: return\\n\\n        if self._size[gx] < self._size[gy]:\\n            self._parent[gx] = gy\\n            self._size[gy] += self._size[gx]\\n        else:\\n            self._parent[gy] = gx\\n            self._size[gx] += self._size[gy]\\n\\n    def get_size(self, x):\\n        return self._size[self.find_root(x)]\\n\\n    def is_same_group(self, x, y):\\n        return self.find_root(x) == self.find_root(y)\\n\\n    def calc_group_num(self):\\n        N = len(self._parent)\\n        ans = 0\\n        for i in range(N):\\n            if self.find_root(i) == i:\\n                ans += 1\\n        return ans\\n\\nimport sys\\n\\ninput=sys.stdin.readline\\nn=int(input())\\nuf0=UnionFindVerSize(n)\\nuf1=UnionFindVerSize(n)\\nfor i in range(n-1):\\n    x,y,c=map(int,input().split())\\n    if c==0:\\n        uf0.unite(x-1,y-1)\\n    else:\\n        uf1.unite(x-1,y-1)\\n\\ndata=[0]*n\\ncnt=[0]*n\\nfor i in range(n):\\n    root=uf1.find_root(i)\\n    data[root]+=uf0.get_size(i)\\n    cnt[root]+=1\\n\\nans=sum(data[i]*cnt[i] for i in range(n))\\nprint(ans-n)\"]",
        "difficulty": "interview",
        "input": "7\n2 1 1\n3 2 0\n4 2 1\n5 2 0\n6 7 1\n7 2 1\n",
        "output": "34\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1156/D"
    },
    {
        "id": 1064,
        "task_id": 2464,
        "test_case_id": 2,
        "question": "You are given a tree (an undirected connected acyclic graph) consisting of $n$ vertices and $n - 1$ edges. A number is written on each edge, each number is either $0$ (let's call such edges $0$-edges) or $1$ (those are $1$-edges).\n\nLet's call an ordered pair of vertices $(x, y)$ ($x \\ne y$) valid if, while traversing the simple path from $x$ to $y$, we never go through a $0$-edge after going through a $1$-edge. Your task is to calculate the number of valid pairs in the tree.\n\n\n-----Input-----\n\nThe first line contains one integer $n$ ($2 \\le n \\le 200000$) — the number of vertices in the tree.\n\nThen $n - 1$ lines follow, each denoting an edge of the tree. Each edge is represented by three integers $x_i$, $y_i$ and $c_i$ ($1 \\le x_i, y_i \\le n$, $0 \\le c_i \\le 1$, $x_i \\ne y_i$) — the vertices connected by this edge and the number written on it, respectively.\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the number of valid pairs of vertices.\n\n\n-----Example-----\nInput\n7\n2 1 1\n3 2 0\n4 2 1\n5 2 0\n6 7 1\n7 2 1\n\nOutput\n34\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example:\n\n[Image]",
        "solutions": "[\"class UnionFind:\\n    def __init__(self, N):\\n        self.par = [i for i in range(N)]\\n        self.rank = [1 for i in range(N)]\\n        self.rank[0] = 0\\n    def union(self, x, y):\\n        if not self.is_same_set(x, y):\\n            par_x = self.find_par(x)\\n            par_y = self.find_par(y)\\n\\n            if self.rank[par_x] > self.rank[par_y]:\\n                self.rank[par_x] += self.rank[par_y]\\n                self.rank[par_y] = 0\\n                self.par[par_y] = par_x\\n            else:\\n                self.rank[par_y] += self.rank[par_x]\\n                self.rank[par_x] = 0\\n                self.par[par_x] = par_y\\n\\n    def find_par(self, x):\\n        if self.par[x] == x: return x\\n        self.par[x] = self.find_par(self.par[x])\\n        return self.par[x]\\n\\n    def is_same_set(self, x, y):\\n        return self.find_par(x) == self.find_par(y)\\n\\n    def size(self, x):\\n        return self.rank[self.find_par(x)]\\n# 2 unionfind, para 0 e para 1 formando 2 florestas\\n# lista de adj\\n# verificar todos os componentes existentes e adicionar na resposta n * (n-1)\\n\\nn = int(input())\\n\\nadj = [[] for i in range(n+1)]\\n\\nuf0 = UnionFind(n+1)\\nuf1 = UnionFind(n+1)\\n\\nfor i in range(n-1):\\n    x, y, c = list(map(int, input().split()))\\n\\n    if c == 0:\\n        uf0.union(x, y)\\n    else:\\n        uf1.union(x, y)\\n    adj[x].append(y)\\n    adj[y].append(x)\\nfor i in range(n+1):\\n    uf0.find_par(i)\\n    uf1.find_par(i)\\n\\nresp = 0\\n\\nfor i in set(uf0.par):\\n    resp += uf0.rank[i] * (uf0.rank[i] - 1)\\nfor i in set(uf1.par):\\n    resp += uf1.rank[i] * (uf1.rank[i] - 1)\\n\\n# pra cada componente do 0-uf verificar se existe esse vertice na 1-uf e ele for conectado com algu\\u00e9m, se sim, multiplicar (n-1)*(m-1) sendo n o componente da 0-uf e m o componente da 1-f e adicionar na resposta\\n\\nfor i in range(len(uf0.par)):\\n    if uf0.rank[uf0.find_par(i)] > 1:\\n        if uf1.rank[uf1.find_par(i)] > 1:\\n            resp += (uf0.rank[uf0.find_par(i)] - 1) * (uf1.rank[uf1.find_par(i)] - 1)\\nprint(resp)\\n\", \"class UnionFind:\\n    def __init__(self, N):\\n        self.par = [i for i in range(N)]\\n        self.rank = [1 for i in range(N)]\\n        self.rank[0] = 0\\n    def union(self, x, y):\\n        if not self.is_same_set(x, y):\\n            par_x = self.find_par(x)\\n            par_y = self.find_par(y)\\n\\n            if self.rank[par_x] > self.rank[par_y]:\\n                self.rank[par_x] += self.rank[par_y]\\n                self.rank[par_y] = 0\\n                self.par[par_y] = par_x\\n            else:\\n                self.rank[par_y] += self.rank[par_x]\\n                self.rank[par_x] = 0\\n                self.par[par_x] = par_y\\n\\n    def find_par(self, x):\\n        if self.par[x] == x: return x\\n        self.par[x] = self.find_par(self.par[x])\\n        return self.par[x]\\n\\n    def is_same_set(self, x, y):\\n        return self.find_par(x) == self.find_par(y)\\n\\n    def size(self, x):\\n        return self.rank[self.find_par(x)]\\n# 2 unionfind, para 0 e para 1 formando 2 florestas\\n# lista de adj\\n# verificar todos os componentes existentes e adicionar na resposta n * (n-1)\\n\\nn = int(input())\\n\\nadj = [[] for i in range(n+1)]\\n\\nuf0 = UnionFind(n+1)\\nuf1 = UnionFind(n+1)\\n\\nfor i in range(n-1):\\n    x, y, c = list(map(int, input().split()))\\n\\n    if c == 0:\\n        uf0.union(x, y)\\n    else:\\n        uf1.union(x, y)\\n    adj[x].append(y)\\n    adj[y].append(x)\\n\\n    uf0.find_par(x)\\n    uf1.find_par(y)\\n\\nresp = 0\\n\\nresp += sum([uf0.rank[i] * (uf0.rank[i] - 1) for i in set(uf0.par)])\\nresp += sum([uf1.rank[i] * (uf1.rank[i] - 1) for i in set(uf1.par)])\\n# pra cada componente do 0-uf verificar se existe esse vertice na 1-uf e ele for conectado com algu\\u00e9m, se sim, multiplicar (n-1)*(m-1) sendo n o componente da 0-uf e m o componente da 1-f e adicionar na resposta\\n#ja_visto = set()\\nfor i in range(len(uf0.par)):\\n    if uf0.rank[uf0.find_par(i)] > 1: #and not uf0.find_par(i) in ja_visto:\\n        #ja_visto.add(uf0.find_par(i))\\n        if uf1.rank[uf1.find_par(i)] > 1:\\n            resp += (uf0.rank[uf0.find_par(i)] - 1) * (uf1.rank[uf1.find_par(i)] - 1)\\nprint(resp)\\n\", \"from collections import deque\\n\\ndef bfs(source, graph, mark, num, fcount):\\n\\tvisited = [source]\\n\\tq = deque()\\n\\tmark[source] = True\\n\\tq.append(source)\\n\\twhile q:\\n\\t\\tu = q.popleft()\\n\\t\\tfor v, c in g[u]:\\n\\t\\t\\tif c == num and not mark[v]:\\n\\t\\t\\t\\tmark[v] = True\\n\\t\\t\\t\\tvisited.append(v)\\n\\t\\t\\t\\tq.append(v)\\n\\tif len(visited) > 1:\\n\\t\\tfor u in visited:\\n\\t\\t\\tfcount[u] = len(visited)\\n\\nn = int(input())\\nedges = [tuple(map(int, input().split())) for _ in range(0, n - 1)]\\ng = [[] for _ in range(0, n)]\\ncnt = [[0 for _ in range(0, n)] for _ in range(0, 2)]\\nfor u, v, c in edges:\\n\\tg[u - 1].append((v - 1, c))\\n\\tg[v - 1].append((u - 1, c))\\n\\nres = 0\\nfor link in range(0, 2):\\n\\tmark = [False] * n\\n\\tfor u in range(0, n):\\n\\t\\tif not mark[u]:\\n\\t\\t\\tbfs(u, g, mark, link, cnt[link])\\n\\t\\t\\tres += cnt[link][u] * (cnt[link][u] - 1)\\n\\nfor i in range(0, n):\\n\\tif cnt[0][i] > 0 and cnt[1][i] > 1:\\n\\t\\tres += (cnt[0][i] - 1) * (cnt[1][i] - 1)\\n\\nprint(int(res))\", \"class Deque:\\n\\tdef __init__(self):\\n\\t\\tself.buff = [0] * 400000\\n\\t\\tself.l = 0\\n\\t\\tself.r = 0\\n\\tdef append(self, x):\\n\\t\\tself.buff[self.r] = x\\n\\t\\tself.r = self.r + 1\\n\\tdef popleft(self):\\n\\t\\told_left = self.l\\n\\t\\tself.l = self.l + 1\\n\\t\\treturn self.buff[old_left]\\n\\tdef __bool__(self):\\n\\t\\treturn self.l != self.r\\n\\nq = Deque()\\n\\ndef bfs(source, graph, mark, num, fcount):\\n\\tvisited = [source]\\n\\tmark[source] = True\\n\\tq.append(source)\\n\\twhile q:\\n\\t\\tu = q.popleft()\\n\\t\\tfor v, c in g[u]:\\n\\t\\t\\tif c == num and not mark[v]:\\n\\t\\t\\t\\tmark[v] = True\\n\\t\\t\\t\\tvisited.append(v)\\n\\t\\t\\t\\tq.append(v)\\n\\tif len(visited) > 1:\\n\\t\\tfor u in visited:\\n\\t\\t\\tfcount[u] = len(visited)\\n\\nn = int(input())\\nedges = [tuple(map(int, input().split())) for _ in range(0, n - 1)]\\ng = [[] for _ in range(0, n)]\\ncnt = [[0 for _ in range(0, n)] for _ in range(0, 2)]\\nfor u, v, c in edges:\\n\\tg[u - 1].append((v - 1, c))\\n\\tg[v - 1].append((u - 1, c))\\n\\nres = 0\\nfor link in range(0, 2):\\n\\tmark = [False] * n\\n\\tfor u in range(0, n):\\n\\t\\tif not mark[u]:\\n\\t\\t\\tbfs(u, g, mark, link, cnt[link])\\n\\t\\t\\tres += cnt[link][u] * (cnt[link][u] - 1)\\n\\nfor i in range(0, n):\\n\\tif cnt[0][i] > 0 and cnt[1][i] > 1:\\n\\t\\tres += (cnt[0][i] - 1) * (cnt[1][i] - 1)\\n\\nprint(int(res))\", \"from collections import defaultdict\\n\\ndef colour(a, graph, cur, i):\\n    top = set()\\n    top.add(i)\\n    while len(top):\\n        x = top.pop()\\n        a[x] = cur\\n        for y in graph[x]:\\n            if a[y] == 0:\\n                top.add(y)\\n\\ndef colour_graph(a, graph, n):\\n    cur = 0\\n    for i in range(1, n + 1):\\n        if a[i] or i not in graph:\\n            continue\\n        else:\\n            cur += 1\\n            colour(a, graph, cur, i)\\n\\ndef count(col):\\n    ans = 0\\n    for el in col:\\n        if col[el] > 1:\\n            ans += col[el]*(col[el] - 1)\\n    return ans\\n\\nn = int(input())\\n\\n\\ngraph0 = defaultdict(set)\\ngraph1 = defaultdict(set)\\n\\n\\nvertex0 = set()\\n\\na0 = [0]*(n + 1)  \\na1 = [0]*(n + 1)\\n\\nfor i in range(n - 1):\\n    x, y, c = list(map(int, input().split()))\\n    if c == 0:\\n        graph0[x].add(y)\\n        graph0[y].add(x)\\n        vertex0.add(x)\\n        vertex0.add(y)\\n    else:\\n        graph1[x].add(y)\\n        graph1[y].add(x)\\n\\n\\ncolour_graph(a0, graph0, n)\\ncolour_graph(a1, graph1, n)\\n\\nanswer = 0\\ncol0 = defaultdict(int)\\ncol1 = defaultdict(int)\\n\\nfor i in range(n + 1):\\n    if a0[i]:\\n        col0[a0[i]] += 1\\n    if a1[i]:\\n        col1[a1[i]] += 1\\n\\nanswer += count(col0) + count(col1)\\n\\ncol = defaultdict(int)\\nfor v in vertex0:\\n    col[a1[v]] += col0[a0[v]] - 1\\nfor el in col:\\n    if el:\\n        answer += col[el]*(col1[el] - 1)\\n        \\nprint(answer)\\n\", \"class dsu:\\n\\tdef __init__(self,n):\\n\\t\\tself.arr = [i for i in range(n)]\\n\\t\\tself.size = [1 for i in range(n)]\\n\\t\\tself.s = n\\n\\tdef find(self,i):\\n\\t\\twhile(i!=self.arr[i]):\\n\\t\\t\\ti = self.arr[i]\\n\\t\\treturn i\\n\\t\\tif(self.arr[i]!=i):\\n\\t\\t\\tself.arr[i] = self.find(self.arr[i])\\n\\t\\treturn self.arr[i]\\n\\tdef union(self,i,j):\\n\\t\\tfa = self.find(i)\\n\\t\\tfb = self.find(j)\\n\\t\\tif(fa == fb):\\n\\t\\t\\treturn\\n\\t\\ts1 = self.size[fa]\\n\\t\\ts2 = self.size[fb]\\n\\t\\tif(s1<s2):\\n\\t\\t\\tself.arr[fa] = fb\\n\\t\\t\\tself.size[fb] += self.size[fa]\\n\\t\\telse:\\n\\t\\t\\tself.arr[fb] = fa\\n\\t\\t\\tself.size[fa] += self.size[fb]\\nn = int(input())\\nzero = dsu(n)\\none = dsu(n)\\nfor i in range(n-1):\\n\\ta = [int(i) for i in input().split(' ')]\\n\\tx = a[0]-1\\n\\ty = a[1]-1\\n\\tc = a[2]\\n\\tif(c==0):\\n\\t\\tzero.union(x,y)\\n\\telse:\\n\\t\\tone.union(x,y)\\ncount = 0\\nfor i in range(n):\\n\\tif(zero.arr[i] == i):\\n\\t\\tcount+=zero.size[i]*(zero.size[i]-1)\\n\\tif(one.arr[i] == i):\\n\\t\\tcount+=one.size[i]*(one.size[i]-1)\\n\\tcount += (zero.size[zero.find(i)]-1)*(one.size[one.find(i)]-1)\\nprint(count)\", \"import sys\\n\\nclass Disjoint:\\n    def __init__(self, n):\\n        self.n = n\\n        self.size = [1] * self.n\\n        self.parent = [i for i in range(self.n)]\\n    \\n    def root(self, node):\\n        self.parent[node] = node if self.parent[node] == node else self.root(self.parent[node])\\n        return self.parent[node]\\n    \\n    def join(self, a, b):\\n        a = self.root(a)\\n        b = self.root(b)\\n        if a == b:\\n            return False\\n        if self.size[a] > self.size[b]:\\n            a, b = b, a\\n            \\n        self.size[b] += self.size[a]\\n        self.parent[a] = b\\n        return True\\n    \\n    \\n\\ninp = [int(x) for x in sys.stdin.read().split()]\\nn = inp[0]\\n\\nwhite = Disjoint(n)\\nblack = Disjoint(n)\\n\\ninp_idx = 1\\nfor i in range(n - 1):\\n    x, y, c = inp[inp_idx], inp[inp_idx + 1], inp[inp_idx + 2]\\n    x -= 1\\n    y -= 1\\n    inp_idx += 3\\n    \\n    if c == 0:\\n        white.join(x, y)\\n    else:\\n        black.join(x, y)\\n    \\nans = 0\\nfor i in range(n):\\n    rootW = white.root(i)\\n    rootB = black.root(i)\\n    ans += white.size[rootW] - 1\\n    ans += black.size[rootB] - 1\\n    \\n    ans += (white.size[rootW] - 1) * (black.size[rootB] - 1)\\n    \\nprint(ans)\\n\", \"class UnionFind():\\n\\n    def __init__(self, n):\\n        self.n = n\\n\\n        self.root = [-1]*(n+1)\\n\\n        self.rnk = [0]*(n+1)\\n\\n\\n    def Find_Root(self, x):\\n        if(self.root[x] < 0):\\n            return x\\n        else:\\n\\n            self.root[x] = self.Find_Root(self.root[x])\\n            return self.root[x]\\n\\n    def Unite(self, x, y):\\n\\n        x = self.Find_Root(x)\\n        y = self.Find_Root(y)\\n\\n        if(x == y):\\n            return \\n\\n        elif(self.rnk[x] > self.rnk[y]):\\n            self.root[x] += self.root[y]\\n            self.root[y] = x\\n\\n        else:\\n            self.root[y] += self.root[x]\\n            self.root[x] = y\\n\\n            if(self.rnk[x] == self.rnk[y]):\\n                self.rnk[y] += 1\\n\\n    def isSameGroup(self, x, y):\\n        return self.Find_Root(x) == self.Find_Root(y)\\n\\n\\n    def Count(self, x):\\n        return -self.root[self.Find_Root(x)]\\n\\nimport sys\\ninput = sys.stdin.readline\\n\\nN = int(input())\\nuni0 = UnionFind(N)\\nuni1 = UnionFind(N)\\nfor _ in range(N-1):\\n    a, b, c = map(int, input().split())\\n    if c == 0:\\n        uni0.Unite(a-1, b-1)\\n    else:\\n        uni1.Unite(a-1, b-1)\\n\\ng0 = {}\\ng1 = {}\\nfor i in range(N):\\n    if uni0.Count(i) != 1:\\n        r = uni0.Find_Root(i)\\n        if not r in g0.keys():\\n            g0[r] = [i]\\n        else:\\n            g0[r].append(i)\\n    if uni1.Count(i) != 1:\\n        r = uni1.Find_Root(i)\\n        if not r in g1.keys():\\n            g1[r] = [i]\\n        else:\\n            g1[r].append(i)\\n\\nans = 0\\nfor v_list in g1.values():\\n    c = 0\\n    for n in v_list:\\n        if uni0.Count(n) == 1:\\n            c += 1\\n    l = len(v_list)\\n    ans += c*(l-1)\\n\\nfor v_list in g0.values():\\n    c = 0\\n    for n in v_list:\\n        if uni1.Count(n) != 1:\\n            r = uni1.Find_Root(n)\\n            c += len(g1[r])-1\\n    c += len(v_list)-1\\n    ans += len(v_list)*c\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.parent = [-1] * n\\n        self.cnt = n\\n\\n    def root(self, x):\\n        if self.parent[x] < 0:\\n            return x\\n        else:\\n            self.parent[x] = self.root(self.parent[x])\\n            return self.parent[x]\\n\\n    def merge(self, x, y):\\n        x = self.root(x)\\n        y = self.root(y)\\n        if x == y:\\n            return\\n        if self.parent[x] > self.parent[y]:\\n            x, y = y, x\\n        self.parent[x] += self.parent[y]\\n        self.parent[y] = x\\n        self.cnt -= 1\\n        \\n    def is_same(self, x, y):\\n        return self.root(x) == self.root(y)\\n\\n    def get_size(self, x):\\n        return -self.parent[self.root(x)]\\n    \\n    def get_cnt(self):\\n        return self.cnt\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\nuf0 = UnionFind(n)\\nuf1 = UnionFind(n)\\ntree0 = [[] for i in range(n)]\\ntree1 = [[] for i in range(n)]\\nfor i in range(n - 1):\\n    a, b, cost = info[i]\\n    a -= 1\\n    b -= 1\\n    if cost == 0:\\n        tree0[a].append(b)\\n        tree0[b].append(a)\\n        uf0.merge(a, b)\\n    else:\\n        tree1[a].append(b)\\n        tree1[b].append(a)\\n        uf1.merge(a, b)\\n\\nans0 = [0] * n\\nans1 = [0] * n\\nfor i in range(n):\\n    ans0[i] = uf0.get_size(i) - 1\\n    ans1[i] = uf1.get_size(i) - 1\\n\\nans = 0\\nfor i in range(n):\\n    ans += ans0[i]\\n    ans += ans1[i]\\n    ans += ans0[i] * ans1[i]\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\nn = int(input())\\nabd = [list(map(int,input().split())) for i in range(n-1)]\\ngraph = [[] for i in range(n+1)]\\ndeg = [0]*(n+1)\\nif n == 2:\\n  print(2)\\n  return\\nfor a,b,d in abd:\\n  graph[a].append((b,d))\\n  graph[b].append((a,d))\\n  deg[a] += 1\\n  deg[b] += 1\\ndp = [[0]*3 for i in range(n+1)]\\nstack = []\\nroot = 0\\nfor i in range(1,n+1):\\n  if deg[i] == 1:\\n    stack.append(i)\\n  elif not root:\\n    root = i\\n    deg[root] += 1\\nwhile stack:\\n  x = stack.pop()\\n  if dp[x][0] == dp[x][1] == 0:\\n    for y,d in graph[x]:\\n      dp[y][d] += 1\\n      deg[y] -= 1\\n      if deg[y] == 1:\\n        stack.append(y)\\n  else:\\n    for y,d in graph[x]:\\n      if deg[y] > 1:\\n        dp[y][d] += dp[x][d]+1\\n        if d == 1:\\n          dp[y][2] += dp[x][0]+dp[x][2]\\n        deg[y] -= 1\\n        if deg[y] == 1:\\n          stack.append(y)\\nstack = [root]\\ndeg[root] -= 1\\nwhile stack:\\n  x = stack.pop()\\n  for y,d in graph[x]:\\n    if deg[y] == 1:\\n      deg[y] -= 1\\n      dp[y][d] += dp[x][d]-dp[y][d]\\n      if d == 1:\\n        dp[y][2] += dp[x][2]+dp[x][0]-dp[y][0]-dp[y][2]\\n      stack.append(y)\\nans = 0\\nfor i,j,k in dp:\\n  ans += i+j+k\\nprint(ans)\", \"def addEdge(arr: [], x, y):\\n    x, y = getRoot(arr, x), getRoot(arr, y)\\n    if arr[x] <= arr[y]:\\n        arr[x] += arr[y]\\n        arr[y] = x\\n    else:\\n        arr[y] += arr[x]\\n        arr[x] = y\\n\\n\\ndef getRoot(arr: [], id):\\n    while arr[id] > 0:\\n        id = arr[id]\\n    return id\\n\\n\\ndef solve(zeroArr: [], oneArr: []):\\n    Ans = 0\\n    for i in range(1, len(zeroArr)):\\n        if zeroArr[i] < 0:\\n            Ans += zeroArr[i] * (-1) * (zeroArr[i] * (-1) - 1)\\n        if oneArr[i] < 0:\\n            Ans += oneArr[i] * (-1) * (oneArr[i] * (-1) - 1)\\n        if (zeroArr[i] != -1 and oneArr[i] != -1):\\n            one, zero = getRoot(oneArr, i), getRoot(zeroArr, i)\\n            Ans += (oneArr[one] * (-1) - 1) * (zeroArr[zero] * (-1) - 1)\\n    return Ans\\n\\nn = int(input())\\nzeroArr, oneArr = [-1] * (n + 1), [-1] * (n + 1)\\nwhile n > 1:\\n    a, b, w = [int(x) for x in input().split()]\\n    if w == 0:\\n        addEdge(zeroArr, a, b)\\n    else:\\n        addEdge(oneArr, a, b)\\n    n -= 1\\nprint(solve(zeroArr, oneArr))\\n\", \"mod = 1000000007\\neps = 10**-9\\n\\n\\ndef main():\\n    import sys\\n    from collections import deque\\n    input = sys.stdin.readline\\n\\n    class UnionFind():\\n        def __init__(self, n):\\n            self.n = n\\n            self.root = [-1] * (n + 1)\\n            self.rnk = [0] * (n + 1)\\n\\n        def find_root(self, x):\\n            while self.root[x] >= 0:\\n                x = self.root[x]\\n            return x\\n\\n        def unite(self, x, y):\\n            x = self.find_root(x)\\n            y = self.find_root(y)\\n            if x == y:\\n                return\\n            elif self.rnk[x] > self.rnk[y]:\\n                self.root[x] += self.root[y]\\n                self.root[y] = x\\n            else:\\n                self.root[y] += self.root[x]\\n                self.root[x] = y\\n                if self.rnk[x] == self.rnk[y]:\\n                    self.rnk[y] += 1\\n\\n        def isSameGroup(self, x, y):\\n            return self.find_root(x) == self.find_root(y)\\n\\n        def size(self, x):\\n            return -self.root[self.find_root(x)]\\n\\n    N = int(input())\\n    adj = [[] for _ in range(N+1)]\\n    UF0 = UnionFind(N+1)\\n    UF1 = UnionFind(N+1)\\n    for _ in range(N-1):\\n        a, b, c = list(map(int, input().split()))\\n        adj[a].append((b, c))\\n        adj[b].append((a, c))\\n        if c == 0:\\n            UF0.unite(a, b)\\n        else:\\n            UF1.unite(a, b)\\n\\n    ans = 0\\n    roots = set()\\n    for v in range(1, N+1):\\n        r = UF1.find_root(v)\\n        if r not in roots:\\n            roots.add(r)\\n            s = -UF1.root[r]\\n            ans += s * (s-1)\\n    #print(ans)\\n\\n    roots = set()\\n    for v in range(1, N+1):\\n        r = UF0.find_root(v)\\n        if r not in roots:\\n            roots.add(r)\\n            s = -UF0.root[r]\\n            ans += s * (s-1)\\n    #print(ans)\\n\\n    for v in range(1, N+1):\\n        W = 0\\n        B = 0\\n        flg0 = 0\\n        flg1 = 0\\n        for u, c in adj[v]:\\n            if flg0 and flg1:\\n                break\\n            if c == 0:\\n                if flg0:\\n                    continue\\n                W += UF0.size(u) - 1\\n                flg0 = 1\\n            else:\\n                if flg1:\\n                    continue\\n                B += UF1.size(u) - 1\\n                flg1 = 1\\n        ans += W * B\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(2*10**5)\\nI = lambda : list(map(int,input().split()))\\n\\nn,=I()\\nr=[[1 for i in range(n)],[1 for i in range(n)]]\\np=[[i for i in range(n)],[i for i in range(n)]]\\ndef find(x,c):\\n\\tif x!=p[c][x]:\\n\\t\\tp[c][x]=find(p[c][x],c)\\n\\treturn p[c][x]\\n\\ndef union(a,b,c):\\n\\tx=find(a,c)\\n\\ty=find(b,c)\\n\\tmm=min(x,y)\\n\\tif x!=y:\\n\\t\\tp[c][y]=p[c][x]=mm\\n\\t\\tr[c][mm]+=r[c][max(x,y)]\\nan=0\\nfor i in range(n-1):\\n\\ta,b,c=I()\\n\\tunion(a-1,b-1,c)\\nvis=[0]*n\\ncc=[]\\nfor i in range(n):\\n\\ts0=r[0][i]\\n\\ts1=r[1][i]\\n\\tif p[0][i]==i:\\n\\t\\tan+=(s0-1)*s0\\n\\tif p[1][i]==i:\\n\\t\\tan+=(s1-1)*s1\\n\\tan+=(r[1][find(i,1)]-1)*(r[0][find(i,0)]-1)\\nprint(an)\", \"import sys;sys.setrecursionlimit(10**9)\\nclass UnionFind:\\n  def __init__(self,n):\\n    self.n=[-1]*n\\n    self.r=[0]*n\\n    self.siz=n\\n  def find_root(self,x):\\n    if self.n[x]<0:\\n      return x\\n    else:\\n      self.n[x]=self.find_root(self.n[x])\\n      return self.n[x]\\n  def unite(self,x,y):\\n    x=self.find_root(x)\\n    y=self.find_root(y)\\n    if x==y:return\\n    elif self.r[x]>self.r[y]:\\n      self.n[x]+=self.n[y]\\n      self.n[y]=x\\n    else:\\n      self.n[y]+=self.n[x]\\n      self.n[x]=y\\n      if self.r[x]==self.r[y]:\\n        self.r[y]+=1\\n    self.siz-=1\\n  def root_same(self,x,y):\\n    return self.find_root(x)==self.find_root(y)\\n  def count(self,x):\\n    return -self.n[self.find_root(x)]\\n  def size(self):\\n    return self.siz\\n\\nn=int(input())\\nouf=UnionFind(n)\\nzuf=UnionFind(n)\\nfor _ in range(n-1):\\n  a,b,c=map(int,input().split())\\n  a-=1\\n  b-=1\\n  if c==0:\\n    zuf.unite(a,b)\\n  else:\\n    ouf.unite(a,b)\\nans=0\\nfor i in range(n):\\n  m=zuf.count(i)\\n  if zuf.find_root(i)==i:ans+=m*(m-1)\\n  mm=ouf.count(i)\\n  if ouf.find_root(i)==i:ans+=mm*(mm-1)\\n  ans+=(m-1)*(mm-1)\\nprint(ans)\", \"class UnionFindVerSize():\\n    def __init__(self, N):\\n        self._parent = [n for n in range(0, N)]\\n        self._size = [1] * N\\n\\n    def find_root(self, x):\\n        if self._parent[x] == x: return x\\n        self._parent[x] = self.find_root(self._parent[x])\\n        return self._parent[x]\\n\\n    def unite(self, x, y):\\n        gx = self.find_root(x)\\n        gy = self.find_root(y)\\n        if gx == gy: return\\n\\n        if self._size[gx] < self._size[gy]:\\n            self._parent[gx] = gy\\n            self._size[gy] += self._size[gx]\\n        else:\\n            self._parent[gy] = gx\\n            self._size[gx] += self._size[gy]\\n\\n    def get_size(self, x):\\n        return self._size[self.find_root(x)]\\n\\n    def is_same_group(self, x, y):\\n        return self.find_root(x) == self.find_root(y)\\n\\n    def calc_group_num(self):\\n        N = len(self._parent)\\n        ans = 0\\n        for i in range(N):\\n            if self.find_root(i) == i:\\n                ans += 1\\n        return ans\\n\\nimport sys\\n\\ninput=sys.stdin.readline\\nn=int(input())\\nuf0=UnionFindVerSize(n)\\nuf1=UnionFindVerSize(n)\\nfor i in range(n-1):\\n    x,y,c=map(int,input().split())\\n    if c==0:\\n        uf0.unite(x-1,y-1)\\n    else:\\n        uf1.unite(x-1,y-1)\\n\\ndata=[0]*n\\ncnt=[0]*n\\nfor i in range(n):\\n    root=uf1.find_root(i)\\n    data[root]+=uf0.get_size(i)\\n    cnt[root]+=1\\n\\nans=sum(data[i]*cnt[i] for i in range(n))\\nprint(ans-n)\"]",
        "difficulty": "interview",
        "input": "2\n1 2 1\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1156/D"
    },
    {
        "id": 1065,
        "task_id": 2464,
        "test_case_id": 3,
        "question": "You are given a tree (an undirected connected acyclic graph) consisting of $n$ vertices and $n - 1$ edges. A number is written on each edge, each number is either $0$ (let's call such edges $0$-edges) or $1$ (those are $1$-edges).\n\nLet's call an ordered pair of vertices $(x, y)$ ($x \\ne y$) valid if, while traversing the simple path from $x$ to $y$, we never go through a $0$-edge after going through a $1$-edge. Your task is to calculate the number of valid pairs in the tree.\n\n\n-----Input-----\n\nThe first line contains one integer $n$ ($2 \\le n \\le 200000$) — the number of vertices in the tree.\n\nThen $n - 1$ lines follow, each denoting an edge of the tree. Each edge is represented by three integers $x_i$, $y_i$ and $c_i$ ($1 \\le x_i, y_i \\le n$, $0 \\le c_i \\le 1$, $x_i \\ne y_i$) — the vertices connected by this edge and the number written on it, respectively.\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the number of valid pairs of vertices.\n\n\n-----Example-----\nInput\n7\n2 1 1\n3 2 0\n4 2 1\n5 2 0\n6 7 1\n7 2 1\n\nOutput\n34\n\n\n\n-----Note-----\n\nThe picture corresponding to the first example:\n\n[Image]",
        "solutions": "[\"class UnionFind:\\n    def __init__(self, N):\\n        self.par = [i for i in range(N)]\\n        self.rank = [1 for i in range(N)]\\n        self.rank[0] = 0\\n    def union(self, x, y):\\n        if not self.is_same_set(x, y):\\n            par_x = self.find_par(x)\\n            par_y = self.find_par(y)\\n\\n            if self.rank[par_x] > self.rank[par_y]:\\n                self.rank[par_x] += self.rank[par_y]\\n                self.rank[par_y] = 0\\n                self.par[par_y] = par_x\\n            else:\\n                self.rank[par_y] += self.rank[par_x]\\n                self.rank[par_x] = 0\\n                self.par[par_x] = par_y\\n\\n    def find_par(self, x):\\n        if self.par[x] == x: return x\\n        self.par[x] = self.find_par(self.par[x])\\n        return self.par[x]\\n\\n    def is_same_set(self, x, y):\\n        return self.find_par(x) == self.find_par(y)\\n\\n    def size(self, x):\\n        return self.rank[self.find_par(x)]\\n# 2 unionfind, para 0 e para 1 formando 2 florestas\\n# lista de adj\\n# verificar todos os componentes existentes e adicionar na resposta n * (n-1)\\n\\nn = int(input())\\n\\nadj = [[] for i in range(n+1)]\\n\\nuf0 = UnionFind(n+1)\\nuf1 = UnionFind(n+1)\\n\\nfor i in range(n-1):\\n    x, y, c = list(map(int, input().split()))\\n\\n    if c == 0:\\n        uf0.union(x, y)\\n    else:\\n        uf1.union(x, y)\\n    adj[x].append(y)\\n    adj[y].append(x)\\nfor i in range(n+1):\\n    uf0.find_par(i)\\n    uf1.find_par(i)\\n\\nresp = 0\\n\\nfor i in set(uf0.par):\\n    resp += uf0.rank[i] * (uf0.rank[i] - 1)\\nfor i in set(uf1.par):\\n    resp += uf1.rank[i] * (uf1.rank[i] - 1)\\n\\n# pra cada componente do 0-uf verificar se existe esse vertice na 1-uf e ele for conectado com algu\\u00e9m, se sim, multiplicar (n-1)*(m-1) sendo n o componente da 0-uf e m o componente da 1-f e adicionar na resposta\\n\\nfor i in range(len(uf0.par)):\\n    if uf0.rank[uf0.find_par(i)] > 1:\\n        if uf1.rank[uf1.find_par(i)] > 1:\\n            resp += (uf0.rank[uf0.find_par(i)] - 1) * (uf1.rank[uf1.find_par(i)] - 1)\\nprint(resp)\\n\", \"class UnionFind:\\n    def __init__(self, N):\\n        self.par = [i for i in range(N)]\\n        self.rank = [1 for i in range(N)]\\n        self.rank[0] = 0\\n    def union(self, x, y):\\n        if not self.is_same_set(x, y):\\n            par_x = self.find_par(x)\\n            par_y = self.find_par(y)\\n\\n            if self.rank[par_x] > self.rank[par_y]:\\n                self.rank[par_x] += self.rank[par_y]\\n                self.rank[par_y] = 0\\n                self.par[par_y] = par_x\\n            else:\\n                self.rank[par_y] += self.rank[par_x]\\n                self.rank[par_x] = 0\\n                self.par[par_x] = par_y\\n\\n    def find_par(self, x):\\n        if self.par[x] == x: return x\\n        self.par[x] = self.find_par(self.par[x])\\n        return self.par[x]\\n\\n    def is_same_set(self, x, y):\\n        return self.find_par(x) == self.find_par(y)\\n\\n    def size(self, x):\\n        return self.rank[self.find_par(x)]\\n# 2 unionfind, para 0 e para 1 formando 2 florestas\\n# lista de adj\\n# verificar todos os componentes existentes e adicionar na resposta n * (n-1)\\n\\nn = int(input())\\n\\nadj = [[] for i in range(n+1)]\\n\\nuf0 = UnionFind(n+1)\\nuf1 = UnionFind(n+1)\\n\\nfor i in range(n-1):\\n    x, y, c = list(map(int, input().split()))\\n\\n    if c == 0:\\n        uf0.union(x, y)\\n    else:\\n        uf1.union(x, y)\\n    adj[x].append(y)\\n    adj[y].append(x)\\n\\n    uf0.find_par(x)\\n    uf1.find_par(y)\\n\\nresp = 0\\n\\nresp += sum([uf0.rank[i] * (uf0.rank[i] - 1) for i in set(uf0.par)])\\nresp += sum([uf1.rank[i] * (uf1.rank[i] - 1) for i in set(uf1.par)])\\n# pra cada componente do 0-uf verificar se existe esse vertice na 1-uf e ele for conectado com algu\\u00e9m, se sim, multiplicar (n-1)*(m-1) sendo n o componente da 0-uf e m o componente da 1-f e adicionar na resposta\\n#ja_visto = set()\\nfor i in range(len(uf0.par)):\\n    if uf0.rank[uf0.find_par(i)] > 1: #and not uf0.find_par(i) in ja_visto:\\n        #ja_visto.add(uf0.find_par(i))\\n        if uf1.rank[uf1.find_par(i)] > 1:\\n            resp += (uf0.rank[uf0.find_par(i)] - 1) * (uf1.rank[uf1.find_par(i)] - 1)\\nprint(resp)\\n\", \"from collections import deque\\n\\ndef bfs(source, graph, mark, num, fcount):\\n\\tvisited = [source]\\n\\tq = deque()\\n\\tmark[source] = True\\n\\tq.append(source)\\n\\twhile q:\\n\\t\\tu = q.popleft()\\n\\t\\tfor v, c in g[u]:\\n\\t\\t\\tif c == num and not mark[v]:\\n\\t\\t\\t\\tmark[v] = True\\n\\t\\t\\t\\tvisited.append(v)\\n\\t\\t\\t\\tq.append(v)\\n\\tif len(visited) > 1:\\n\\t\\tfor u in visited:\\n\\t\\t\\tfcount[u] = len(visited)\\n\\nn = int(input())\\nedges = [tuple(map(int, input().split())) for _ in range(0, n - 1)]\\ng = [[] for _ in range(0, n)]\\ncnt = [[0 for _ in range(0, n)] for _ in range(0, 2)]\\nfor u, v, c in edges:\\n\\tg[u - 1].append((v - 1, c))\\n\\tg[v - 1].append((u - 1, c))\\n\\nres = 0\\nfor link in range(0, 2):\\n\\tmark = [False] * n\\n\\tfor u in range(0, n):\\n\\t\\tif not mark[u]:\\n\\t\\t\\tbfs(u, g, mark, link, cnt[link])\\n\\t\\t\\tres += cnt[link][u] * (cnt[link][u] - 1)\\n\\nfor i in range(0, n):\\n\\tif cnt[0][i] > 0 and cnt[1][i] > 1:\\n\\t\\tres += (cnt[0][i] - 1) * (cnt[1][i] - 1)\\n\\nprint(int(res))\", \"class Deque:\\n\\tdef __init__(self):\\n\\t\\tself.buff = [0] * 400000\\n\\t\\tself.l = 0\\n\\t\\tself.r = 0\\n\\tdef append(self, x):\\n\\t\\tself.buff[self.r] = x\\n\\t\\tself.r = self.r + 1\\n\\tdef popleft(self):\\n\\t\\told_left = self.l\\n\\t\\tself.l = self.l + 1\\n\\t\\treturn self.buff[old_left]\\n\\tdef __bool__(self):\\n\\t\\treturn self.l != self.r\\n\\nq = Deque()\\n\\ndef bfs(source, graph, mark, num, fcount):\\n\\tvisited = [source]\\n\\tmark[source] = True\\n\\tq.append(source)\\n\\twhile q:\\n\\t\\tu = q.popleft()\\n\\t\\tfor v, c in g[u]:\\n\\t\\t\\tif c == num and not mark[v]:\\n\\t\\t\\t\\tmark[v] = True\\n\\t\\t\\t\\tvisited.append(v)\\n\\t\\t\\t\\tq.append(v)\\n\\tif len(visited) > 1:\\n\\t\\tfor u in visited:\\n\\t\\t\\tfcount[u] = len(visited)\\n\\nn = int(input())\\nedges = [tuple(map(int, input().split())) for _ in range(0, n - 1)]\\ng = [[] for _ in range(0, n)]\\ncnt = [[0 for _ in range(0, n)] for _ in range(0, 2)]\\nfor u, v, c in edges:\\n\\tg[u - 1].append((v - 1, c))\\n\\tg[v - 1].append((u - 1, c))\\n\\nres = 0\\nfor link in range(0, 2):\\n\\tmark = [False] * n\\n\\tfor u in range(0, n):\\n\\t\\tif not mark[u]:\\n\\t\\t\\tbfs(u, g, mark, link, cnt[link])\\n\\t\\t\\tres += cnt[link][u] * (cnt[link][u] - 1)\\n\\nfor i in range(0, n):\\n\\tif cnt[0][i] > 0 and cnt[1][i] > 1:\\n\\t\\tres += (cnt[0][i] - 1) * (cnt[1][i] - 1)\\n\\nprint(int(res))\", \"from collections import defaultdict\\n\\ndef colour(a, graph, cur, i):\\n    top = set()\\n    top.add(i)\\n    while len(top):\\n        x = top.pop()\\n        a[x] = cur\\n        for y in graph[x]:\\n            if a[y] == 0:\\n                top.add(y)\\n\\ndef colour_graph(a, graph, n):\\n    cur = 0\\n    for i in range(1, n + 1):\\n        if a[i] or i not in graph:\\n            continue\\n        else:\\n            cur += 1\\n            colour(a, graph, cur, i)\\n\\ndef count(col):\\n    ans = 0\\n    for el in col:\\n        if col[el] > 1:\\n            ans += col[el]*(col[el] - 1)\\n    return ans\\n\\nn = int(input())\\n\\n\\ngraph0 = defaultdict(set)\\ngraph1 = defaultdict(set)\\n\\n\\nvertex0 = set()\\n\\na0 = [0]*(n + 1)  \\na1 = [0]*(n + 1)\\n\\nfor i in range(n - 1):\\n    x, y, c = list(map(int, input().split()))\\n    if c == 0:\\n        graph0[x].add(y)\\n        graph0[y].add(x)\\n        vertex0.add(x)\\n        vertex0.add(y)\\n    else:\\n        graph1[x].add(y)\\n        graph1[y].add(x)\\n\\n\\ncolour_graph(a0, graph0, n)\\ncolour_graph(a1, graph1, n)\\n\\nanswer = 0\\ncol0 = defaultdict(int)\\ncol1 = defaultdict(int)\\n\\nfor i in range(n + 1):\\n    if a0[i]:\\n        col0[a0[i]] += 1\\n    if a1[i]:\\n        col1[a1[i]] += 1\\n\\nanswer += count(col0) + count(col1)\\n\\ncol = defaultdict(int)\\nfor v in vertex0:\\n    col[a1[v]] += col0[a0[v]] - 1\\nfor el in col:\\n    if el:\\n        answer += col[el]*(col1[el] - 1)\\n        \\nprint(answer)\\n\", \"class dsu:\\n\\tdef __init__(self,n):\\n\\t\\tself.arr = [i for i in range(n)]\\n\\t\\tself.size = [1 for i in range(n)]\\n\\t\\tself.s = n\\n\\tdef find(self,i):\\n\\t\\twhile(i!=self.arr[i]):\\n\\t\\t\\ti = self.arr[i]\\n\\t\\treturn i\\n\\t\\tif(self.arr[i]!=i):\\n\\t\\t\\tself.arr[i] = self.find(self.arr[i])\\n\\t\\treturn self.arr[i]\\n\\tdef union(self,i,j):\\n\\t\\tfa = self.find(i)\\n\\t\\tfb = self.find(j)\\n\\t\\tif(fa == fb):\\n\\t\\t\\treturn\\n\\t\\ts1 = self.size[fa]\\n\\t\\ts2 = self.size[fb]\\n\\t\\tif(s1<s2):\\n\\t\\t\\tself.arr[fa] = fb\\n\\t\\t\\tself.size[fb] += self.size[fa]\\n\\t\\telse:\\n\\t\\t\\tself.arr[fb] = fa\\n\\t\\t\\tself.size[fa] += self.size[fb]\\nn = int(input())\\nzero = dsu(n)\\none = dsu(n)\\nfor i in range(n-1):\\n\\ta = [int(i) for i in input().split(' ')]\\n\\tx = a[0]-1\\n\\ty = a[1]-1\\n\\tc = a[2]\\n\\tif(c==0):\\n\\t\\tzero.union(x,y)\\n\\telse:\\n\\t\\tone.union(x,y)\\ncount = 0\\nfor i in range(n):\\n\\tif(zero.arr[i] == i):\\n\\t\\tcount+=zero.size[i]*(zero.size[i]-1)\\n\\tif(one.arr[i] == i):\\n\\t\\tcount+=one.size[i]*(one.size[i]-1)\\n\\tcount += (zero.size[zero.find(i)]-1)*(one.size[one.find(i)]-1)\\nprint(count)\", \"import sys\\n\\nclass Disjoint:\\n    def __init__(self, n):\\n        self.n = n\\n        self.size = [1] * self.n\\n        self.parent = [i for i in range(self.n)]\\n    \\n    def root(self, node):\\n        self.parent[node] = node if self.parent[node] == node else self.root(self.parent[node])\\n        return self.parent[node]\\n    \\n    def join(self, a, b):\\n        a = self.root(a)\\n        b = self.root(b)\\n        if a == b:\\n            return False\\n        if self.size[a] > self.size[b]:\\n            a, b = b, a\\n            \\n        self.size[b] += self.size[a]\\n        self.parent[a] = b\\n        return True\\n    \\n    \\n\\ninp = [int(x) for x in sys.stdin.read().split()]\\nn = inp[0]\\n\\nwhite = Disjoint(n)\\nblack = Disjoint(n)\\n\\ninp_idx = 1\\nfor i in range(n - 1):\\n    x, y, c = inp[inp_idx], inp[inp_idx + 1], inp[inp_idx + 2]\\n    x -= 1\\n    y -= 1\\n    inp_idx += 3\\n    \\n    if c == 0:\\n        white.join(x, y)\\n    else:\\n        black.join(x, y)\\n    \\nans = 0\\nfor i in range(n):\\n    rootW = white.root(i)\\n    rootB = black.root(i)\\n    ans += white.size[rootW] - 1\\n    ans += black.size[rootB] - 1\\n    \\n    ans += (white.size[rootW] - 1) * (black.size[rootB] - 1)\\n    \\nprint(ans)\\n\", \"class UnionFind():\\n\\n    def __init__(self, n):\\n        self.n = n\\n\\n        self.root = [-1]*(n+1)\\n\\n        self.rnk = [0]*(n+1)\\n\\n\\n    def Find_Root(self, x):\\n        if(self.root[x] < 0):\\n            return x\\n        else:\\n\\n            self.root[x] = self.Find_Root(self.root[x])\\n            return self.root[x]\\n\\n    def Unite(self, x, y):\\n\\n        x = self.Find_Root(x)\\n        y = self.Find_Root(y)\\n\\n        if(x == y):\\n            return \\n\\n        elif(self.rnk[x] > self.rnk[y]):\\n            self.root[x] += self.root[y]\\n            self.root[y] = x\\n\\n        else:\\n            self.root[y] += self.root[x]\\n            self.root[x] = y\\n\\n            if(self.rnk[x] == self.rnk[y]):\\n                self.rnk[y] += 1\\n\\n    def isSameGroup(self, x, y):\\n        return self.Find_Root(x) == self.Find_Root(y)\\n\\n\\n    def Count(self, x):\\n        return -self.root[self.Find_Root(x)]\\n\\nimport sys\\ninput = sys.stdin.readline\\n\\nN = int(input())\\nuni0 = UnionFind(N)\\nuni1 = UnionFind(N)\\nfor _ in range(N-1):\\n    a, b, c = map(int, input().split())\\n    if c == 0:\\n        uni0.Unite(a-1, b-1)\\n    else:\\n        uni1.Unite(a-1, b-1)\\n\\ng0 = {}\\ng1 = {}\\nfor i in range(N):\\n    if uni0.Count(i) != 1:\\n        r = uni0.Find_Root(i)\\n        if not r in g0.keys():\\n            g0[r] = [i]\\n        else:\\n            g0[r].append(i)\\n    if uni1.Count(i) != 1:\\n        r = uni1.Find_Root(i)\\n        if not r in g1.keys():\\n            g1[r] = [i]\\n        else:\\n            g1[r].append(i)\\n\\nans = 0\\nfor v_list in g1.values():\\n    c = 0\\n    for n in v_list:\\n        if uni0.Count(n) == 1:\\n            c += 1\\n    l = len(v_list)\\n    ans += c*(l-1)\\n\\nfor v_list in g0.values():\\n    c = 0\\n    for n in v_list:\\n        if uni1.Count(n) != 1:\\n            r = uni1.Find_Root(n)\\n            c += len(g1[r])-1\\n    c += len(v_list)-1\\n    ans += len(v_list)*c\\n\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\n\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.parent = [-1] * n\\n        self.cnt = n\\n\\n    def root(self, x):\\n        if self.parent[x] < 0:\\n            return x\\n        else:\\n            self.parent[x] = self.root(self.parent[x])\\n            return self.parent[x]\\n\\n    def merge(self, x, y):\\n        x = self.root(x)\\n        y = self.root(y)\\n        if x == y:\\n            return\\n        if self.parent[x] > self.parent[y]:\\n            x, y = y, x\\n        self.parent[x] += self.parent[y]\\n        self.parent[y] = x\\n        self.cnt -= 1\\n        \\n    def is_same(self, x, y):\\n        return self.root(x) == self.root(y)\\n\\n    def get_size(self, x):\\n        return -self.parent[self.root(x)]\\n    \\n    def get_cnt(self):\\n        return self.cnt\\n\\n\\nn = int(input())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\n\\nuf0 = UnionFind(n)\\nuf1 = UnionFind(n)\\ntree0 = [[] for i in range(n)]\\ntree1 = [[] for i in range(n)]\\nfor i in range(n - 1):\\n    a, b, cost = info[i]\\n    a -= 1\\n    b -= 1\\n    if cost == 0:\\n        tree0[a].append(b)\\n        tree0[b].append(a)\\n        uf0.merge(a, b)\\n    else:\\n        tree1[a].append(b)\\n        tree1[b].append(a)\\n        uf1.merge(a, b)\\n\\nans0 = [0] * n\\nans1 = [0] * n\\nfor i in range(n):\\n    ans0[i] = uf0.get_size(i) - 1\\n    ans1[i] = uf1.get_size(i) - 1\\n\\nans = 0\\nfor i in range(n):\\n    ans += ans0[i]\\n    ans += ans1[i]\\n    ans += ans0[i] * ans1[i]\\nprint(ans)\", \"import sys\\ninput = sys.stdin.readline\\nn = int(input())\\nabd = [list(map(int,input().split())) for i in range(n-1)]\\ngraph = [[] for i in range(n+1)]\\ndeg = [0]*(n+1)\\nif n == 2:\\n  print(2)\\n  return\\nfor a,b,d in abd:\\n  graph[a].append((b,d))\\n  graph[b].append((a,d))\\n  deg[a] += 1\\n  deg[b] += 1\\ndp = [[0]*3 for i in range(n+1)]\\nstack = []\\nroot = 0\\nfor i in range(1,n+1):\\n  if deg[i] == 1:\\n    stack.append(i)\\n  elif not root:\\n    root = i\\n    deg[root] += 1\\nwhile stack:\\n  x = stack.pop()\\n  if dp[x][0] == dp[x][1] == 0:\\n    for y,d in graph[x]:\\n      dp[y][d] += 1\\n      deg[y] -= 1\\n      if deg[y] == 1:\\n        stack.append(y)\\n  else:\\n    for y,d in graph[x]:\\n      if deg[y] > 1:\\n        dp[y][d] += dp[x][d]+1\\n        if d == 1:\\n          dp[y][2] += dp[x][0]+dp[x][2]\\n        deg[y] -= 1\\n        if deg[y] == 1:\\n          stack.append(y)\\nstack = [root]\\ndeg[root] -= 1\\nwhile stack:\\n  x = stack.pop()\\n  for y,d in graph[x]:\\n    if deg[y] == 1:\\n      deg[y] -= 1\\n      dp[y][d] += dp[x][d]-dp[y][d]\\n      if d == 1:\\n        dp[y][2] += dp[x][2]+dp[x][0]-dp[y][0]-dp[y][2]\\n      stack.append(y)\\nans = 0\\nfor i,j,k in dp:\\n  ans += i+j+k\\nprint(ans)\", \"def addEdge(arr: [], x, y):\\n    x, y = getRoot(arr, x), getRoot(arr, y)\\n    if arr[x] <= arr[y]:\\n        arr[x] += arr[y]\\n        arr[y] = x\\n    else:\\n        arr[y] += arr[x]\\n        arr[x] = y\\n\\n\\ndef getRoot(arr: [], id):\\n    while arr[id] > 0:\\n        id = arr[id]\\n    return id\\n\\n\\ndef solve(zeroArr: [], oneArr: []):\\n    Ans = 0\\n    for i in range(1, len(zeroArr)):\\n        if zeroArr[i] < 0:\\n            Ans += zeroArr[i] * (-1) * (zeroArr[i] * (-1) - 1)\\n        if oneArr[i] < 0:\\n            Ans += oneArr[i] * (-1) * (oneArr[i] * (-1) - 1)\\n        if (zeroArr[i] != -1 and oneArr[i] != -1):\\n            one, zero = getRoot(oneArr, i), getRoot(zeroArr, i)\\n            Ans += (oneArr[one] * (-1) - 1) * (zeroArr[zero] * (-1) - 1)\\n    return Ans\\n\\nn = int(input())\\nzeroArr, oneArr = [-1] * (n + 1), [-1] * (n + 1)\\nwhile n > 1:\\n    a, b, w = [int(x) for x in input().split()]\\n    if w == 0:\\n        addEdge(zeroArr, a, b)\\n    else:\\n        addEdge(oneArr, a, b)\\n    n -= 1\\nprint(solve(zeroArr, oneArr))\\n\", \"mod = 1000000007\\neps = 10**-9\\n\\n\\ndef main():\\n    import sys\\n    from collections import deque\\n    input = sys.stdin.readline\\n\\n    class UnionFind():\\n        def __init__(self, n):\\n            self.n = n\\n            self.root = [-1] * (n + 1)\\n            self.rnk = [0] * (n + 1)\\n\\n        def find_root(self, x):\\n            while self.root[x] >= 0:\\n                x = self.root[x]\\n            return x\\n\\n        def unite(self, x, y):\\n            x = self.find_root(x)\\n            y = self.find_root(y)\\n            if x == y:\\n                return\\n            elif self.rnk[x] > self.rnk[y]:\\n                self.root[x] += self.root[y]\\n                self.root[y] = x\\n            else:\\n                self.root[y] += self.root[x]\\n                self.root[x] = y\\n                if self.rnk[x] == self.rnk[y]:\\n                    self.rnk[y] += 1\\n\\n        def isSameGroup(self, x, y):\\n            return self.find_root(x) == self.find_root(y)\\n\\n        def size(self, x):\\n            return -self.root[self.find_root(x)]\\n\\n    N = int(input())\\n    adj = [[] for _ in range(N+1)]\\n    UF0 = UnionFind(N+1)\\n    UF1 = UnionFind(N+1)\\n    for _ in range(N-1):\\n        a, b, c = list(map(int, input().split()))\\n        adj[a].append((b, c))\\n        adj[b].append((a, c))\\n        if c == 0:\\n            UF0.unite(a, b)\\n        else:\\n            UF1.unite(a, b)\\n\\n    ans = 0\\n    roots = set()\\n    for v in range(1, N+1):\\n        r = UF1.find_root(v)\\n        if r not in roots:\\n            roots.add(r)\\n            s = -UF1.root[r]\\n            ans += s * (s-1)\\n    #print(ans)\\n\\n    roots = set()\\n    for v in range(1, N+1):\\n        r = UF0.find_root(v)\\n        if r not in roots:\\n            roots.add(r)\\n            s = -UF0.root[r]\\n            ans += s * (s-1)\\n    #print(ans)\\n\\n    for v in range(1, N+1):\\n        W = 0\\n        B = 0\\n        flg0 = 0\\n        flg1 = 0\\n        for u, c in adj[v]:\\n            if flg0 and flg1:\\n                break\\n            if c == 0:\\n                if flg0:\\n                    continue\\n                W += UF0.size(u) - 1\\n                flg0 = 1\\n            else:\\n                if flg1:\\n                    continue\\n                B += UF1.size(u) - 1\\n                flg1 = 1\\n        ans += W * B\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(2*10**5)\\nI = lambda : list(map(int,input().split()))\\n\\nn,=I()\\nr=[[1 for i in range(n)],[1 for i in range(n)]]\\np=[[i for i in range(n)],[i for i in range(n)]]\\ndef find(x,c):\\n\\tif x!=p[c][x]:\\n\\t\\tp[c][x]=find(p[c][x],c)\\n\\treturn p[c][x]\\n\\ndef union(a,b,c):\\n\\tx=find(a,c)\\n\\ty=find(b,c)\\n\\tmm=min(x,y)\\n\\tif x!=y:\\n\\t\\tp[c][y]=p[c][x]=mm\\n\\t\\tr[c][mm]+=r[c][max(x,y)]\\nan=0\\nfor i in range(n-1):\\n\\ta,b,c=I()\\n\\tunion(a-1,b-1,c)\\nvis=[0]*n\\ncc=[]\\nfor i in range(n):\\n\\ts0=r[0][i]\\n\\ts1=r[1][i]\\n\\tif p[0][i]==i:\\n\\t\\tan+=(s0-1)*s0\\n\\tif p[1][i]==i:\\n\\t\\tan+=(s1-1)*s1\\n\\tan+=(r[1][find(i,1)]-1)*(r[0][find(i,0)]-1)\\nprint(an)\", \"import sys;sys.setrecursionlimit(10**9)\\nclass UnionFind:\\n  def __init__(self,n):\\n    self.n=[-1]*n\\n    self.r=[0]*n\\n    self.siz=n\\n  def find_root(self,x):\\n    if self.n[x]<0:\\n      return x\\n    else:\\n      self.n[x]=self.find_root(self.n[x])\\n      return self.n[x]\\n  def unite(self,x,y):\\n    x=self.find_root(x)\\n    y=self.find_root(y)\\n    if x==y:return\\n    elif self.r[x]>self.r[y]:\\n      self.n[x]+=self.n[y]\\n      self.n[y]=x\\n    else:\\n      self.n[y]+=self.n[x]\\n      self.n[x]=y\\n      if self.r[x]==self.r[y]:\\n        self.r[y]+=1\\n    self.siz-=1\\n  def root_same(self,x,y):\\n    return self.find_root(x)==self.find_root(y)\\n  def count(self,x):\\n    return -self.n[self.find_root(x)]\\n  def size(self):\\n    return self.siz\\n\\nn=int(input())\\nouf=UnionFind(n)\\nzuf=UnionFind(n)\\nfor _ in range(n-1):\\n  a,b,c=map(int,input().split())\\n  a-=1\\n  b-=1\\n  if c==0:\\n    zuf.unite(a,b)\\n  else:\\n    ouf.unite(a,b)\\nans=0\\nfor i in range(n):\\n  m=zuf.count(i)\\n  if zuf.find_root(i)==i:ans+=m*(m-1)\\n  mm=ouf.count(i)\\n  if ouf.find_root(i)==i:ans+=mm*(mm-1)\\n  ans+=(m-1)*(mm-1)\\nprint(ans)\", \"class UnionFindVerSize():\\n    def __init__(self, N):\\n        self._parent = [n for n in range(0, N)]\\n        self._size = [1] * N\\n\\n    def find_root(self, x):\\n        if self._parent[x] == x: return x\\n        self._parent[x] = self.find_root(self._parent[x])\\n        return self._parent[x]\\n\\n    def unite(self, x, y):\\n        gx = self.find_root(x)\\n        gy = self.find_root(y)\\n        if gx == gy: return\\n\\n        if self._size[gx] < self._size[gy]:\\n            self._parent[gx] = gy\\n            self._size[gy] += self._size[gx]\\n        else:\\n            self._parent[gy] = gx\\n            self._size[gx] += self._size[gy]\\n\\n    def get_size(self, x):\\n        return self._size[self.find_root(x)]\\n\\n    def is_same_group(self, x, y):\\n        return self.find_root(x) == self.find_root(y)\\n\\n    def calc_group_num(self):\\n        N = len(self._parent)\\n        ans = 0\\n        for i in range(N):\\n            if self.find_root(i) == i:\\n                ans += 1\\n        return ans\\n\\nimport sys\\n\\ninput=sys.stdin.readline\\nn=int(input())\\nuf0=UnionFindVerSize(n)\\nuf1=UnionFindVerSize(n)\\nfor i in range(n-1):\\n    x,y,c=map(int,input().split())\\n    if c==0:\\n        uf0.unite(x-1,y-1)\\n    else:\\n        uf1.unite(x-1,y-1)\\n\\ndata=[0]*n\\ncnt=[0]*n\\nfor i in range(n):\\n    root=uf1.find_root(i)\\n    data[root]+=uf0.get_size(i)\\n    cnt[root]+=1\\n\\nans=sum(data[i]*cnt[i] for i in range(n))\\nprint(ans-n)\"]",
        "difficulty": "interview",
        "input": "2\n1 2 0\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1156/D"
    },
    {
        "id": 1066,
        "task_id": 2517,
        "test_case_id": 1,
        "question": "There are N towns in the State of Atcoder, connected by M bidirectional roads.\nThe i-th road connects Town A_i and B_i and has a length of C_i.\nJoisino is visiting R towns in the state, r_1,r_2,..,r_R (not necessarily in this order).\nShe will fly to the first town she visits, and fly back from the last town she visits, but for the rest of the trip she will have to travel by road.\nIf she visits the towns in the order that minimizes the distance traveled by road, what will that distance be?\n\n-----Constraints-----\n - 2≤N≤200\n - 1≤M≤N×(N-1)/2\n - 2≤R≤min(8,N) (min(8,N) is the smaller of 8 and N.)\n - r_i≠r_j (i≠j)\n - 1≤A_i,B_i≤N, A_i≠B_i\n - (A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)\n - 1≤C_i≤100000\n - Every town can be reached from every town by road.\n - All input values are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN M R\nr_1 ... r_R\nA_1 B_1 C_1\n:\nA_M B_M C_M\n\n-----Output-----\nPrint the distance traveled by road if Joisino visits the towns in the order that minimizes it.\n\n-----Sample Input-----\n3 3 3\n1 2 3\n1 2 1\n2 3 1\n3 1 4\n\n-----Sample Output-----\n2\n\nFor example, if she visits the towns in the order of 1, 2, 3, the distance traveled will be 2, which is the minimum possible.",
        "solutions": "[\"from itertools import permutations as p\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nn, m, r = map(int, input().split())\\nR = list(map(int, input().split()))\\nl = [[0]*n for _ in range(n)]\\nfor _ in range(m):\\n  a, b, c = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  l[a][b] = c\\n  l[b][a] = c\\nF = floyd_warshall(l)\\n\\nans = float(\\\"inf\\\")\\nfor v in p(R):\\n  temp = 0\\n  for i in range(r-1):\\n    temp += F[v[i]-1][v[i+1]-1]\\n  ans = min(ans, temp)\\n  \\nprint(int(ans))\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import floyd_warshall\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = floyd_warshall(G, directed=False)\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"def main():\\n    import itertools\\n    n,m,r=list(map(int,input().split()))\\n    rx=[int(i) for i in input().split()]\\n    inf=100000000\\n    wf=[[inf]*n for i in range(n)]\\n    for i in range(n):\\n        wf[i][i]=0\\n\\n    for i in range(m):\\n        a,b,c=list(map(int,input().split()))\\n        wf[a-1][b-1]=c\\n        wf[b-1][a-1]=c\\n\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if wf[i][j]>wf[i][k]+wf[k][j]:\\n                    wf[i][j]=wf[i][k]+wf[k][j]\\n    cnt=0\\n    l=list(itertools.permutations([i for i in range(r)]))\\n    cnt=inf\\n    for i in l:\\n        cnt_sub=0\\n        for j in range(r-1):\\n            cnt_sub+=wf[rx[i[j]]-1][rx[i[j+1]]-1]\\n        cnt=min(cnt,cnt_sub)\\n    print(cnt)\\nmain()\\n\", \"import itertools as it\\nn,m,r=map(int,input().split())\\nll=list(map(int,input().split()))\\nd=[[float(\\\"inf\\\") for _ in range(n)] for _ in range(n)] #\\u5404\\u9802\\u70b9\\u304b\\u3089\\u5404\\u9802\\u70b9\\u3078\\u306e\\u6700\\u5c0f\\u30b3\\u30b9\\u30c8\\nfor i in range(m): #\\u91cd\\u8907\\u306a\\u3057ver\\n    a,b,t=map(int,input().split())\\n    d[a-1][b-1]=t\\n    d[b-1][a-1]=t #\\u6709\\u5411\\u30b0\\u30e9\\u30d5\\u306e\\u5834\\u5408\\u6d88\\u3059\\nfor i in range(n):\\n    d[i][i]=0\\nfor k in range(n):\\n    for i in range(n):\\n        for j in range(n):\\n            if d[i][j]>d[i][k]+d[k][j]:\\n                d[i][j]=d[i][k]+d[k][j]\\nans=10**9\\nfor p in it.permutations(ll):\\n    tmp=0\\n    for j in range(r-1):\\n        s,g=p[j]-1,p[j+1]-1\\n        tmp+=d[s][g]\\n    ans=min(ans,tmp)\\nprint(ans)\", \"import sys, re\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees#, log2\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left\\nfrom fractions import gcd\\nfrom heapq import heappush, heappop, heapify\\nfrom functools import reduce\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nr = LIST()\\nABC = [LIST() for _ in range(M)]\\n\\ndist_matrix = [[0]*N for _ in range(N)]\\nfor A, B, C in ABC:\\n\\tdist_matrix[A-1][B-1] = C\\n\\tdist_matrix[B-1][A-1] = C\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\n#\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5  \\u6ce8\\u610f PyPy\\u975e\\u5bfe\\u5fdc!!!!\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\ngraph = csr_matrix(dist_matrix) #\\u96a3\\u63a5\\u884c\\u5217\\u304b\\u3089csr\\u884c\\u5217\\u3092\\u3064\\u304f\\u308b\\ndist_matrix = floyd_warshall(graph, directed=False) #\\u7121\\u5411\\u306e\\u5834\\u5408directed=False\\n#dist_matrix\\u306e\\u4e2d\\u8eab\\u306ffloat\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u306e\\u3067\\u6ce8\\u610f\\uff01\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\nans = INF\\nfor x in permutations(r):\\n\\tdis = 0\\n\\tfor i in range(1, R):\\n\\t\\tdis += dist_matrix[x[i-1]-1][x[i]-1]\\n\\tans = min(ans, dis)\\n\\nprint((int(ans)))\\n\\n\\n\\t\\n\", \"# ABC073 D - joisino's travel\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\n\\ninf = 10**18\\ndist = [[inf]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = map(int, input().split())\\n    dist[A-1][B-1] = dist[B-1][A-1] = C\\n\\ndist = floyd_warshall(dist)\\n\\nans = inf\\nfor rr in permutations(r):\\n    cost = 0\\n    for i in range(R-1):\\n        cost += dist[rr[i]-1][rr[i+1]-1]\\n    if cost < ans:\\n        ans = cost\\nprint(int(ans))\", \"import itertools\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nN, M, R = list(map(int, input().split()))\\nr = tuple(map(int, input().split()))\\n\\nINF = 10**10\\n\\nd = [[INF] * N for _ in range(N)]\\n\\nfor i in range(N):\\n    d[i][i] = 0\\n\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    if d[a][b] > c:\\n        d[a][b] = c\\n        d[b][a] = c\\n\\n# for k in range(N):\\n#     for i in range(N):\\n#         for j in range(N):\\n#             d[i][j] = min(d[i][j], d[i][k] + d[k][j])\\n\\nd = floyd_warshall(d)\\n\\n\\nans = INF\\nfor p in itertools.permutations(r):\\n    dist = 0\\n    for i in range(R-1):\\n        dist += d[p[i]-1][p[i+1]-1]\\n\\n    ans = min(ans, dist)\\n\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\n\\nd = [[0] * (N + 1) for _ in range(N + 1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    d[A][B] = C\\n    d[B][A] = C\\ng = csgraph_from_dense(d)\\ng = floyd_warshall(g)\\n\\nresult = 1000000000\\nfor p in permutations(r):\\n    t = 0\\n    for i in range(R - 1):\\n        t += g[p[i]][p[i + 1]]\\n    result = min(result, t)\\nprint((int(result)))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heapify,heappush,heappop\\nreadline=sys.stdin.readline\\n\\ndef main():\\n    N,M,R=list(map(int,readline().split()))\\n    r=list(map(int,readline().split()))\\n    ew=[tuple(map(int,readline().split())) for _ in range(M)]\\n    edges=defaultdict(list)\\n    wt=defaultdict(int)\\n    for e in ew:\\n        edges[e[0]].append(e[1])\\n        edges[e[1]].append(e[0])\\n        wt[(e[0],e[1])]=e[2]\\n        wt[(e[1],e[0])]=e[2]\\n    inf=float('inf')\\n    def dijkstra(v):\\n        dist=[inf]*(N+1)\\n        dist[v]=0\\n        vw=[(0,v)]\\n        heapify(vw)\\n        while vw:\\n            cvw=heappop(vw)\\n            if cvw[0]>dist[cvw[1]]:\\n                continue\\n            for w in edges[cvw[1]]:\\n                cand=cvw[0]+wt[(cvw[1],w)]\\n                if dist[w]>cand:\\n                    dist[w]=cand\\n                    heappush(vw,(cand,w))\\n        return dist\\n    distdict=dict((v,dijkstra(v)) for v in r)\\n    mindist=inf\\n    def permlist(lst):\\n        tmp=[]\\n        if not lst:\\n            return [[]]\\n        for i,x in enumerate(lst):\\n            lstx=lst[:i]+lst[i+1:]\\n            ret=permlist(lstx)\\n            for e in ret:\\n                e.append(x)\\n            tmp.extend(ret)\\n        return tmp\\n    #print(distdict,permlist(r))\\n    pathlist=[sum([distdict[v][w] for v,w in zip(lst,lst[1:])]) for lst in permlist(r)]\\n    print((min(pathlist)))\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\ndef warshallFloyd():\\n    for k in range(N):\\n        for i in range(N-1):\\n            for j in range(i, N):\\n                tmp = min(E[i][j], E[i][k] + E[k][j])\\n                E[i][j] = E[j][i] = tmp\\n\\ndef dfs(c, p, d):\\n    if c == R:\\n        ans[0] = min(ans[0], d)\\n        return\\n    for i in range(R):\\n        if used[i]:\\n            continue\\n        used[i] = True\\n        if p == -1:\\n            dfs(c+1, i, 0)\\n        else:\\n            dfs(c+1, i, d + E[S[p]-1][S[i]-1])\\n        used[i] = False\\n\\nN, M, R = map(int, input().split())\\nS = list(map(int, input().split()))\\nE = [[float('inf')] * N for _ in range(N)]\\nfor i in range(N):\\n    E[i][i] = 0\\nfor _ in range(M):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    E[a][b] = E[b][a] = c\\nwarshallFloyd()\\nans = [float('inf')]\\nused = [False] * R\\ndfs(0, -1, 0)\\nprint(*ans)\", \"import sys\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\ninput = sys.stdin.readline\\nans = -1\\n\\nn, m, r = list(map(int, input().split()))\\nr  = list(map(int, input().split()))\\nA, B, C = [], [], []\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    A.append(a-1)\\n    B.append(b-1)\\n    C.append(c)\\n\\nwf = floyd_warshall(csr_matrix((C, (A, B)), shape = (n, n)), directed=False)\\n\\nfor pi in permutations(r):\\n    cand = 0\\n    for r1, r2 in zip(pi, pi[1:]):\\n        cand += int(wf[r1-1][r2-1])\\n    if ans < 0 or ans > cand:\\n        ans = cand\\n\\nprint(ans)\\n\", \"#!/usr/bin/env python3\\nimport sys\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    import numpy as np    \\n    from scipy.sparse import coo_matrix    \\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations, islice\\n    from functools import reduce\\n    # coo_matrix((data, (i, j)), [shape=(M, N)])\\n    mat = floyd_warshall(coo_matrix((C, (A, B)), shape=(N+1, N+1), dtype=np.int32).tocsr(), directed=False)\\n    f = lambda a, b: (a[0]+mat[a[1]][b], b)\\n    return int(min(reduce(f, q, (0, s))[0] for s, *q in permutations(r)))\\n\\n# Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    print((solve(N, M, R, r, A, B, C)))\\n\\ndef test():\\n    import doctest\\n    doctest.testmod()\\n\\ndef __starting_point():\\n    #test()\\n    main()\\n\\n__starting_point()\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nn,m,r=map(int,input().split())\\nvis=list(map(int,input().split()))\\nINF=10**18\\npath=[[INF]*n for i in range(n)]\\nfor i in range(m):\\n  a,b,c=map(int,input().split())\\n  path[a-1][b-1]=c\\n  path[b-1][a-1]=c\\npath=floyd_warshall(path)\\nans=10**18\\nfor root in permutations(vis):\\n  cnt=0\\n  for j in range(1,r):\\n    cnt+=path[root[j-1]-1][root[j]-1]\\n  ans=min(ans,cnt)\\nprint(int(ans))\", \"import sys\\nfrom itertools import permutations\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n\\n    def warshall_floyd(d):\\n        # d[i][j]: i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n        for k in range(N):\\n            for i in range(N):\\n                for j in range(N):\\n                    tmp = d[i][k] + d[k][j]\\n                    if d[i][j] > tmp:\\n                        d[i][j] = tmp\\n        return d\\n\\n    r = list(map(int, input().split()))\\n    d = [[float(\\\"inf\\\")] * N for i in range(N)]\\n    for i in range(M):\\n        x, y, z = list(map(int, input().split()))\\n        d[x - 1][y - 1] = z\\n        d[y - 1][x - 1] = z\\n    for i in range(N):\\n        d[i][i] = 0  # \\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\n    D = warshall_floyd(d)\\n    P = list(permutations(r))\\n    cost = float(\\\"inf\\\")\\n    for p in P:\\n        c = 0\\n        for i in range(R - 1):\\n            c += D[p[i] - 1][p[i + 1] - 1]\\n        if c < cost:\\n            cost = c\\n    print(cost)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"#!/usr/bin/env python3\\nimport sys\\nimport itertools\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    from scipy.sparse import csr_matrix\\n    from scipy.sparse.csgraph import floyd_warshall\\n    matrix = [[0]*N for _ in range(N)]\\n    for i in range(M):\\n        matrix[A[i]-1][B[i]-1] = C[i]\\n        matrix[B[i]-1][A[i]-1] = C[i]\\n    \\n    dist_matrix = floyd_warshall(csgraph=matrix, directed=False)\\n\\n    perm = list(itertools.permutations(r)) \\n\\n    answer = float('inf')\\n    for p in perm:\\n        a = 0\\n        for index in range(len(p)-1):\\n            a += dist_matrix[p[index]-1][p[index+1]-1]\\n        answer = min(answer,a)\\n    print((int(answer)))\\n    return\\n\\n\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    solve(N, M, R, r, A, B, C)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\nn,m,r,*L=map(int,open(0).read().split())\\nf=floyd_warshall(csr_matrix((L[r+2::3],(L[r::3],L[r+1::3])),(n+1,n+1)),0)\\nprint(int(min(sum(f[s,t]for s,t in zip(o,o[1:]))for o in permutations(L[:r]))))\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path\\n\\nN,M,R = map(int, input().split())\\nrs = np.array(list(map(int, input().split()))) -1\\n\\nnodes = np.full((N,N), np.inf) \\nfor i in range(M):\\n  a,b,c = map(int, input().split())\\n  nodes[a-1,b-1] = nodes[b-1,a-1] = c\\nmin_ds = shortest_path(nodes)\\nres = float('inf')\\nfor p in itertools.permutations(rs):\\n  prev = p[0]\\n  tmp = 0\\n  for next in p[1:]:\\n    tmp += min_ds[prev,next]\\n    prev = next\\n  res = min(res, tmp)\\n\\nprint(int(res))\", \"def main():\\n  import numpy as np\\n  import itertools\\n  from scipy.sparse import csr_matrix\\n  from scipy.sparse.csgraph import floyd_warshall\\n  import sys\\n  readline = sys.stdin.readline\\n  readlines = sys.stdin.readlines\\n\\n  N, M, R= map(int, readline().split())\\n  r = list(map(int,input().split()))\\n  lines = readlines()\\n  edge = np.array([line.split() for line in lines], dtype = np.int64).T\\n  graph = csr_matrix((edge[2], (edge[:2] - 1)), (N, N))\\n  distance_mat = floyd_warshall(graph,directed = False)\\n  #print(distance_mat)\\n  ans = float(\\\"inf\\\")\\n  for town in itertools.permutations(r,R):\\n    #print(town)\\n    tmp = 0\\n    for i in range(R-1):\\n      tmp += distance_mat[town[i]-1][town[i+1]-1]\\n    ans = min(ans,tmp)\\n  print(int(ans))\\nmain()\", \"import sys\\nimport numpy as np \\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nI = np.array(sys.stdin.read().split(), dtype=np.int64)\\nn, m, R = I[:3]\\nr = I[3:3+R] - 1\\na, b, c = I[3+R:].reshape(-1, 3).T\\na -= 1; b -= 1\\ngraph = csr_matrix((c, (a, b)), shape=(n, n))\\n\\ndef main():\\n    dist = floyd_warshall(graph, directed=False).astype(np.int64)\\n    *perms, = permutations(r)\\n    perms = np.array(perms)\\n    res = dist[perms[:, :-1], perms[:, 1:]]\\n    ans = np.amin(np.sum(res, axis=1))\\n    return ans\\n\\ndef __starting_point():\\n    ans = main()\\n    print(ans)\\n__starting_point()\", \"import itertools\\nfrom scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nN, M, R = list(map(int, input().split()))\\nr = [int(i) for i in input().split()]\\n\\nG = [[10**8]*(N+1) for _ in range(N+1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    G[A][B] = C\\n    G[B][A] = C\\n\\nDG = csgraph_from_dense(G, null_value=10**8)\\nd = floyd_warshall(DG)\\n\\nans = 10**8\\nfor route in itertools.permutations(r):\\n    c = 0\\n    for i in range(len(route)-1):\\n        c += d[route[i]][route[i+1]]\\n    ans = min(ans, c)\\nprint((int(ans)))\\n\", \"import heapq\\nfrom itertools import permutations\\nn, m, r = list(map(int, input().split()))\\nR = list([int(x)-1 for x in input().split()])\\nedges = [[]for _ in range(n)]\\nfor _ in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    edges[a].append((c, b))\\n    edges[b].append((c, a))\\n\\n\\ndef dijkstra(init_v):\\n    next_v = [(0, init_v)]\\n    INF = 10**18\\n    dist = [INF]*n\\n    dist[init_v] = 0\\n    while next_v:\\n        d, v = heapq.heappop(next_v)\\n        if dist[v] < d:\\n            continue\\n        for d, v2 in edges[v]:\\n            if dist[v2] <= dist[v]+d:\\n                continue\\n            dist[v2] = dist[v]+d\\n            heapq.heappush(next_v, (dist[v2], v2))\\n    return dist\\n\\n\\ndists = []\\nfor x in R:\\n    dist = dijkstra(x)\\n    dists.append(dist)\\n\\nINF = 10**18\\nans = INF\\nfor subset in permutations(list(range(r))):\\n    d = 0\\n    for v, v2 in zip(subset, subset[1:]):\\n        d += dists[v][R[v2]]\\n    if d < ans:\\n        ans = d\\n\\nprint(ans)\\n\", \"# ABC073 D - joisino's travel\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\n\\ninf = 10**18\\ndist = [[inf]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = map(int, input().split())\\n    dist[A-1][B-1] = dist[B-1][A-1] = C\\n\\ndist = floyd_warshall(dist)\\n\\nans = inf\\nfor rr in permutations(r):\\n    cost = 0\\n    for i in range(R-1):\\n        cost += dist[rr[i]-1][rr[i+1]-1]\\n    ans = min(ans, cost)\\nprint(int(ans))\", \"from itertools import permutations\\n\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph._shortest_path import floyd_warshall\\n\\n\\ndef solve():\\n    N, M, R = list(map(int, input().split()))\\n    r = list([int(x)-1 for x in input().split()])\\n    dist = [[0 for _ in range(N)] for _ in range(N)]\\n    for _ in range(M):\\n        A, B, C = list(map(int, input().split()))\\n        dist[A-1][B-1] = C\\n        dist[B-1][A-1] = C\\n    fw = floyd_warshall(csr_matrix(dist))\\n    \\n    ans = 10**12\\n    for visit in permutations(r, R):\\n        m_sum = 0\\n        for i in range(R-1):\\n            m_sum += fw[visit[i]][visit[i+1]]\\n        ans = min(ans, m_sum)\\n    print((int(ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"import numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nn, m, r = map(int,input().split()) #n:\\u9802\\u70b9\\u6570\\u3000m:\\u8fba\\u306e\\u6570 r:\\u8a2a\\u308c\\u308b\\u753a\\u306e\\u6570\\nR = list(map(int,input().split()))\\n\\nd = np.array([[float(\\\"inf\\\") for i in range(n)] for i in range(n)])\\n#d[u][v] : \\u8fbauv\\u306e\\u30b3\\u30b9\\u30c8(\\u5b58\\u5728\\u3057\\u306a\\u3044\\u3068\\u304d\\u306finf)\\nfor i in range(m):\\n    x, y, z = map(int,input().split())\\n    d[x-1][y-1] = z\\n    d[y-1][x-1] = z\\nfor i in range(n):\\n    d[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\nwa = floyd_warshall(d)\\n\\n# \\u8a2a\\u308c\\u308b\\u753a\\u306e\\u9806\\u756a\\u3092\\u9806\\u5217\\u3067\\u7528\\u610f\\nfrom itertools import permutations\\nptn = permutations(R)\\n\\nans = 10**19\\nfor p in ptn:\\n  cost = 0\\n  roots = [(j-1, i-1) for i, j in zip(p, p[1:])]\\n  for i, j in roots:\\n    cost += wa[i][j]\\n  ans = min(ans, cost)\\n  \\nprint(int(ans))\", \"import itertools\\n\\n\\ndef dijkstra(v, G):\\n    import heapq\\n\\n    ret = [10 ** 10] * len(G)\\n    ret[v] = 0\\n    q = [(ret[i], i) for i in range(len(G))]\\n    heapq.heapify(q)\\n\\n    while len(q):\\n        tmpr, u = heapq.heappop(q)\\n        if tmpr == ret[u]:\\n            for w, l in G[u]:\\n                if ret[w] > ret[u] + l:\\n                    ret[w] = ret[u] + l\\n                    heapq.heappush(q, (ret[w], w))\\n    return ret\\n\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\nNE = [[] for _ in range(N)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    NE[A - 1].append((B - 1, C))\\n    NE[B - 1].append((A - 1, C))\\n\\nRE = [[0] * R for _ in range(R)]\\nfor iR in range(R):\\n    for jR in range(iR + 1, R):\\n        lR = dijkstra(r[iR] - 1, NE)[r[jR] - 1]\\n        RE[iR][jR] = lR\\n        RE[jR][iR] = lR\\n\\nans = 10 ** 10\\nfor p in itertools.permutations(list(range(R))):\\n    ans = min(ans, sum([RE[p[i]][p[i + 1]] for i in range(R - 1)]))\\n\\nprint(ans)\\n\", \"# solution\\n\\nimport itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nint1 = lambda x: int(x) - 1\\n\\nN, M, R = map(int, input().split())\\nT = sorted(list(map(int1, input().split())))\\nA = np.array([tuple(map(int, input().split())) for _ in range(M)]).T\\n\\nmatr = csr_matrix((A[2], (A[0] - 1, A[1] - 1)), shape=(N, N))\\nway = [dijkstra(matr, indices=t, directed=False)[T].astype(int).tolist() for t in T]\\n\\nresult = float('inf')\\nfor t in itertools.permutations(range(R)):\\n    tmp = 0\\n    for i in range(R - 1):\\n        tmp += way[t[i]][t[i + 1]]\\n    result = min(result, tmp)\\nprint(result)\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nsys.setrecursionlimit(10 ** 9)\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nA = [a-1 for a in LIST()]\\nG = list2d(N, N, INF)\\nfor i in range(M):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    G[a][b] = c\\n    G[b][a] = c\\n\\n# \\u5168\\u4f53\\u30b0\\u30e9\\u30d5\\u306e\\u6700\\u77ed\\u8ddd\\u96e2(\\u3053\\u3053\\u304b\\u3089\\u5fc5\\u8981\\u306a\\u9802\\u70b9\\u9593\\u3060\\u3051\\u4f7f\\u3046)\\nwf = floyd_warshall(G)\\n\\nans = INF\\nfor r in range(R):\\n    # TSP(\\u5de1\\u56de\\u30bb\\u30fc\\u30eb\\u30b9\\u30de\\u30f3)\\n    dp = list2d(1<<R, R, INF)\\n    dp[1<<r][r] = 0\\n    for bit in range(1, (1<<R)-1):\\n        for i in range(R):\\n            if not (bit >> i & 1):\\n                continue\\n            for j in range(R):\\n                if bit >> j & 1:\\n                    continue\\n                a, b = A[i], A[j]\\n                dp[bit|1<<j][j] = min(dp[bit|1<<j][j], dp[bit][i] + int(wf[a,b]))\\n    ans = min(ans, min(dp[-1]))\\nprint(ans)\\n\", \"import numpy as np\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\nI = [int(_) for _ in open(0).read().split()]\\nN, M, R = I[:3]\\nr = np.array(I[3:3 + R])\\nABC = I[3 + R:]\\nF = floyd_warshall(csr_matrix((ABC[2::3], (ABC[::3], ABC[1::3])), (N + 1, N + 1)), 0).astype(np.int64)\\nans = float('inf')\\nfor root in itertools.permutations(r, R):\\n    ans = min(ans, sum(F[i,j] for i, j in zip(root, root[1:])))\\nprint(ans)\\n\", \"#!/usr/bin python3\\n# -*- coding: utf-8 -*-\\n\\n# \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5\\n# \\u5168\\u9802\\u70b9\\u9593\\u6700\\u77ed\\u8def\\n# d[i][j]\\u306f2\\u9802\\u70b9\\u9593i, j\\u9593\\u306e\\u79fb\\u52d5\\u30b3\\u30b9\\u30c8\\u3092\\u683c\\u7d0d, M\\u306f\\u9802\\u70b9\\u6570\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\n\\ndef main():\\n    N, M, R = map(int,input().split()) #N:\\u9802\\u70b9\\u6570\\u3000M:\\u8fba\\u306e\\u6570\\n    d = [[float(\\\"inf\\\")]*N for i in range(N)]\\n    #d[u][v] : \\u8fbauv\\u306e\\u30b3\\u30b9\\u30c8(\\u5b58\\u5728\\u3057\\u306a\\u3044\\u3068\\u304d\\u306finf)\\n    r = list(map(int,input().split()))\\n    for i in range(M):\\n        u, v, w = map(int,input().split())\\n        d[u-1][v-1] = w\\n        d[v-1][u-1] = w\\n    for i in range(N):\\n        d[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\n    cost = floyd_warshall(d)\\n    L = list(permutations(r, R))\\n    ret = 10**9\\n    for rt in L:\\n        cos = 0\\n        for i in range(1,R):\\n            cos += cost[rt[i]-1][rt[i-1]-1]\\n        ret = min(ret, cos)\\n    print(int(ret))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nINF = 1001001001\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\ngraph = [[INF]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    graph[A-1][B-1] = C\\n    graph[B-1][A-1] = C\\ndist_matrix = floyd_warshall(graph)\\n\\nans = INF\\nfor root in permutations(r):\\n    cnt = 0\\n    for j in range(R-1):\\n        cnt += dist_matrix[root[j]-1][root[j+1]-1]\\n    ans = min(ans, cnt)\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\nabc = [list(map(int, input().split())) for _ in range(M)]\\n\\nr = [e - 1 for e in r]\\n\\ng = [[0] * N for _ in range(N)]\\nfor a, b, c in abc:\\n\\ta -= 1\\n\\tb -= 1\\n\\tg[a][b] = c\\n\\tg[b][a] = c\\n\\ndist = floyd_warshall(csr_matrix(g))\\n\\nans = float(\\\"inf\\\")\\nfor pat in permutations(r, len(r)):\\n\\tsm = 0\\n\\tfor e1, e2 in zip(pat, pat[1:]):\\n\\t\\tsm += dist[e1][e2]\\n\\n\\tans = min(ans, sm)\\n\\nans = int(ans)\\nprint(ans)\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\n\\ns = [[10**9]*N for i in range(N)]\\nt = list(map(int, input().split()))\\nfor i in range(M):\\n    a, b, c = map(int, input().split())\\n    s[a-1][b-1] = c\\n    s[b-1][a-1] = c\\n    \\ns = floyd_warshall(s)\\nans = float('inf')\\nfor i in permutations(t):\\n    A_n = 0\\n    for j in range(R-1):\\n        A_n += s[i[j] - 1][i[j+1] - 1]\\n    ans = min(ans, A_n)\\n    \\nprint(int(ans))\", \"import sys\\nfrom itertools import permutations\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\nMOD = 1000000007\\n\\n\\ndef main():\\n    N, M, R = list(map(int, readline().split()))\\n    city = [int(s) - 1 for s in readline().split()]\\n    ABC = list(map(int, read().split()))\\n\\n    G = [[INF] * N for _ in range(N)]\\n    for a, b, c in zip(*[iter(ABC)] * 3):\\n        G[a - 1][b - 1] = G[b - 1][a - 1] = c\\n\\n    for i in range(N):\\n        G[i][i] = 0\\n\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if G[i][j] > G[i][k] + G[k][j]:\\n                    G[i][j] = G[i][k] + G[k][j]\\n\\n    ans = INF\\n    for plan in permutations(city):\\n        res = 0\\n        for i in range(R - 1):\\n            res += G[plan[i]][plan[i + 1]]\\n        if ans > res:\\n            ans = res\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"#!/usr/bin/env python3\\nimport sys\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    from scipy.sparse import coo_matrix    \\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations\\n    from functools import reduce\\n    # coo_matrix((data, (i, j)), [shape=(M, N)])\\n    mat = floyd_warshall(coo_matrix((C, (A, B)), shape=(N+1, N+1)).tocsr(), directed=False)\\n    f = lambda a, b: (a[0]+mat[a[1]][b], b)\\n    return int(min(reduce(f, q, (0, s))[0] for s, *q in permutations(r)))\\n\\n# Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    print(solve(N, M, R, r, A, B, C))\\n\\ndef test():\\n    import doctest\\n    doctest.testmod()\\n\\ndef __starting_point():\\n    #test()\\n    main()\\n__starting_point()\", \"n,m,r_=list(map(int,input().split()))\\nr=list(map(int,input().split()))\\ng=[[]for _ in range(n)]\\nfor _ in range(m):\\n  a,b,c=list(map(int,input().split()))\\n  a-=1\\n  b-=1\\n  g[a].append([b,c])\\n  g[b].append([a,c])\\nfrom heapq import heapify,heappush,heappop\\ndef dks(t0,t1):\\n  kyori=[-1]*n\\n  todo=[[0,t0]]\\n  heapify(todo)\\n  while todo:\\n    d,t=heappop(todo)\\n    kyori[t]=d\\n    if t==t1:break\\n    l=g[t]\\n    for li,c in l:\\n      if kyori[li]==-1:\\n        heappush(todo,[c+d,li])\\n  return kyori[t1]\\nans=pow(10,9)\\nallr=[]\\nimport sys\\nsys.setrecursionlimit(10**7)\\ndef dfs(s,k): #s:list, k:set\\n  if len(k)==1:\\n    s.append(k.pop())\\n    return [s]\\n  ret=[]\\n  for ki in k:\\n    ret.extend(dfs(s+[ki],k-{ki}))\\n  return ret\\nallr=dfs([],set(r))\\nd={}\\nfor i in range(r_-1):\\n  for j in range(i+1,r_):\\n    k=dks(r[i]-1,r[j]-1)\\n    d[r[i]-1,r[j]-1]=k\\n    d[r[j]-1,r[i]-1]=k\\n\\nfor j in range(len(allr)):\\n  r1=allr[j]\\n  ansi=0\\n  for i in range(r_-1):\\n    ansi+=d[(r1[i]-1,r1[i+1]-1)]\\n  ans=min(ans,ansi)\\nprint(ans)\\n\", \"#!/usr/bin/env python3\\n\\nfrom itertools import permutations\\nimport numpy as np\\n\\nHUGE = 10 ** 18\\n\\ndef main():\\n    n, m, r = list(map(int, input().split()))\\n    togo = list(map(int, input().split()))\\n    adj_mat = [[HUGE for j in range(n)] for i in range(n)]\\n    for i in range(n):\\n        adj_mat[i][i] = HUGE\\n    for i in range(m):\\n        a, b, c = list(map(int, input().split()))\\n        a0 = a - 1\\n        b0 = b - 1\\n        adj_mat[a0][b0] = c\\n        adj_mat[b0][a0] = c\\n    adj_mat = np.array(adj_mat)\\n\\n    wf(adj_mat, n)\\n\\n    res = HUGE\\n    for way in permutations(togo):\\n        assert len(way) > 1\\n        x = 0\\n        for i in range(len(way) - 1):\\n            x += adj_mat[way[i] - 1][way[i + 1] - 1]\\n        res = min(res, x)\\n\\n    print(res)\\n\\ndef wf(adj_mat, n):\\n    for k in range(n):\\n        for i in range(n):\\n            adj_mat[i, :] = np.minimum(adj_mat[i, :], adj_mat[i][k] + adj_mat[k, :])\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys, re\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees#, log2\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left\\nfrom fractions import gcd\\nfrom heapq import heappush, heappop, heapify\\nfrom functools import reduce\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nr = LIST()\\nABC = [LIST() for _ in range(M)]\\n\\ndist_matrix = [[INF]*N for _ in range(N)]\\n\\nfor A, B, C in ABC:\\n\\tdist_matrix[A-1][B-1] = C\\n\\tdist_matrix[B-1][A-1] = C\\n\\nfor i in range(N):\\n\\tdist_matrix[i][i] = 0\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\n#\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5  \\u6ce8\\u610f PyPy\\u975e\\u5bfe\\u5fdc!!!!\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\ngraph = csr_matrix(dist_matrix) #\\u96a3\\u63a5\\u884c\\u5217\\u304b\\u3089csr\\u884c\\u5217\\u3092\\u3064\\u304f\\u308b\\ndist_matrix = floyd_warshall(graph, directed=False) #\\u7121\\u5411\\u306e\\u5834\\u5408directed=False\\n#dist_matrix\\u306e\\u4e2d\\u8eab\\u306ffloat\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u306e\\u3067\\u6ce8\\u610f\\uff01\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\nans = INF\\nfor x in permutations(r):\\n\\ttmp = 0\\n\\tfor i in range(1, R):\\n\\t\\ttmp += dist_matrix[x[i-1]-1][x[i]-1]\\n\\tans = min(ans, tmp)\\n\\nprint((int(ans)))\\n\\t\\n\", \"from itertools import permutations\\nimport sys\\ninput = sys.stdin.readline\\n\\ndef main():\\n    N, M, RR = map(int, input().split())\\n    R = list(map(int, input().split()))\\n    INF = float(\\\"inf\\\")\\n    T = [[INF] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = tuple(map(int, input().split()))\\n        T[a-1][b-1] = c\\n        T[b-1][a-1] = c\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if T[i][j] > T[i][k]+T[k][j]:\\n                    T[i][j] = T[i][k]+T[k][j]\\n    ans = INF\\n    for rs in permutations(R):\\n        cost = 0\\n        for i in range(RR-1):\\n            cost += T[rs[i]-1][rs[i+1]-1]\\n        ans = min(ans, cost)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations as p\\n \\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\nn, m, r = map(int, input().split())\\nR = list(map(int, input().split()))\\nl = [[0]*n for _ in range(n)]\\nfor _ in range(m):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    l[a][b] = c\\n    l[b][a] = c\\nF = floyd_warshall(l)\\n \\nans = float(\\\"inf\\\")\\nfor v in p(R):\\n    temp = 0\\n    for i in range(r-1):\\n        temp += F[v[i]-1][v[i+1]-1]\\n    ans = min(ans, temp)\\nprint(int(ans))\", \"n,m,R = map(int,input().split())\\nr = list(map(int,input().split()))\\n\\nfrom itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\ncost = np.ones((n,n))*float('inf')\\nfor i in range(m):\\n    a,b,c = map(int,input().split())\\n    cost[a-1][b-1]=cost[b-1][a-1]=c\\nd = floyd_warshall(cost)\\ndef main():\\n    m = float('inf')\\n\\n    for j in permutations(r):\\n        l = 0\\n        for i in range(len(j)-1):\\n            s,t = j[i],j[i+1]\\n            l += d[s-1][t-1]\\n        if l<m:\\n            m = l\\n    print(int(m))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# https://atcoder.jp/contests/abc073/tasks/abc073_d\\n\\nimport sys\\nread = sys.stdin.readline\\n\\n\\ndef read_ints():\\n    return list(map(int, read().split()))\\n\\n\\n\\n# default import\\nfrom itertools import product, permutations, combinations\\n\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import dijkstra\\n\\n# \\u30c0\\u30a4\\u30af\\u30b9\\u30c8\\u30e9\\u304b\\uff1f\\n# \\u5168\\u70b9\\u9593\\u306e\\u6700\\u5c0f\\u8ddd\\u96e2\\u3092\\u53d6\\u5f97\\n# r\\u306b\\u3064\\u3044\\u3066permutation\\u3057\\u3066\\u3002\\u305d\\u306e\\u901a\\u308a\\u306b\\u8a2a\\u308c\\u305f\\u3068\\u304d\\u306e\\u8ddd\\u96e2\\u3092\\u30b7\\u30df\\u30e5\\u30ec\\u30fc\\u30b7\\u30e7\\u30f3\\n# \\u6700\\u5c0f\\u5024\\u3092\\u9078\\u3076\\n\\nN, M, r = read_ints()\\nR = [x - 1 for x in read_ints()]\\nadj_mat = [[0] * N for _ in range(N)]\\nfor _ in range(M):\\n    a, b, c = read_ints()\\n    a -= 1\\n    b -= 1\\n    adj_mat[a][b] = c\\n    adj_mat[b][a] = c\\n\\n# print(csr_matrix(adj_mat, dtype='int'))\\nD = dijkstra(csr_matrix(adj_mat, dtype='int'), directed=False)\\n# print(D)\\n# \\u5168\\u63a2\\u7d22\\u30d1\\u30fc\\u30c8\\nans = 2 ** 31\\n\\n\\ndef get_kyori(p):\\n    ret = 0\\n    for ps, pt in zip(p[:-1], p[1:]):\\n        ret += D[ps, pt]\\n    return ret\\n\\n\\nfor p in permutations(R):\\n    ans = min(ans, get_kyori(p))\\nprint((int(ans)))\\n\", \"import sys\\nimport heapq, math\\nfrom itertools import zip_longest, permutations, combinations, combinations_with_replacement\\nfrom itertools import accumulate, dropwhile, takewhile, groupby\\nfrom functools import lru_cache\\nfrom copy import deepcopy\\n\\nN, M, R = list(map(int, input().split()))\\n\\nRS = list(map(int, input().split()))\\n\\nG = [[] for _ in range(N + 1)]\\n\\nfor i in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    G[A].append((B, C))\\n    G[B].append((A, C))\\n\\n\\ndef dijkstra(s):\\n    d = [1 << 28] * (N + 1)\\n    d[s] = 0\\n    q = []\\n    heapq.heappush(q, (0, s))\\n\\n    while q:\\n        cur = heapq.heappop(q)\\n\\n        if d[cur[1]] != cur[0]:\\n            continue\\n\\n        for nx in G[cur[1]]:\\n            if d[nx[0]] > cur[0] + nx[1]:\\n                d[nx[0]] = cur[0] + nx[1]\\n                heapq.heappush(q, (cur[0] + nx[1], nx[0]))\\n\\n    return d\\n\\n\\nDIRS = [dijkstra(s) for s in RS]\\nMAT = [[d[r] for r in RS] for d in DIRS]\\n\\nmem = {}\\n\\n\\ndef solve(idx, st):\\n    if st == (1 << R) - 1:\\n        return 0\\n    if (idx, st) in mem:\\n        return mem[(idx, st)]\\n\\n    ret = 1 << 28\\n    for i in range(R):\\n        if ((st >> i) & 1) == 0:\\n            ret = min(ret, solve(i, st | (1 << i)) + MAT[idx][i])\\n\\n    mem[(idx, st)] = ret\\n    return ret\\n\\n\\nprint((min([solve(i, 1 << i) for i in range(R)])))\\n\", \"from scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list(map(int, input().split()))\\n    r = [i-1 for i in r]\\n    l = [[0] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        l[a][b] = c\\n        l[b][a] = c\\n    G = csr_matrix(l)\\n    dd = dijkstra(G, directed=False)\\n    ans = 10**100\\n    for i in permutations(r, len(r)):\\n        t = 0\\n        for j in range(len(r)-1):\\n            t += dd[i[j]][i[j+1]]\\n        ans = min(ans, t)\\n    return int(ans)\\nprint((main()))\\n\", \"import sys\\nfrom itertools import permutations\\n\\nsys.setrecursionlimit(10 ** 6)\\nINF = float(\\\"inf\\\")\\nMOD = 10 ** 9 + 7\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list([int(x) - 1 for x in input().split()])\\n    P = permutations(r)\\n\\n    dp = [[float(\\\"inf\\\")] * N for _ in range(N)]\\n    for i in range(N):\\n        dp[i][i] = 0\\n\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        dp[a][b] = c\\n        dp[b][a] = c\\n\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                tmp = dp[i][k] + dp[k][j]\\n                if tmp < dp[i][j]:\\n                    dp[i][j] = tmp\\n\\n    ans = float(\\\"inf\\\")\\n    for p in P:\\n        tmp = 0\\n        for i in range(R - 1):\\n            tmp += dp[p[i]][p[i + 1]]\\n\\n        if tmp < ans:\\n            ans = tmp\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import heapq\\nimport itertools\\nfrom collections import defaultdict\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\n\\nedges = [[] for i in range(n+1)]\\nfor _ in range(m):\\n    _from, to, distance = list(map(int, input().split()))\\n    edges[_from].append([to, distance])\\n    edges[to].append([_from, distance])\\n\\n\\ncomp_edges = defaultdict(dict)\\nfor _from in visiting_town:\\n    seen = [False] * (n+1)\\n    todo = []\\n    for to, dist in edges[_from]:\\n        heapq.heappush(todo, [dist, to])\\n\\n    while todo:\\n        dist, node = heapq.heappop(todo)\\n        if seen[node]:\\n            continue\\n\\n        seen[node] = dist\\n        for to, add_dist in edges[node]:\\n            if seen[to] or to == _from:\\n                continue\\n            new_dist = dist + add_dist\\n            heapq.heappush(todo, [new_dist, to])\\n\\n    for to in visiting_town:\\n        if to == _from:\\n            continue\\n        comp_edges[_from][to] = seen[to]\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        if to in comp_edges[_from]:\\n            dist = comp_edges[_from][to]\\n            result += dist\\n        else:\\n            break\\n    candidates.append(result)\\n\\nprint((min(candidates)))\\n\", \"import sys\\nimport numpy as np \\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nI = np.array(sys.stdin.read().split(), dtype=np.int64)\\nn, m, R = I[:3]\\nr = I[3:3+R] - 1\\na, b, c = I[3+R:].reshape(-1, 3).T\\ngraph = csr_matrix((c, (a-1, b-1)), shape=(n, n))\\n\\ndef main():\\n    dist = floyd_warshall(graph, directed=False).astype(np.int64)\\n    \\n    perms = np.array(list(permutations(r)))\\n    res = dist[perms[:, :-1], perms[:, 1:]]\\n    ans = np.amin(np.sum(res, axis=1))\\n    return ans\\n\\ndef __starting_point():\\n    ans = main()\\n    print(ans)\\n__starting_point()\", \"import itertools\\nimport sys\\n\\ndef main():\\n  input = sys.stdin.readline\\n  n, m, r = map(int, input().split())\\n  R = [int(x) for x in input().split()]\\n  inf = pow(10, 9)+7\\n\\n  roads = [[inf]*n for _ in range(n)]\\n  for i in range(m):\\n    a, b, c = map(int, input().split())\\n    roads[a-1][b-1] = c\\n    roads[b-1][a-1] = c\\n\\n  for k in range(n):\\n    for i in range(n):\\n      for j in range(n):\\n        if roads[i][j] > roads[i][k]+roads[k][j]:\\n          roads[i][j] = roads[i][k]+roads[k][j]\\n\\n  ans = inf\\n  for value in itertools.permutations(R):\\n    sub = 0\\n    for k in range(r-1):\\n      sub += roads[value[k]-1][value[k+1]-1]\\n    if ans > sub:\\n      ans = sub\\n\\n  print(ans)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from scipy.sparse.csgraph import csgraph_from_dense,dijkstra,floyd_warshall\\nN,M,R = map(int,input().split())\\n\\nmachi = list(map(int,input().split()))\\n\\nList = [[10**6]*(N) for i in range(N)]\\nfor i in range(M):\\n  a,b,c = map(int,input().split())\\n  \\n  if List[a-1][b-1] > c:\\n    List[a-1][b-1] = c\\n    List[b-1][a-1] = c\\nG = csgraph_from_dense(List, null_value=10**6)\\nd = floyd_warshall(G)\\n\\nimport itertools\\na = list(itertools.permutations(machi))\\narr = []\\nfor i in a:\\n  ans = 0\\n  for j in range(R):\\n    if j != R-1 :\\n      ans += d[i[j]-1][i[j+1]-1]\\n  arr.append(ans) \\n  \\nprint(int(min(arr)))\", \"from itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nN, M, R = map(int, input().split())\\nR = list(map(int, input().split()))\\nA, B, C = np.array([input().split() for _ in range(M)], dtype=int).T\\nG = csr_matrix((C, (A - 1, B - 1)), shape=(N, N))\\npath = dijkstra(G, directed=False)\\npath = (path + 0.5).astype(int)\\n\\nans = 10 ** 18\\nfor p in permutations(R):\\n    cnt = 0\\n    for x, y in zip(p[:-1], p[1:]):\\n        x -= 1\\n        y -= 1\\n        cnt += path[x][y]\\n    ans = min(ans, cnt)\\n\\nprint(ans)\", \"import numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nn,m,R = map(int, input().split())\\nr = list(map(int, input().split()))\\nr = [i-1 for i in r]\\nabc = [list(map(int, input().split())) for i in range(m)]\\nedge = [[10 ** 9] * (n) for i in range(n)]\\nfor a,b,c in abc:\\n  a-=1\\n  b-=1\\n  edge[a][b] = c\\n  edge[b][a] = c\\nd = floyd_warshall(edge)\\nans = 10 ** 18\\nfor i in permutations(r):\\n  tmp = 0\\n  for j in range(R-1):\\n    tmp += d[i[j]][i[j+1]]\\n  ans = min(ans,tmp)\\nprint(int(ans))\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nINF = 10 ** 18\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\ndp = [[INF for _ in range(N)] for _ in range(N)]\\nfor i in range(N):\\n    for j in range(N):\\n        if i == j:\\n            dp[i][j] = 0\\n\\nfor i in range(M):\\n    a, b, c = map(int, input().split())\\n    dp[a - 1][b - 1] = c\\n    dp[b - 1][a - 1] = c\\n\\nwf = floyd_warshall(dp)\\nans = INF\\nfor i in permutations(r):\\n    tmp = 0\\n    for j in range(1, len(i)):\\n        tmp += wf[i[j - 1] - 1][i[j] - 1]\\n    ans = min(ans, tmp)\\n\\nprint(int(ans))\", \"from itertools import permutations, combinations\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import dijkstra\\nN,M,R = map(int,input().split())\\nr = list(map(int,input().split()))\\nr = [r[i]-1 for i in range(R)]\\nvve = [list(map(int,input().split())) for i in range(M)]\\nv1,v2,edge = zip(*vve)\\nv1 = list(v1)\\nv2 = list(v2)\\nfor i in range(M):\\n  v1[i] -= 1\\n  v2[i] -= 1\\ncsr = csr_matrix((edge,(v1,v2)),shape = (N,N))\\ndist = [[0 for j in range(R)] for i in range(R)]\\nfor i,j in combinations(range(R),2):\\n  r1,r2=r[i],r[j]\\n  dist[i][j] = int(dijkstra(csr, directed = False,indices = r1)[r2])\\n  dist[j][i] = dist[i][j]\\nans = 10**18\\nfor x in permutations(range(R)):\\n  distsum = 0\\n  for i in range(1,R):\\n    distsum += dist[x[i]][x[i-1]]\\n  ans = min(ans,distsum)\\nprint(ans)\", \"def main():\\n    import sys\\n    input = sys.stdin.readline\\n    sys.setrecursionlimit(10**7)\\n    from collections import Counter, deque\\n    #from collections import defaultdict\\n    from itertools import combinations, permutations, accumulate, groupby\\n    #from itertools import product\\n    from bisect import bisect_left,bisect_right\\n    from heapq import heapify, heappop, heappush\\n    from math import floor, ceil\\n    #from operator import itemgetter\\n\\n    inf = 10**17\\n    #mod = 10**9 + 7\\n\\n    N,M,R = map(int, input().split())\\n    r = list(map(int, input().split()))\\n    for i in range(R):\\n        r[i] -= 1\\n\\n    def dijkstra_heap(start,edge):\\n        #\\u59cb\\u70b9\\u304b\\u3089\\u5404\\u9802\\u70b9\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2(\\u9802\\u70b9\\u756a\\u53f7:0~N-1)\\n        d = [inf]*N\\n        used = [False]*N\\n        d[start] = 0\\n        used[start] = True\\n        edgelist = []\\n        #a:\\u91cd\\u307f(//), b:\\u6b21\\u306e\\u9802\\u70b9(%)\\n        for a,b in edge[start]:\\n            heappush(edgelist,a*(10**6)+b)\\n\\n        while len(edgelist):\\n            #\\u307e\\u3060\\u6700\\u77ed\\u8ddd\\u96e2\\u304c\\u6c7a\\u307e\\u3063\\u3066\\u3044\\u306a\\u3044\\u9802\\u70b9\\u306e\\u4e2d\\u304b\\u3089\\u6700\\u5c0f\\u306e\\u8ddd\\u96e2\\u306e\\u3082\\u306e\\u3092\\u63a2\\u3059\\n            minedge = heappop(edgelist)\\n            if used[minedge%(10**6)]:\\n                continue\\n            node = minedge%(10**6)\\n            d[node] = minedge//(10**6)\\n            used[node] = True\\n\\n            for e in edge[node]:\\n                if not used[e[1]]:\\n                    heappush(edgelist,(e[0]+d[node])*(10**6)+e[1])\\n        return d\\n\\n    edge = [[] for i in range(N)]\\n    #edge[i] : i\\u304b\\u3089\\u51fa\\u308b\\u9053\\u306e[\\u91cd\\u307f,\\u884c\\u5148]\\u306e\\u914d\\u5217\\n    for _ in range(M):\\n        x,y,z = map(int,input().split())\\n        edge[x-1].append((z,y-1))\\n        edge[y-1].append((z,x-1))\\n\\n    kyori = []\\n    for i in r:\\n        kyori.append(dijkstra_heap(i,edge))\\n    \\n    res = inf\\n    for i in permutations(range(R), R):\\n        l = list(i)\\n        path = 0\\n        for j in range(R-1):\\n            path += kyori[l[j]][r[l[j+1]]]\\n        res = min(res, path)\\n\\n    print(res)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\n\\ndef warshal_floyd(d):\\n    # d[i][j] := i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n    nonlocal N\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                d[i][j] = min(d[i][j], d[i][k] + d[k][j])\\n    # \\\"\\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bi\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\\"\\u3068\\\"\\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bi\\u304b\\u3089k\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2 + \\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bk\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2 \\u306e\\u548c\\\"\\u3092\\u6bd4\\u8f03\\u3059\\u308b\\n    return d\\n\\n\\nN, M, R = list(map(int, input().split()))  # N\\u500b\\u306e\\u753a, M\\u672c\\u306e\\u9053, R\\u500b\\u306e\\u753a\\u3092\\u8a2a\\u308c\\u308b\\u3053\\u3068\\u306b\\u306a\\u3063\\u305f\\ntown = list([int(x)-1 for x in input().split()])\\ndistant = [[float('inf')]*N for _ in range(N)]\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    distant[a][b] = c\\n    distant[b][a] = c\\nfor i in range(N):\\n    distant[i][i] = 0\\ndistant = floyd_warshall(distant)\\n# warshal_floyd(distant)\\n# print(distant)  # R\\u306f\\u6700\\u59278\\u3060\\u304b\\u3089\\u5168\\u3066\\u3092\\u8a66\\u3057\\u3066\\u3082\\u5927\\u4e08\\u592b\\nans = 10**9\\nfor x in itertools.permutations(town):\\n    x = list(x)\\n    now = x[0]\\n    tmp = 0\\n    for i in range(1, len(x)):\\n        tmp += distant[now][x[i]]\\n        now = x[i]\\n    ans = min(ans, tmp)\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\nr = [ri-1 for ri in r]\\n\\ngraph = [[0]*N for i in range(N)]\\nfor i in range(M):\\n  a, b, c = map(int, input().split())\\n  a -= 1; b -= 1\\n  graph[a][b] = graph[b][a] = c\\n\\ndist = floyd_warshall(graph).astype(int)\\norders = list(itertools.permutations(r, R))\\n\\nans = float('inf')\\nfor order in orders:\\n  d = 0\\n  for u, v in zip(order, order[1:]):\\n    d += dist[u][v]\\n  ans = min(ans, d)\\n\\nprint(ans)\", \"import numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split(' ')))\\n    towns = list(map(lambda t: int(t) - 1, input().split(' ')))\\n    A = np.zeros((N, N), dtype=int)\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split(' ')))\\n        A[a-1][b-1] = c\\n        A[b-1][a-1] = c\\n    fw = floyd_warshall(A)\\n    d = fw[towns, :][:, towns]\\n    min_d = 10**12\\n    for trip in itertools.permutations(range(R)):\\n        min_d = min([min_d, sum([d[trip[i]][trip[i+1]] for i in range(len(trip) - 1)])])\\n    print(int(min_d))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nfrom itertools import permutations, combinations\\nn,m,R = map(int,input().split())\\nr=list(map(int,input().split()))\\nfor i in range(R):\\n    r[i]-=1\\nd = np.zeros((n,n)) \\n# \\u5165\\u529b\\nfor i in range(m):\\n    a,b,c = map(int,input().split())\\n    a,b = a-1, b-1\\n    d[a,b] = c\\ndist =floyd_warshall(d,directed=0).astype(int)\\nans=10**10\\nfor v in permutations(r):\\n    tmp=0\\n    for i in range(R-1):\\n        tmp+=dist[v[i],v[i+1]]\\n    ans=min(ans,tmp)\\nprint(ans)\", \"def main():\\n    from sys import stdin\\n    input = stdin.readline\\n    n, m, r = list(map(int, input().split()))\\n    root = list(map(int, input().split()))\\n    abc = [list(map(int, input().split())) for _ in [0]*m]\\n\\n    import numpy as np\\n    import scipy.sparse.csgraph as sp\\n    inf = float('inf')\\n    d = np.full((n, n), inf)\\n    for i in range(n):\\n        d[i][i] = 0\\n    for a, b, c in abc:\\n        d[a-1][b-1] = c\\n        d[b-1][a-1] = c\\n    # indices=x\\u3067x\\u304b\\u3089\\u306e\\u5358\\u4e00\\u59cb\\u70b9\\u306b,return_predecessors=True\\u3067\\u7d4c\\u8def\\u304c\\u51fa\\u308b\\n    s = sp.shortest_path(d)\\n\\n    from itertools import permutations\\n\\n    p = list(permutations(root))\\n\\n    D = 10**10\\n    for i in p:\\n        d = 0\\n        for j in range(len(i)-1):\\n            d += s[i[j]-1][i[j+1]-1]\\n        D = min(D, d)\\n    print((int(D)))\\n\\n\\nmain()\\n\", \"from scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\n\\ng = [[0] * (N + 1) for _ in range(N + 1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    g[A][B] = C\\n    g[B][A] = C\\ng = floyd_warshall(csgraph_from_dense(g))\\n\\nresult = 1000000000\\nfor p in permutations(r):\\n    t = 0\\n    for i in range(R - 1):\\n        t += g[p[i]][p[i + 1]]\\n    result = min(result, t)\\nprint((int(result)))\\n\", \"import sys\\nfrom itertools import permutations\\n\\nsys.setrecursionlimit(10 ** 6)\\nINF = float(\\\"inf\\\")\\nMOD = 10 ** 9 + 7\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list(map(int, input().split()))\\n    P = permutations(r)\\n\\n    dp = [[INF] * N for _ in range(N)]\\n    for i in range(N):\\n        dp[i][i] = 0\\n\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        dp[a][b] = c\\n        dp[b][a] = c\\n\\n    # \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                tmp = dp[i][k] + dp[k][j]\\n                if tmp < dp[i][j]:\\n                    dp[i][j] = tmp\\n\\n    ans = INF\\n    for p in P:\\n        tmp = 0\\n        for i in range(R - 1):\\n            tmp += dp[p[i] - 1][p[i + 1] - 1]\\n\\n        if tmp < ans:\\n            ans = tmp\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nint1 = lambda x: int(x) - 1\\n\\nN, M, R = map(int, input().split())\\nT = sorted(list(map(int1, input().split())))\\nA = np.array([tuple(map(int, input().split())) for _ in range(M)]).T\\n\\nmatr = csr_matrix((A[2], (A[0] - 1, A[1] - 1)), shape=(N, N))\\nway = [dijkstra(matr, indices=t, directed=False)[T].astype(int).tolist() for t in T]\\n\\nans = float('inf')\\nfor t in itertools.permutations(range(R)):\\n    tmp = 0\\n    for i in range(R - 1):\\n        tmp += way[t[i]][t[i + 1]]\\n    ans = min(ans, tmp)\\nprint(ans)\", \"import heapq\\nimport itertools\\n\\ndef dijkstra(s):\\n    inf = pow(10, 10)\\n    dist = [inf] * (n + 1)\\n    dist[s] = 0\\n    c = [0] * (n + 1)\\n    p = []\\n    heapq.heapify(p)\\n    heapq.heappush(p, (dist[s], s))\\n    while p:\\n        d, u = heapq.heappop(p)\\n        if dist[u] < d:\\n            continue\\n        c[u] = 1\\n        for g in G[u]:\\n            if c[g[0]] == 0 and dist[u] + g[1] < dist[g[0]]:\\n                dist[g[0]] = dist[u] + g[1]\\n                heapq.heappush(p, (dist[g[0]], g[0]))\\n    return dist\\n\\nn, m, r = map(int, input().split())\\nt = list(map(int, input().split()))\\nG = [[] for _ in range(n + 1)]\\nfor _ in range(m):\\n    a, b, c = map(int, input().split())\\n    G[a].append([b, c])\\n    G[b].append([a, c])\\nD = [[0] * r for _ in range(r)]\\nfor i in range(r):\\n    d = dijkstra(t[i])\\n    for j in range(r):\\n        D[i][j] = d[t[j]]\\n\\nx = [i for i in range(r)]\\nans = pow(10, 10)\\nfor y in itertools.permutations(x):\\n    dist = 0\\n    y = list(y)\\n    for i in range(r - 1):\\n        dist += D[y[i]][y[i + 1]]\\n    ans = min(ans, dist)\\nprint(ans)\", \"from itertools import permutations as perm\\n\\ndef warshall(d, n):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if d[i][k] + d[k][j] < d[i][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\nn,m,r = map(int, input().split())\\nrr = [i-1 for i in map(int, input().split())]\\n\\ninf = float('INF')\\nroute = [[inf for j in range(n)] for i in range(n)]\\nfor _ in range(m):\\n    a,b,c = map(int, input().split())\\n    route[a-1][b-1] = route[b-1][a-1] = c\\n\\nwarshall(route, n)\\n\\nans = inf\\nfor tmp in perm(rr):\\n    cost = 0\\n    for i in range(r-1):\\n        cost += route[tmp[i]][tmp[i+1]]\\n    if cost < ans:\\n        ans = cost\\nprint(int(ans))\", \"# coding: utf-8\\nfrom heapq import heappop, heappush\\nfrom itertools import permutations\\n\\nN,M,R=map(int,input().split())\\ntown=list(map(int,input().split()))\\nINF=10**9\\nG=[[INF for i in range(N+1)] for j in range(N+1)]\\n\\nfor i in range(M):\\n    A,B,C=map(int,input().split())\\n    G[A][B]=C\\n    G[B][A]=C\\n\\nd=[[INF*10 for i in range(N+1)] for j in range(R)]\\n\\nfor k in range(R):\\n    r=town[k]\\n    d[k][r]=0\\n    \\n    used=[False for i in range(N+1)]\\n    \\n    heap=[]\\n    heappush(heap,(d[k][r],r))\\n    \\n    while heap:\\n        d_u, u = heappop(heap)\\n\\n        used[u] = True\\n        \\n        if d[k][u] < d_u:\\n            continue\\n        \\n        for v in range(N+1):\\n            if not(used[v]) and d_u + G[u][v] < d[k][v]:\\n                d[k][v] = d_u + G[u][v]\\n                heappush(heap,(d[k][v],v))\\n\\nans=INF\\n\\nL=[i for i in range(R)]\\n\\nfor v in permutations(L,R):\\n    D=0\\n    for i in range(R-1):\\n        D+=d[v[i]][town[v[i+1]]]\\n    ans=min(ans,D)\\n    \\nprint(ans)\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import dijkstra\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = {}\\nfor node in visiting_town:\\n    dist_list = dijkstra(G, directed=False, indices=node-1)\\n    comp_dist[node-1] = dist_list\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\n\\nN,M,R = map(int, input().split())\\nr_list = np.array(list(map(int, input().split()))) -1 \\n\\nnodes = np.full((N,N), np.inf) \\nfor i in range(M):\\n    a,b,c = map(int, input().split())\\n    nodes[a-1,b-1] = nodes[b-1,a-1] = c\\nmin_ds = shortest_path(nodes)\\nres = 10**9\\nfor cmb in itertools.permutations(r_list):\\n    prev = cmb[0]\\n    tmpres = 0\\n    for to in cmb[1:]:\\n        tmpres += min_ds[prev,to]\\n        prev = to\\n    res = min(res, tmpres)\\n\\nprint(int(res))\", \"import numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\n\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int,input().split()))\\nfor i in range(R):\\n    r[i] -= 1\\n\\nconnections = np.zeros((N,N))\\nfor i in range(M):\\n    A,B,C = list(map(int,input().split()))\\n    A,B = A-1,B-1\\n    \\n    connections[A][B] = C\\n    connections[B][A] = C\\n\\nconnections = dijkstra(connections)\\n\\ndef explore(unvisited,total,start):\\n    if unvisited:\\n        ret = float('inf')\\n        for town in unvisited:\\n            ret = min(ret,explore(unvisited-set([town]),\\n                total+connections[start][town],town))\\n        return ret\\n    else:\\n        return total\\n\\nans = float('inf')\\ntargets = set(r)\\nfor s in r:\\n    ans = min(ans,explore(targets-set([s]),0,s))\\nprint(int(ans))\", \"def abc073_d():\\n    from scipy.sparse import lil_matrix\\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations\\n\\n    n, m, r = map(int, input().split())\\n    R = list(map(lambda x: int(x)-1, input().split()))\\n    graph = lil_matrix((n, n), dtype=int)\\n    for _ in range(m):\\n        a, b, c = map(int, input().split())\\n        a -= 1\\n        b -= 1\\n        graph[a, b] = c\\n        graph[b, a] = c\\n\\n    dist = floyd_warshall(csgraph=graph)\\n    #print(dist)\\n\\n    ans = 10**18\\n    for route in permutations(R, r):\\n        tmp = 0\\n        for i in range(r-1):\\n            s = route[i]\\n            t = route[i+1]\\n            tmp += dist[s][t]\\n            if tmp > ans: break\\n        ans = min(ans, tmp)\\n        #print(route, tmp)\\n\\n    print(int(ans))\\n\\ndef __starting_point():\\n    abc073_d()\\n__starting_point()\", \"from itertools import permutations\\nimport heapq\\n\\ndef dijkstra(N, G, s):\\n    INF = 10**40\\n    d = [INF] * N\\n    d[s] = 0\\n    q = [(0, s)]\\n    while q:\\n        c, v = heapq.heappop(q)\\n        if d[v] < c:\\n            continue\\n        for t, co in G[v]:\\n            if d[v] + co < d[t]:\\n                d[t] = d[v] + co\\n                heapq.heappush(q, (d[t], t))\\n    return d\\n\\n\\ndef main():\\n    N, M, _ = list(map(int, input().split()))\\n    R = list(map(int, input().split()))\\n    R = [r - 1 for r in R]\\n    G = [[] for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        G[a - 1].append((b - 1, c))\\n        G[b - 1].append((a - 1, c))\\n    RM = []\\n    for r in R:\\n        d = dijkstra(N, G, r)\\n        RM.append([d[rr] for rr in R])\\n    m = 10 ** 40\\n    for i in permutations(list(range(len(R)))):\\n        t = i[0]\\n        c = 0\\n        for r in i[1:]:\\n            c += RM[t][r]\\n            t = r\\n        m = min(m, c)\\n    return m\\n\\n\\nprint((main()))\\n\", \"import sys\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson, csgraph_from_dense\\nimport itertools\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, M, R = [int(x) for x in input().split()]\\n    r = [int(x) - 1 for x in input().split()]\\n    ABC = [[int(x) for x in input().split()] for _ in range(M)]\\n\\n    EDGE = [[10 ** 9] * N for j in range(N)]\\n\\n    for a, b, c in ABC:\\n        EDGE[a - 1][b - 1] = c\\n        EDGE[b - 1][a - 1] = c\\n\\n    G = csgraph_from_dense(EDGE, null_value=10 ** 9)\\n\\n    d = floyd_warshall(G)\\n\\n    ans = float(\\\"inf\\\")\\n    for a in itertools.permutations(r):\\n        tmp = 0\\n        c = a[0]\\n        for n in a:\\n            tmp += int(d[c][n])\\n            c = n\\n        ans = min(ans, tmp)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n  import numpy as np\\n  import itertools\\n  from scipy.sparse import csr_matrix\\n  from scipy.sparse.csgraph import dijkstra\\n  import sys\\n  readline = sys.stdin.readline\\n  readlines = sys.stdin.readlines\\n\\n  N, M, R= map(int, readline().split())\\n  r = list(map(int,input().split()))\\n  lines = readlines()\\n  edge = np.array([line.split() for line in lines], dtype = np.int64).T\\n  #print(r)\\n  #print(edge)\\n  graph = csr_matrix((edge[2], (edge[:2] - 1)), (N, N))\\n  ans = float(\\\"inf\\\")\\n  distance_mat = {}\\n  for i in r:\\n    distance_mat[i-1] = (dijkstra(graph, directed = False, indices = i-1))\\n\\n  for town in itertools.permutations(r,R):\\n    #print(town)\\n    tmp = 0\\n    for i in range(R-1):\\n      tmp += distance_mat[town[i]-1][town[i+1]-1]\\n    ans = min(ans,tmp)\\n  print(int(ans))\\nmain()\", \"import itertools\\ndef main():\\n    n,m,r=list(map(int,input().split()))\\n    rx=[int(i) for i in input().split()]\\n    inf=100000000\\n    wf=[[inf]*n for i in range(n)]\\n    for i in range(n):\\n        wf[i][i]=0\\n\\n    for i in range(m):\\n        a,b,c=list(map(int,input().split()))\\n        wf[a-1][b-1]=c\\n        wf[b-1][a-1]=c\\n\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if wf[i][j]>wf[i][k]+wf[k][j]:\\n                    wf[i][j]=wf[i][k]+wf[k][j]\\n    cnt=0\\n    l=list(itertools.permutations([i for i in range(r)]))\\n    cnt=inf\\n    for i in l:\\n        cnt_sub=0\\n        for j in range(r-1):\\n            cnt_sub+=wf[rx[i[j]]-1][rx[i[j+1]]-1]\\n        cnt=min(cnt,cnt_sub)\\n    print(cnt)\\nmain()\\n\", \"from collections import deque\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nimport itertools\\n\\nn, m, r = map(int,input().split())\\nrlist = list(map(int,input().split()))\\nfor i in range(r):\\n    rlist[i] -= 1\\ne = np.zeros((n,n), dtype=int)\\n\\nfor i in range(m):\\n    a, b, c = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    e[a][b] = c\\n    e[b][a] = c\\ncsr = csr_matrix(e)\\nf = floyd_warshall(csr)\\n\\nans = 10**18\\nfor v in itertools.permutations(rlist,r):\\n    d = 0\\n    for i in range(r-1):\\n        d += f[v[i]][v[i+1]]\\n    ans = min(ans, d)\\nprint(int(ans))\", \"def main():\\n    from sys import stdin\\n    input = stdin.readline\\n\\n    n, m, r = list(map(int, input().split()))\\n    l = [int(i) - 1 for i in input().split()]\\n    d = [[10**8] * n for _ in range(n)]\\n    for _ in range(m):\\n        i, j, k = list(map(int, input().split()))\\n        d[i-1][j-1] = k\\n        d[j-1][i-1] = k\\n\\n    # Warshall-Floyd algorithm\\n    for k in range(n):\\n        for i in range(n-1):\\n            for j in range(i+1, n):\\n                if d[i][j] > d[i][k] + d[k][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n                    d[j][i] = d[i][j]\\n\\n    # full search\\n    # 8! = 40320\\n    from itertools import permutations\\n\\n    answer = float('inf')\\n    for i in permutations(l):\\n        ans = 0\\n        for j in range(r-1):\\n            ans += d[i[j]][i[j+1]]\\n        if ans < answer:\\n            answer = ans\\n\\n    print(answer)\\n\\nmain()\\n\", \"from scipy.sparse.csgraph import shortest_path\\nimport numpy as np\\nimport itertools\\n\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int, input().split()))\\n\\nedge = np.zeros((N+1,N+1))\\nfor _ in range(M):\\n  A,B,C = list(map(int,input().split()))\\n  edge[A,B] = C  \\ndist = shortest_path(edge, directed = False).astype(int)\\n\\nanswer = 10**18\\nfor visit in itertools.permutations(r):\\n  p = 0\\n  prev = visit[0]\\n  for x in visit[1:]:\\n    if prev != None:\\n      p += dist[prev,x]\\n    prev = x\\n  answer = min(answer,p)\\n  \\nprint(answer)\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nfrom itertools import permutations\\n\\nn,m,r = map(int,input().split())\\nrr = np.array(list(map(int,input().split()))) - 1\\n\\nL = np.zeros((n,n),int)\\nfor _ in range(m):\\n    a,b,c = map(int,input().split())\\n    L[a-1,b-1] = c\\n\\nshortest = floyd_warshall(L, directed = False)\\n\\nans = float('inf')\\n\\nfor route in permutations(rr,len(rr)):\\n    ans = min(ans, shortest[route[:-1],route[1:]].sum())\\nprint(int(ans))\", \"# \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u3067\\u3001\\u5404\\u753a\\u9593\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\u306e\\u66f4\\u65b0\\u306f200^3 = 8,000,000\\n# \\u8a2a\\u308c\\u308b\\u3079\\u304d\\u753aR\\u306f\\u305f\\u304b\\u3060\\u304b8\\u500b\\u306a\\u306e\\u3067\\u3001\\u9806\\u756a\\u306e\\u5168\\u901a\\u308a\\u3092\\u8a66\\u3057\\u30668! = \\u7d0440000\\u901a\\u308a\\n\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall\\nfrom scipy.sparse import csr_matrix\\n\\nN,M,R = map(int,input().split())\\nr = list(map(int,input().split()))\\nr = list(map(lambda x:x-1,r))\\n\\nE = [[0 for j in range(N)] for i in range(N)]\\nfor i in range(M):\\n  a,b,c = map(int,input().split())\\n  E[a-1][b-1] = c\\n  E[b-1][a-1] = c\\n\\nE = np.array(E)\\nE = shortest_path(E,method = \\\"FW\\\")\\n\\n# DFS\\u3067\\u3059\\u3079\\u3066\\u306e\\u6570\\u3092\\u8a66\\u3059\\nstack = []\\nfor i in range(len(r)):\\n  stack.append([r[i],[],0])\\nans = 10 ** 18\\nwhile stack:\\n  v,visited,dist = stack.pop()\\n  if len(visited) != 0:\\n    dist += E[visited[-1]][v]\\n  visited2 = visited.copy()\\n  visited2.append(v)\\n  if len(visited2) == len(r):\\n    if dist < ans:\\n      ans = dist\\n    continue\\n  for i in range(len(r)):\\n    if r[i] not in visited2:\\n      stack.append([r[i],visited2,dist])\\n    \\nprint(int(ans))\", \"from sys import stdin\\nfrom itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\ndef main():\\n    #\\u5165\\u529b\\n    readline=stdin.readline\\n    inf=10**9\\n    n,m,r=map(int,readline().split())\\n    targets=list(map(lambda x:int(x)-1,readline().split()))\\n    G=[[inf]*n for _ in range(n)]\\n    for _ in range(m):\\n        a,b,c=map(int,readline().split())\\n        a-=1\\n        b-=1\\n        G[a][b]=min(G[a][b],c)\\n        G[b][a]=min(G[b][a],c)\\n\\n    #\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\n    G=csgraph_from_dense(G,null_value=10**9)\\n    li=floyd_warshall(G)\\n    li=[list(map(int,li[i])) for i in range(n)]\\n    \\n    #\\u9806\\u5217\\u5168\\u63a2\\u7d22\\n    ans=inf\\n    for p in permutations(targets,r):\\n        tmp=0\\n        for i in range(r-1):\\n            now=p[i]\\n            nex=p[i+1]\\n            tmp+=li[now][nex]\\n        ans=min(ans,tmp)\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import *\\nfrom scipy.sparse.csgraph import *\\nimport numpy as np\\nN,M,R = map(int,input().split())\\nT = list(map(int,input().split()))\\nE = [list(map(int,input().split())) for m in range(M)]\\nG = np.zeros((N,N))\\nA = []\\n\\nfor a,b,c in E:\\n  G[a-1][b-1] = c\\n  G[b-1][a-1] = c\\n\\nF = floyd_warshall(G)\\n\\nfor P in permutations(T):\\n  A+=[sum(F[P[r]-1][P[r+1]-1] for r in range(R-1))]\\n\\nprint(int(min(A)))\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\nfrom sys import stdin\\nnii=lambda:map(int,stdin.readline().split())\\n\\nn,m,r=nii()\\nr=list(nii())\\n\\ns=[[float('inf')]*n for i in range(n)]\\nfor i in range(n):\\n  for j in range(n):\\n    s[i][j]=0\\nfor i in range(m):\\n  a,b,c=nii()\\n  a-=1\\n  b-=1\\n  s[a][b]=c\\n  s[b][a]=c\\n\\nws=floyd_warshall(s)\\n\\nans=10**9\\nfor i in permutations(r):\\n  t_ans=0\\n  for j in range(1,len(i)):\\n    t_ans+=ws[i[j-1]-1][i[j]-1]\\n  ans=min(ans,t_ans)\\nprint(int(ans))\", \"import itertools\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nN, M, R = map(int, input().split())\\nr = tuple(map(int, input().split()))\\n\\nINF = 10**10\\n\\nd = [[INF] * N for _ in range(N)]\\n\\nfor i in range(N):\\n    d[i][i] = 0\\n\\nfor _ in range(M):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    if d[a][b] > c:\\n        d[a][b] = c\\n        d[b][a] = c\\n\\n\\ndef warshall(d):\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if d[i][j] > d[i][k] + d[k][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\n\\n# d = floyd_warshall(d)\\nwarshall(d)\\n\\n\\nans = INF\\nfor p in itertools.permutations(r):\\n    dist = 0\\n    for i in range(R-1):\\n        dist += d[p[i]-1][p[i+1]-1]\\n\\n    if ans > dist:\\n        ans = dist\\n\\nprint(int(ans))\", \"from itertools import permutations as perm\\nfrom scipy.sparse.csgraph import dijkstra as di\\n\\ndef warshall(d, n):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if d[i][k] + d[k][j] < d[i][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\nn,m,r = list(map(int, input().split()))\\nrr = [i-1 for i in map(int, input().split())]\\n\\ninf = float('INF')\\nroute = [[0 for j in range(n)] for i in range(n)]\\nfor _ in range(m):\\n    a,b,c = list(map(int, input().split()))\\n    route[a-1][b-1] = route[b-1][a-1] = c\\n\\nroute = di(route, n)\\n\\nans = inf\\nfor tmp in perm(rr):\\n    cost = 0\\n    for i in range(r-1):\\n        cost += route[tmp[i]][tmp[i+1]]\\n    if cost < ans:\\n        ans = cost\\nprint((int(ans)))\\n\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nN, M, R = list(map(int, input().split()))\\nr = tuple(map(int, input().split()))\\n\\ninf = 10**9\\ngraph = np.ones((N, N), dtype=int)*inf\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    graph[a][b] = c\\n    graph[b][a] = c\\nfy = floyd_warshall(graph)\\n\\nans = 10 ** 9\\nfor p in permutations(r):\\n    tmp = 0\\n    for x, y in zip(p[:-1], p[1:]):\\n        x -= 1\\n        y -= 1\\n        tmp += int(fy[x][y])\\n    ans = min(ans, tmp)\\nprint(ans)\\n\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nn,m,r=map(int,input().split())\\nR=list(map(int,input().split()))\\nl=[[float('inf')]*n for _ in range(n)]\\nfor _ in range(m):\\n    a,b,c,=map(int,input().split())\\n    a-=1\\n    b-=1\\n    l[a][b]=c\\n    l[b][a]=c\\nfor i in range(n):\\n    l[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\ndef warshall_floyd(d):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                d[i][j]=min(d[i][j],d[i][k]+d[k][j])\\n                \\n    return d\\n#F=warshall_floyd(l)\\nF1 = floyd_warshall(l)\\nans=float('inf')\\nfor v in permutations(R):\\n    temp=0\\n    for i in range(r-1):\\n        temp+=F1[v[i]-1][v[i+1]-1]\\n    ans=min(ans,temp)\\nprint(int(ans))\", \"# coding: utf-8\\n# Your code here!\\n\\n#\\u4fdd\\u5b58\\nimport heapq\\nimport sys\\nimport itertools\\n \\nsys.setrecursionlimit(10**7)\\n\\ndef bfs(cost,node):\\n    root[node]=min(root[node],cost)\\n    for next_c,next_n in way[node]:\\n        if root[next_n]==10**9:\\n            heapq.heappush(q,[cost+next_c,next_n])\\n    return\\n    \\nN,M,R=list(map(int,input().split()))\\nr=list(map(int,input().split()))\\n\\nway=[[] for i in range(N)]\\n\\nfor _ in range(M):\\n    A,B,C=list(map(int,input().split()))\\n    way[A-1].append([C,B-1])\\n    way[B-1].append([C,A-1])\\n\\nmade=[]\\n\\n#print(q)\\nfor start in r:\\n    root=[10**9]*N\\n    q=[[0,start-1]]\\n    heapq.heapify(q)#cost\\u3068node\\n    while q:\\n        temp=heapq.heappop(q)\\n        #print(temp)\\n        if root[temp[1]]==10**9:\\n            bfs(temp[0],temp[1])\\n    made.append(root)\\n#print(made)\\n\\nans=10**9\\nfor order in itertools.permutations([i for i in range(len(r))]):\\n    #print(order)\\n    temp=0\\n    for i in range(len(order)-1):\\n        fro=r[order[i]]\\n        aft=r[order[i+1]]\\n        #print(\\\"YES\\\")\\n        #print(order[i]-1,order[i+1]-1)\\n        #print(root[order[i]-1][0])\\n        temp+=abs(made[order[i]][aft-1]-made[order[i]][fro-1])\\n    ans=min(ans,temp)\\nprint(ans)\\n\\n\\n\", \"import time\\nimport itertools\\n\\ndef main():\\n    N,M,R = tuple([int(x) for x in input().split()])\\n\\n    r = [int(x)-1 for x in input().split()]\\n    \\n    w_e = [tuple([int(x)for x in input().split()]) for _ in [0]*M]\\n    w_e = [(a-1,b-1,w) for (a,b,w) in w_e]\\n\\n\\n    g = Graph(list(range(N)),w_e)\\n\\n    minimums = {}\\n    for r_i in r:\\n        minimums[r_i] = g.dijkstra(r_i)[0]\\n\\n    bf = itertools.permutations(r)\\n\\n    candidates = []\\n    for bf_i in bf:\\n        distance = 0\\n        for i in range(R-1):\\n            distance += minimums[bf_i[i]][bf_i[i+1]]\\n        candidates.append(distance)\\n\\n    print(min(candidates))\\n\\ndef speedtest(func,*args):\\n    b = time.perf_counter()\\n    res = func(*args)\\n    e = time.perf_counter()\\n    elapsed = e-b\\n    print(\\\"{} (sec)\\\".format(elapsed))\\n\\n    return res\\n\\nclass Graph:\\n    def __init__(self,w_v,w_e):\\n        super().__init__()\\n        self.size = len(w_v)\\n        self.w_v = [v_i for v_i in w_v]#value of vartex\\n        self.w_e = [{} for _ in [0]*self.size]\\n\\n        self.neighbor = [[] for _ in [0]*self.size]\\n        for a_i,b_i,w_i in w_e:\\n            self.w_e[a_i][b_i] = w_i#weight of edge\\n            self.w_e[b_i][a_i] = w_i#weight of edge\\n\\n            self.neighbor[a_i].append(b_i)\\n            self.neighbor[b_i].append(a_i)\\n\\n    def dijkstra(self,v_n):\\n        d = [-1]*self.size\\n        temp_d = [(10**9,i)for i in range(self.size)]\\n        temp_d[v_n] = (0,v_n)\\n        prev = [-1]*self.size\\n\\n        q = set(range(self.size))\\n        \\n        while len(q)>0:\\n            u = min([temp_d[q_i] for q_i in q])[1]\\n            d[u] = temp_d[u][0]\\n            q.discard(u)\\n            for v in self.neighbor[u]:\\n                temp = temp_d[u][0]+self.w_e[u][v]\\n                if temp_d[v][0]>temp:\\n                    temp_d[v] = temp,v\\n                    prev[v] = u\\n\\n        return d,prev\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations\\nimport sys\\ninput = sys.stdin.readline\\n\\ndef main():\\n    N, M, RR = map(int, input().split())\\n    R = list(map(int, input().split()))\\n    INF = float(\\\"inf\\\")\\n    T = [[INF] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = tuple(map(int, input().split()))\\n        T[a-1][b-1] = c\\n        T[b-1][a-1] = c\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if T[i][j] > T[i][k]+T[k][j]:\\n                    T[i][j] = T[i][k]+T[k][j]\\n    print(min(sum(T[rs[i]-1][rs[i+1]-1] for i in range(RR-1)) for rs in permutations(R)))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import dijkstra\\nimport numpy as np\\nfrom itertools import permutations\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int,input().split()))\\n\\n\\\"\\\"\\\"\\nhttps://juppy.hatenablog.com/entry/2018/11/01/%E8%9F%BB%E6%9C%AC_python_%E5%85%A8%E7%82%B9%E5%AF%BE%E6%9C%80%E7%9F%AD%E7%B5%8C%E8%B7%AF%E6%B3%95%EF%BC%88%E3%83%AF%E3%83%BC%E3%82%B7%E3%83%A3%E3%83%AB%E3%83%95%E3%83%AD%E3%82%A4%E3%83%89%E6%B3%95\\n\\\"\\\"\\\"\\n\\ndef warshall_floyd(d):\\n    n = len(d)\\n    #d[i][j]: i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                d[i][j] = min(d[i][j],d[i][k] + d[k][j])\\n    return d\\n\\ngraph = [[float(\\\"inf\\\") for i in range(N)] for j in range(N)] \\nfor i in range(M):\\n    A,B,C = list(map(int,input().split()))\\n    graph[A-1][B-1] = C\\n    graph[B-1][A-1] = C\\n\\ndist = dijkstra(graph)\\n\\nans = 1e10\\nfor i in permutations(r):\\n    flag = True\\n    way = list(i)\\n    tmp = 0\\n    for i in range(len(way)-1):\\n        if dist[way[i]-1][way[i+1]-1] < 0:\\n            flag = False\\n            break\\n        tmp += dist[way[i]-1][way[i+1]-1]\\n    if flag:\\n        ans = min(tmp,ans)\\n\\nprint((int(ans)))\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time, copy,bisect\\nfrom operator import itemgetter\\n#from heapq import heappush, heappop\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\n#from scipy.sparse import csr_matrix\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\nmod = 10**9 + 7\\n\\nstdin = sys.stdin\\n\\nni = lambda: int(ns())\\nnf = lambda: float(ns())\\nna = lambda: list(map(int, stdin.readline().split()))\\nnb = lambda: list(map(float, stdin.readline().split()))\\nns = lambda: stdin.readline().rstrip()  # ignore trailing spaces\\n\\nN, M, R = na()\\nr = na()\\ng = [[0] * N for _ in range(N)]\\nfor i in range(M):\\n    a, b, c = na()\\n    a -= 1\\n    b -= 1\\n    g[a][b] = c\\n    g[b][a] = c\\n\\ng = np.array(g)\\nd = shortest_path(g)\\n\\nR = len(r)\\nans = inf\\nfor x in itertools.permutations(r):\\n    tmp = 0\\n    for i in range(R-1):\\n        tmp += d[x[i] - 1][x[i+1] - 1]\\n    ans = min(ans, tmp)\\nprint((int(ans)))\\n\\n\\n\\n\\n\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import dijkstra\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = dijkstra(G, directed=False)\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"from itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\n\\nwith open(0) as f:\\n    N, M, R = map(int, f.readline().split())\\n    r = list(map(int, f.readline().split()))\\n    path = [tuple(map(int, line.split())) for line in f.readlines()]\\n\\ngraph = np.full((N,N), np.inf)\\nfor a, b, c in path:\\n    graph[a-1, b-1] = c\\n    graph[b-1, a-1] = c\\n\\ngraph = dijkstra(graph, directed=False)\\nans = np.inf\\npermutation = list(permutations(r))\\nfor p in permutation:\\n    way = 0\\n    for x,y in zip(p[:len(r)], p[1:]):\\n        way += graph[x-1, y-1]\\n    ans = min(ans, way)\\nprint(int(ans))\", \"n,m,r  = list(map(int,input().split()))\\nr_list = list(map(int,input().split()))\\nmatrix = [[float(\\\"inf\\\") for i in range(n)] for j in range(n)]\\nfor c in range(n):\\n  matrix[c][c] = 0\\nfor v in range(m):\\n  a,b,c = list(map(int,input().split()))\\n  if matrix[a-1][b-1] > c:\\n    matrix[a-1][b-1] = c\\n    matrix[b-1][a-1] = c\\nimport numpy as np\\nmatrix = np.array(matrix)\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nload = shortest_path(matrix, directed=False)\\nimport itertools\\nans_list = []\\nfor balls in itertools.permutations(r_list):\\n    balls = list(balls)\\n    ans = 0\\n    for i in range(1,len(r_list)):\\n      ans += load[balls[i-1]-1][balls[i]-1]\\n    ans_list.append(int(ans))\\n    \\nprint((min(ans_list)))\\n      \\n\\n\\n\", \"N, M, K = (int(x) for x in input().split())\\nR = [int(x)-1 for x in input().split()]\\n\\ndist = [[10e8 for _ in range(N)] for _ in range(N)]\\nfor i in range(N): dist[i][i] = 0\\nfor _ in range(M):\\n    a, b, c = (int(x) for x in input().split())\\n    dist[a-1][b-1] = dist[b-1][a-1] = c \\ndel a, b, c, K\\n\\nfor k in range(N):\\n    for i in range(N):\\n        for j in range(i,N):\\n            dist[i][j] = dist[j][i] = min(dist[i][j], dist[i][k]+dist[k][j])\\n\\ndef mindist(x, X):\\n    nonlocal dist\\n    Y = X.copy()\\n    Y.remove(x)\\n    if len(Y) == 0: return 0\\n    return min([dist[x][y] + mindist(y,Y) for y in Y])\\n\\nans = min([mindist(r, R) for r in R])\\nprint(ans)\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\ninf = 10 ** 10\\nn, m, r = map(int,input().split())\\nR = list(map(int,input().split()))\\n\\nedge = [[inf] * n for i in range(n)]\\nfor i in range(m):\\n    a, b, c = map(int,input().split())\\n    edge[a - 1][b - 1] = c\\n\\nG = csgraph_from_dense(edge, null_value = inf)\\nd = floyd_warshall(G, False)\\n\\nrec = inf\\nfor i in permutations(R, r):\\n    temp = 0\\n    #print(i)\\n    for j in range(r):\\n        if j == r - 1:continue\\n        temp += d[i[j] - 1][i[j + 1] - 1]\\n    rec = min(rec, temp)\\n\\nprint(int(rec))\"]",
        "difficulty": "interview",
        "input": "3 3 3\n1 2 3\n1 2 1\n2 3 1\n3 1 4\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc073/tasks/abc073_d"
    },
    {
        "id": 1067,
        "task_id": 2517,
        "test_case_id": 2,
        "question": "There are N towns in the State of Atcoder, connected by M bidirectional roads.\nThe i-th road connects Town A_i and B_i and has a length of C_i.\nJoisino is visiting R towns in the state, r_1,r_2,..,r_R (not necessarily in this order).\nShe will fly to the first town she visits, and fly back from the last town she visits, but for the rest of the trip she will have to travel by road.\nIf she visits the towns in the order that minimizes the distance traveled by road, what will that distance be?\n\n-----Constraints-----\n - 2≤N≤200\n - 1≤M≤N×(N-1)/2\n - 2≤R≤min(8,N) (min(8,N) is the smaller of 8 and N.)\n - r_i≠r_j (i≠j)\n - 1≤A_i,B_i≤N, A_i≠B_i\n - (A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)\n - 1≤C_i≤100000\n - Every town can be reached from every town by road.\n - All input values are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN M R\nr_1 ... r_R\nA_1 B_1 C_1\n:\nA_M B_M C_M\n\n-----Output-----\nPrint the distance traveled by road if Joisino visits the towns in the order that minimizes it.\n\n-----Sample Input-----\n3 3 3\n1 2 3\n1 2 1\n2 3 1\n3 1 4\n\n-----Sample Output-----\n2\n\nFor example, if she visits the towns in the order of 1, 2, 3, the distance traveled will be 2, which is the minimum possible.",
        "solutions": "[\"from itertools import permutations as p\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nn, m, r = map(int, input().split())\\nR = list(map(int, input().split()))\\nl = [[0]*n for _ in range(n)]\\nfor _ in range(m):\\n  a, b, c = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  l[a][b] = c\\n  l[b][a] = c\\nF = floyd_warshall(l)\\n\\nans = float(\\\"inf\\\")\\nfor v in p(R):\\n  temp = 0\\n  for i in range(r-1):\\n    temp += F[v[i]-1][v[i+1]-1]\\n  ans = min(ans, temp)\\n  \\nprint(int(ans))\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import floyd_warshall\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = floyd_warshall(G, directed=False)\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"def main():\\n    import itertools\\n    n,m,r=list(map(int,input().split()))\\n    rx=[int(i) for i in input().split()]\\n    inf=100000000\\n    wf=[[inf]*n for i in range(n)]\\n    for i in range(n):\\n        wf[i][i]=0\\n\\n    for i in range(m):\\n        a,b,c=list(map(int,input().split()))\\n        wf[a-1][b-1]=c\\n        wf[b-1][a-1]=c\\n\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if wf[i][j]>wf[i][k]+wf[k][j]:\\n                    wf[i][j]=wf[i][k]+wf[k][j]\\n    cnt=0\\n    l=list(itertools.permutations([i for i in range(r)]))\\n    cnt=inf\\n    for i in l:\\n        cnt_sub=0\\n        for j in range(r-1):\\n            cnt_sub+=wf[rx[i[j]]-1][rx[i[j+1]]-1]\\n        cnt=min(cnt,cnt_sub)\\n    print(cnt)\\nmain()\\n\", \"import itertools as it\\nn,m,r=map(int,input().split())\\nll=list(map(int,input().split()))\\nd=[[float(\\\"inf\\\") for _ in range(n)] for _ in range(n)] #\\u5404\\u9802\\u70b9\\u304b\\u3089\\u5404\\u9802\\u70b9\\u3078\\u306e\\u6700\\u5c0f\\u30b3\\u30b9\\u30c8\\nfor i in range(m): #\\u91cd\\u8907\\u306a\\u3057ver\\n    a,b,t=map(int,input().split())\\n    d[a-1][b-1]=t\\n    d[b-1][a-1]=t #\\u6709\\u5411\\u30b0\\u30e9\\u30d5\\u306e\\u5834\\u5408\\u6d88\\u3059\\nfor i in range(n):\\n    d[i][i]=0\\nfor k in range(n):\\n    for i in range(n):\\n        for j in range(n):\\n            if d[i][j]>d[i][k]+d[k][j]:\\n                d[i][j]=d[i][k]+d[k][j]\\nans=10**9\\nfor p in it.permutations(ll):\\n    tmp=0\\n    for j in range(r-1):\\n        s,g=p[j]-1,p[j+1]-1\\n        tmp+=d[s][g]\\n    ans=min(ans,tmp)\\nprint(ans)\", \"import sys, re\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees#, log2\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left\\nfrom fractions import gcd\\nfrom heapq import heappush, heappop, heapify\\nfrom functools import reduce\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nr = LIST()\\nABC = [LIST() for _ in range(M)]\\n\\ndist_matrix = [[0]*N for _ in range(N)]\\nfor A, B, C in ABC:\\n\\tdist_matrix[A-1][B-1] = C\\n\\tdist_matrix[B-1][A-1] = C\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\n#\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5  \\u6ce8\\u610f PyPy\\u975e\\u5bfe\\u5fdc!!!!\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\ngraph = csr_matrix(dist_matrix) #\\u96a3\\u63a5\\u884c\\u5217\\u304b\\u3089csr\\u884c\\u5217\\u3092\\u3064\\u304f\\u308b\\ndist_matrix = floyd_warshall(graph, directed=False) #\\u7121\\u5411\\u306e\\u5834\\u5408directed=False\\n#dist_matrix\\u306e\\u4e2d\\u8eab\\u306ffloat\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u306e\\u3067\\u6ce8\\u610f\\uff01\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\nans = INF\\nfor x in permutations(r):\\n\\tdis = 0\\n\\tfor i in range(1, R):\\n\\t\\tdis += dist_matrix[x[i-1]-1][x[i]-1]\\n\\tans = min(ans, dis)\\n\\nprint((int(ans)))\\n\\n\\n\\t\\n\", \"# ABC073 D - joisino's travel\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\n\\ninf = 10**18\\ndist = [[inf]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = map(int, input().split())\\n    dist[A-1][B-1] = dist[B-1][A-1] = C\\n\\ndist = floyd_warshall(dist)\\n\\nans = inf\\nfor rr in permutations(r):\\n    cost = 0\\n    for i in range(R-1):\\n        cost += dist[rr[i]-1][rr[i+1]-1]\\n    if cost < ans:\\n        ans = cost\\nprint(int(ans))\", \"import itertools\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nN, M, R = list(map(int, input().split()))\\nr = tuple(map(int, input().split()))\\n\\nINF = 10**10\\n\\nd = [[INF] * N for _ in range(N)]\\n\\nfor i in range(N):\\n    d[i][i] = 0\\n\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    if d[a][b] > c:\\n        d[a][b] = c\\n        d[b][a] = c\\n\\n# for k in range(N):\\n#     for i in range(N):\\n#         for j in range(N):\\n#             d[i][j] = min(d[i][j], d[i][k] + d[k][j])\\n\\nd = floyd_warshall(d)\\n\\n\\nans = INF\\nfor p in itertools.permutations(r):\\n    dist = 0\\n    for i in range(R-1):\\n        dist += d[p[i]-1][p[i+1]-1]\\n\\n    ans = min(ans, dist)\\n\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\n\\nd = [[0] * (N + 1) for _ in range(N + 1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    d[A][B] = C\\n    d[B][A] = C\\ng = csgraph_from_dense(d)\\ng = floyd_warshall(g)\\n\\nresult = 1000000000\\nfor p in permutations(r):\\n    t = 0\\n    for i in range(R - 1):\\n        t += g[p[i]][p[i + 1]]\\n    result = min(result, t)\\nprint((int(result)))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heapify,heappush,heappop\\nreadline=sys.stdin.readline\\n\\ndef main():\\n    N,M,R=list(map(int,readline().split()))\\n    r=list(map(int,readline().split()))\\n    ew=[tuple(map(int,readline().split())) for _ in range(M)]\\n    edges=defaultdict(list)\\n    wt=defaultdict(int)\\n    for e in ew:\\n        edges[e[0]].append(e[1])\\n        edges[e[1]].append(e[0])\\n        wt[(e[0],e[1])]=e[2]\\n        wt[(e[1],e[0])]=e[2]\\n    inf=float('inf')\\n    def dijkstra(v):\\n        dist=[inf]*(N+1)\\n        dist[v]=0\\n        vw=[(0,v)]\\n        heapify(vw)\\n        while vw:\\n            cvw=heappop(vw)\\n            if cvw[0]>dist[cvw[1]]:\\n                continue\\n            for w in edges[cvw[1]]:\\n                cand=cvw[0]+wt[(cvw[1],w)]\\n                if dist[w]>cand:\\n                    dist[w]=cand\\n                    heappush(vw,(cand,w))\\n        return dist\\n    distdict=dict((v,dijkstra(v)) for v in r)\\n    mindist=inf\\n    def permlist(lst):\\n        tmp=[]\\n        if not lst:\\n            return [[]]\\n        for i,x in enumerate(lst):\\n            lstx=lst[:i]+lst[i+1:]\\n            ret=permlist(lstx)\\n            for e in ret:\\n                e.append(x)\\n            tmp.extend(ret)\\n        return tmp\\n    #print(distdict,permlist(r))\\n    pathlist=[sum([distdict[v][w] for v,w in zip(lst,lst[1:])]) for lst in permlist(r)]\\n    print((min(pathlist)))\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\ndef warshallFloyd():\\n    for k in range(N):\\n        for i in range(N-1):\\n            for j in range(i, N):\\n                tmp = min(E[i][j], E[i][k] + E[k][j])\\n                E[i][j] = E[j][i] = tmp\\n\\ndef dfs(c, p, d):\\n    if c == R:\\n        ans[0] = min(ans[0], d)\\n        return\\n    for i in range(R):\\n        if used[i]:\\n            continue\\n        used[i] = True\\n        if p == -1:\\n            dfs(c+1, i, 0)\\n        else:\\n            dfs(c+1, i, d + E[S[p]-1][S[i]-1])\\n        used[i] = False\\n\\nN, M, R = map(int, input().split())\\nS = list(map(int, input().split()))\\nE = [[float('inf')] * N for _ in range(N)]\\nfor i in range(N):\\n    E[i][i] = 0\\nfor _ in range(M):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    E[a][b] = E[b][a] = c\\nwarshallFloyd()\\nans = [float('inf')]\\nused = [False] * R\\ndfs(0, -1, 0)\\nprint(*ans)\", \"import sys\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\ninput = sys.stdin.readline\\nans = -1\\n\\nn, m, r = list(map(int, input().split()))\\nr  = list(map(int, input().split()))\\nA, B, C = [], [], []\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    A.append(a-1)\\n    B.append(b-1)\\n    C.append(c)\\n\\nwf = floyd_warshall(csr_matrix((C, (A, B)), shape = (n, n)), directed=False)\\n\\nfor pi in permutations(r):\\n    cand = 0\\n    for r1, r2 in zip(pi, pi[1:]):\\n        cand += int(wf[r1-1][r2-1])\\n    if ans < 0 or ans > cand:\\n        ans = cand\\n\\nprint(ans)\\n\", \"#!/usr/bin/env python3\\nimport sys\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    import numpy as np    \\n    from scipy.sparse import coo_matrix    \\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations, islice\\n    from functools import reduce\\n    # coo_matrix((data, (i, j)), [shape=(M, N)])\\n    mat = floyd_warshall(coo_matrix((C, (A, B)), shape=(N+1, N+1), dtype=np.int32).tocsr(), directed=False)\\n    f = lambda a, b: (a[0]+mat[a[1]][b], b)\\n    return int(min(reduce(f, q, (0, s))[0] for s, *q in permutations(r)))\\n\\n# Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    print((solve(N, M, R, r, A, B, C)))\\n\\ndef test():\\n    import doctest\\n    doctest.testmod()\\n\\ndef __starting_point():\\n    #test()\\n    main()\\n\\n__starting_point()\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nn,m,r=map(int,input().split())\\nvis=list(map(int,input().split()))\\nINF=10**18\\npath=[[INF]*n for i in range(n)]\\nfor i in range(m):\\n  a,b,c=map(int,input().split())\\n  path[a-1][b-1]=c\\n  path[b-1][a-1]=c\\npath=floyd_warshall(path)\\nans=10**18\\nfor root in permutations(vis):\\n  cnt=0\\n  for j in range(1,r):\\n    cnt+=path[root[j-1]-1][root[j]-1]\\n  ans=min(ans,cnt)\\nprint(int(ans))\", \"import sys\\nfrom itertools import permutations\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n\\n    def warshall_floyd(d):\\n        # d[i][j]: i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n        for k in range(N):\\n            for i in range(N):\\n                for j in range(N):\\n                    tmp = d[i][k] + d[k][j]\\n                    if d[i][j] > tmp:\\n                        d[i][j] = tmp\\n        return d\\n\\n    r = list(map(int, input().split()))\\n    d = [[float(\\\"inf\\\")] * N for i in range(N)]\\n    for i in range(M):\\n        x, y, z = list(map(int, input().split()))\\n        d[x - 1][y - 1] = z\\n        d[y - 1][x - 1] = z\\n    for i in range(N):\\n        d[i][i] = 0  # \\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\n    D = warshall_floyd(d)\\n    P = list(permutations(r))\\n    cost = float(\\\"inf\\\")\\n    for p in P:\\n        c = 0\\n        for i in range(R - 1):\\n            c += D[p[i] - 1][p[i + 1] - 1]\\n        if c < cost:\\n            cost = c\\n    print(cost)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"#!/usr/bin/env python3\\nimport sys\\nimport itertools\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    from scipy.sparse import csr_matrix\\n    from scipy.sparse.csgraph import floyd_warshall\\n    matrix = [[0]*N for _ in range(N)]\\n    for i in range(M):\\n        matrix[A[i]-1][B[i]-1] = C[i]\\n        matrix[B[i]-1][A[i]-1] = C[i]\\n    \\n    dist_matrix = floyd_warshall(csgraph=matrix, directed=False)\\n\\n    perm = list(itertools.permutations(r)) \\n\\n    answer = float('inf')\\n    for p in perm:\\n        a = 0\\n        for index in range(len(p)-1):\\n            a += dist_matrix[p[index]-1][p[index+1]-1]\\n        answer = min(answer,a)\\n    print((int(answer)))\\n    return\\n\\n\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    solve(N, M, R, r, A, B, C)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\nn,m,r,*L=map(int,open(0).read().split())\\nf=floyd_warshall(csr_matrix((L[r+2::3],(L[r::3],L[r+1::3])),(n+1,n+1)),0)\\nprint(int(min(sum(f[s,t]for s,t in zip(o,o[1:]))for o in permutations(L[:r]))))\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path\\n\\nN,M,R = map(int, input().split())\\nrs = np.array(list(map(int, input().split()))) -1\\n\\nnodes = np.full((N,N), np.inf) \\nfor i in range(M):\\n  a,b,c = map(int, input().split())\\n  nodes[a-1,b-1] = nodes[b-1,a-1] = c\\nmin_ds = shortest_path(nodes)\\nres = float('inf')\\nfor p in itertools.permutations(rs):\\n  prev = p[0]\\n  tmp = 0\\n  for next in p[1:]:\\n    tmp += min_ds[prev,next]\\n    prev = next\\n  res = min(res, tmp)\\n\\nprint(int(res))\", \"def main():\\n  import numpy as np\\n  import itertools\\n  from scipy.sparse import csr_matrix\\n  from scipy.sparse.csgraph import floyd_warshall\\n  import sys\\n  readline = sys.stdin.readline\\n  readlines = sys.stdin.readlines\\n\\n  N, M, R= map(int, readline().split())\\n  r = list(map(int,input().split()))\\n  lines = readlines()\\n  edge = np.array([line.split() for line in lines], dtype = np.int64).T\\n  graph = csr_matrix((edge[2], (edge[:2] - 1)), (N, N))\\n  distance_mat = floyd_warshall(graph,directed = False)\\n  #print(distance_mat)\\n  ans = float(\\\"inf\\\")\\n  for town in itertools.permutations(r,R):\\n    #print(town)\\n    tmp = 0\\n    for i in range(R-1):\\n      tmp += distance_mat[town[i]-1][town[i+1]-1]\\n    ans = min(ans,tmp)\\n  print(int(ans))\\nmain()\", \"import sys\\nimport numpy as np \\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nI = np.array(sys.stdin.read().split(), dtype=np.int64)\\nn, m, R = I[:3]\\nr = I[3:3+R] - 1\\na, b, c = I[3+R:].reshape(-1, 3).T\\na -= 1; b -= 1\\ngraph = csr_matrix((c, (a, b)), shape=(n, n))\\n\\ndef main():\\n    dist = floyd_warshall(graph, directed=False).astype(np.int64)\\n    *perms, = permutations(r)\\n    perms = np.array(perms)\\n    res = dist[perms[:, :-1], perms[:, 1:]]\\n    ans = np.amin(np.sum(res, axis=1))\\n    return ans\\n\\ndef __starting_point():\\n    ans = main()\\n    print(ans)\\n__starting_point()\", \"import itertools\\nfrom scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nN, M, R = list(map(int, input().split()))\\nr = [int(i) for i in input().split()]\\n\\nG = [[10**8]*(N+1) for _ in range(N+1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    G[A][B] = C\\n    G[B][A] = C\\n\\nDG = csgraph_from_dense(G, null_value=10**8)\\nd = floyd_warshall(DG)\\n\\nans = 10**8\\nfor route in itertools.permutations(r):\\n    c = 0\\n    for i in range(len(route)-1):\\n        c += d[route[i]][route[i+1]]\\n    ans = min(ans, c)\\nprint((int(ans)))\\n\", \"import heapq\\nfrom itertools import permutations\\nn, m, r = list(map(int, input().split()))\\nR = list([int(x)-1 for x in input().split()])\\nedges = [[]for _ in range(n)]\\nfor _ in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    edges[a].append((c, b))\\n    edges[b].append((c, a))\\n\\n\\ndef dijkstra(init_v):\\n    next_v = [(0, init_v)]\\n    INF = 10**18\\n    dist = [INF]*n\\n    dist[init_v] = 0\\n    while next_v:\\n        d, v = heapq.heappop(next_v)\\n        if dist[v] < d:\\n            continue\\n        for d, v2 in edges[v]:\\n            if dist[v2] <= dist[v]+d:\\n                continue\\n            dist[v2] = dist[v]+d\\n            heapq.heappush(next_v, (dist[v2], v2))\\n    return dist\\n\\n\\ndists = []\\nfor x in R:\\n    dist = dijkstra(x)\\n    dists.append(dist)\\n\\nINF = 10**18\\nans = INF\\nfor subset in permutations(list(range(r))):\\n    d = 0\\n    for v, v2 in zip(subset, subset[1:]):\\n        d += dists[v][R[v2]]\\n    if d < ans:\\n        ans = d\\n\\nprint(ans)\\n\", \"# ABC073 D - joisino's travel\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\n\\ninf = 10**18\\ndist = [[inf]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = map(int, input().split())\\n    dist[A-1][B-1] = dist[B-1][A-1] = C\\n\\ndist = floyd_warshall(dist)\\n\\nans = inf\\nfor rr in permutations(r):\\n    cost = 0\\n    for i in range(R-1):\\n        cost += dist[rr[i]-1][rr[i+1]-1]\\n    ans = min(ans, cost)\\nprint(int(ans))\", \"from itertools import permutations\\n\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph._shortest_path import floyd_warshall\\n\\n\\ndef solve():\\n    N, M, R = list(map(int, input().split()))\\n    r = list([int(x)-1 for x in input().split()])\\n    dist = [[0 for _ in range(N)] for _ in range(N)]\\n    for _ in range(M):\\n        A, B, C = list(map(int, input().split()))\\n        dist[A-1][B-1] = C\\n        dist[B-1][A-1] = C\\n    fw = floyd_warshall(csr_matrix(dist))\\n    \\n    ans = 10**12\\n    for visit in permutations(r, R):\\n        m_sum = 0\\n        for i in range(R-1):\\n            m_sum += fw[visit[i]][visit[i+1]]\\n        ans = min(ans, m_sum)\\n    print((int(ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"import numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nn, m, r = map(int,input().split()) #n:\\u9802\\u70b9\\u6570\\u3000m:\\u8fba\\u306e\\u6570 r:\\u8a2a\\u308c\\u308b\\u753a\\u306e\\u6570\\nR = list(map(int,input().split()))\\n\\nd = np.array([[float(\\\"inf\\\") for i in range(n)] for i in range(n)])\\n#d[u][v] : \\u8fbauv\\u306e\\u30b3\\u30b9\\u30c8(\\u5b58\\u5728\\u3057\\u306a\\u3044\\u3068\\u304d\\u306finf)\\nfor i in range(m):\\n    x, y, z = map(int,input().split())\\n    d[x-1][y-1] = z\\n    d[y-1][x-1] = z\\nfor i in range(n):\\n    d[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\nwa = floyd_warshall(d)\\n\\n# \\u8a2a\\u308c\\u308b\\u753a\\u306e\\u9806\\u756a\\u3092\\u9806\\u5217\\u3067\\u7528\\u610f\\nfrom itertools import permutations\\nptn = permutations(R)\\n\\nans = 10**19\\nfor p in ptn:\\n  cost = 0\\n  roots = [(j-1, i-1) for i, j in zip(p, p[1:])]\\n  for i, j in roots:\\n    cost += wa[i][j]\\n  ans = min(ans, cost)\\n  \\nprint(int(ans))\", \"import itertools\\n\\n\\ndef dijkstra(v, G):\\n    import heapq\\n\\n    ret = [10 ** 10] * len(G)\\n    ret[v] = 0\\n    q = [(ret[i], i) for i in range(len(G))]\\n    heapq.heapify(q)\\n\\n    while len(q):\\n        tmpr, u = heapq.heappop(q)\\n        if tmpr == ret[u]:\\n            for w, l in G[u]:\\n                if ret[w] > ret[u] + l:\\n                    ret[w] = ret[u] + l\\n                    heapq.heappush(q, (ret[w], w))\\n    return ret\\n\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\nNE = [[] for _ in range(N)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    NE[A - 1].append((B - 1, C))\\n    NE[B - 1].append((A - 1, C))\\n\\nRE = [[0] * R for _ in range(R)]\\nfor iR in range(R):\\n    for jR in range(iR + 1, R):\\n        lR = dijkstra(r[iR] - 1, NE)[r[jR] - 1]\\n        RE[iR][jR] = lR\\n        RE[jR][iR] = lR\\n\\nans = 10 ** 10\\nfor p in itertools.permutations(list(range(R))):\\n    ans = min(ans, sum([RE[p[i]][p[i + 1]] for i in range(R - 1)]))\\n\\nprint(ans)\\n\", \"# solution\\n\\nimport itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nint1 = lambda x: int(x) - 1\\n\\nN, M, R = map(int, input().split())\\nT = sorted(list(map(int1, input().split())))\\nA = np.array([tuple(map(int, input().split())) for _ in range(M)]).T\\n\\nmatr = csr_matrix((A[2], (A[0] - 1, A[1] - 1)), shape=(N, N))\\nway = [dijkstra(matr, indices=t, directed=False)[T].astype(int).tolist() for t in T]\\n\\nresult = float('inf')\\nfor t in itertools.permutations(range(R)):\\n    tmp = 0\\n    for i in range(R - 1):\\n        tmp += way[t[i]][t[i + 1]]\\n    result = min(result, tmp)\\nprint(result)\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nsys.setrecursionlimit(10 ** 9)\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nA = [a-1 for a in LIST()]\\nG = list2d(N, N, INF)\\nfor i in range(M):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    G[a][b] = c\\n    G[b][a] = c\\n\\n# \\u5168\\u4f53\\u30b0\\u30e9\\u30d5\\u306e\\u6700\\u77ed\\u8ddd\\u96e2(\\u3053\\u3053\\u304b\\u3089\\u5fc5\\u8981\\u306a\\u9802\\u70b9\\u9593\\u3060\\u3051\\u4f7f\\u3046)\\nwf = floyd_warshall(G)\\n\\nans = INF\\nfor r in range(R):\\n    # TSP(\\u5de1\\u56de\\u30bb\\u30fc\\u30eb\\u30b9\\u30de\\u30f3)\\n    dp = list2d(1<<R, R, INF)\\n    dp[1<<r][r] = 0\\n    for bit in range(1, (1<<R)-1):\\n        for i in range(R):\\n            if not (bit >> i & 1):\\n                continue\\n            for j in range(R):\\n                if bit >> j & 1:\\n                    continue\\n                a, b = A[i], A[j]\\n                dp[bit|1<<j][j] = min(dp[bit|1<<j][j], dp[bit][i] + int(wf[a,b]))\\n    ans = min(ans, min(dp[-1]))\\nprint(ans)\\n\", \"import numpy as np\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\nI = [int(_) for _ in open(0).read().split()]\\nN, M, R = I[:3]\\nr = np.array(I[3:3 + R])\\nABC = I[3 + R:]\\nF = floyd_warshall(csr_matrix((ABC[2::3], (ABC[::3], ABC[1::3])), (N + 1, N + 1)), 0).astype(np.int64)\\nans = float('inf')\\nfor root in itertools.permutations(r, R):\\n    ans = min(ans, sum(F[i,j] for i, j in zip(root, root[1:])))\\nprint(ans)\\n\", \"#!/usr/bin python3\\n# -*- coding: utf-8 -*-\\n\\n# \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5\\n# \\u5168\\u9802\\u70b9\\u9593\\u6700\\u77ed\\u8def\\n# d[i][j]\\u306f2\\u9802\\u70b9\\u9593i, j\\u9593\\u306e\\u79fb\\u52d5\\u30b3\\u30b9\\u30c8\\u3092\\u683c\\u7d0d, M\\u306f\\u9802\\u70b9\\u6570\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\n\\ndef main():\\n    N, M, R = map(int,input().split()) #N:\\u9802\\u70b9\\u6570\\u3000M:\\u8fba\\u306e\\u6570\\n    d = [[float(\\\"inf\\\")]*N for i in range(N)]\\n    #d[u][v] : \\u8fbauv\\u306e\\u30b3\\u30b9\\u30c8(\\u5b58\\u5728\\u3057\\u306a\\u3044\\u3068\\u304d\\u306finf)\\n    r = list(map(int,input().split()))\\n    for i in range(M):\\n        u, v, w = map(int,input().split())\\n        d[u-1][v-1] = w\\n        d[v-1][u-1] = w\\n    for i in range(N):\\n        d[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\n    cost = floyd_warshall(d)\\n    L = list(permutations(r, R))\\n    ret = 10**9\\n    for rt in L:\\n        cos = 0\\n        for i in range(1,R):\\n            cos += cost[rt[i]-1][rt[i-1]-1]\\n        ret = min(ret, cos)\\n    print(int(ret))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nINF = 1001001001\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\ngraph = [[INF]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    graph[A-1][B-1] = C\\n    graph[B-1][A-1] = C\\ndist_matrix = floyd_warshall(graph)\\n\\nans = INF\\nfor root in permutations(r):\\n    cnt = 0\\n    for j in range(R-1):\\n        cnt += dist_matrix[root[j]-1][root[j+1]-1]\\n    ans = min(ans, cnt)\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\nabc = [list(map(int, input().split())) for _ in range(M)]\\n\\nr = [e - 1 for e in r]\\n\\ng = [[0] * N for _ in range(N)]\\nfor a, b, c in abc:\\n\\ta -= 1\\n\\tb -= 1\\n\\tg[a][b] = c\\n\\tg[b][a] = c\\n\\ndist = floyd_warshall(csr_matrix(g))\\n\\nans = float(\\\"inf\\\")\\nfor pat in permutations(r, len(r)):\\n\\tsm = 0\\n\\tfor e1, e2 in zip(pat, pat[1:]):\\n\\t\\tsm += dist[e1][e2]\\n\\n\\tans = min(ans, sm)\\n\\nans = int(ans)\\nprint(ans)\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\n\\ns = [[10**9]*N for i in range(N)]\\nt = list(map(int, input().split()))\\nfor i in range(M):\\n    a, b, c = map(int, input().split())\\n    s[a-1][b-1] = c\\n    s[b-1][a-1] = c\\n    \\ns = floyd_warshall(s)\\nans = float('inf')\\nfor i in permutations(t):\\n    A_n = 0\\n    for j in range(R-1):\\n        A_n += s[i[j] - 1][i[j+1] - 1]\\n    ans = min(ans, A_n)\\n    \\nprint(int(ans))\", \"import sys\\nfrom itertools import permutations\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\nMOD = 1000000007\\n\\n\\ndef main():\\n    N, M, R = list(map(int, readline().split()))\\n    city = [int(s) - 1 for s in readline().split()]\\n    ABC = list(map(int, read().split()))\\n\\n    G = [[INF] * N for _ in range(N)]\\n    for a, b, c in zip(*[iter(ABC)] * 3):\\n        G[a - 1][b - 1] = G[b - 1][a - 1] = c\\n\\n    for i in range(N):\\n        G[i][i] = 0\\n\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if G[i][j] > G[i][k] + G[k][j]:\\n                    G[i][j] = G[i][k] + G[k][j]\\n\\n    ans = INF\\n    for plan in permutations(city):\\n        res = 0\\n        for i in range(R - 1):\\n            res += G[plan[i]][plan[i + 1]]\\n        if ans > res:\\n            ans = res\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"#!/usr/bin/env python3\\nimport sys\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    from scipy.sparse import coo_matrix    \\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations\\n    from functools import reduce\\n    # coo_matrix((data, (i, j)), [shape=(M, N)])\\n    mat = floyd_warshall(coo_matrix((C, (A, B)), shape=(N+1, N+1)).tocsr(), directed=False)\\n    f = lambda a, b: (a[0]+mat[a[1]][b], b)\\n    return int(min(reduce(f, q, (0, s))[0] for s, *q in permutations(r)))\\n\\n# Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    print(solve(N, M, R, r, A, B, C))\\n\\ndef test():\\n    import doctest\\n    doctest.testmod()\\n\\ndef __starting_point():\\n    #test()\\n    main()\\n__starting_point()\", \"n,m,r_=list(map(int,input().split()))\\nr=list(map(int,input().split()))\\ng=[[]for _ in range(n)]\\nfor _ in range(m):\\n  a,b,c=list(map(int,input().split()))\\n  a-=1\\n  b-=1\\n  g[a].append([b,c])\\n  g[b].append([a,c])\\nfrom heapq import heapify,heappush,heappop\\ndef dks(t0,t1):\\n  kyori=[-1]*n\\n  todo=[[0,t0]]\\n  heapify(todo)\\n  while todo:\\n    d,t=heappop(todo)\\n    kyori[t]=d\\n    if t==t1:break\\n    l=g[t]\\n    for li,c in l:\\n      if kyori[li]==-1:\\n        heappush(todo,[c+d,li])\\n  return kyori[t1]\\nans=pow(10,9)\\nallr=[]\\nimport sys\\nsys.setrecursionlimit(10**7)\\ndef dfs(s,k): #s:list, k:set\\n  if len(k)==1:\\n    s.append(k.pop())\\n    return [s]\\n  ret=[]\\n  for ki in k:\\n    ret.extend(dfs(s+[ki],k-{ki}))\\n  return ret\\nallr=dfs([],set(r))\\nd={}\\nfor i in range(r_-1):\\n  for j in range(i+1,r_):\\n    k=dks(r[i]-1,r[j]-1)\\n    d[r[i]-1,r[j]-1]=k\\n    d[r[j]-1,r[i]-1]=k\\n\\nfor j in range(len(allr)):\\n  r1=allr[j]\\n  ansi=0\\n  for i in range(r_-1):\\n    ansi+=d[(r1[i]-1,r1[i+1]-1)]\\n  ans=min(ans,ansi)\\nprint(ans)\\n\", \"#!/usr/bin/env python3\\n\\nfrom itertools import permutations\\nimport numpy as np\\n\\nHUGE = 10 ** 18\\n\\ndef main():\\n    n, m, r = list(map(int, input().split()))\\n    togo = list(map(int, input().split()))\\n    adj_mat = [[HUGE for j in range(n)] for i in range(n)]\\n    for i in range(n):\\n        adj_mat[i][i] = HUGE\\n    for i in range(m):\\n        a, b, c = list(map(int, input().split()))\\n        a0 = a - 1\\n        b0 = b - 1\\n        adj_mat[a0][b0] = c\\n        adj_mat[b0][a0] = c\\n    adj_mat = np.array(adj_mat)\\n\\n    wf(adj_mat, n)\\n\\n    res = HUGE\\n    for way in permutations(togo):\\n        assert len(way) > 1\\n        x = 0\\n        for i in range(len(way) - 1):\\n            x += adj_mat[way[i] - 1][way[i + 1] - 1]\\n        res = min(res, x)\\n\\n    print(res)\\n\\ndef wf(adj_mat, n):\\n    for k in range(n):\\n        for i in range(n):\\n            adj_mat[i, :] = np.minimum(adj_mat[i, :], adj_mat[i][k] + adj_mat[k, :])\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys, re\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees#, log2\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left\\nfrom fractions import gcd\\nfrom heapq import heappush, heappop, heapify\\nfrom functools import reduce\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nr = LIST()\\nABC = [LIST() for _ in range(M)]\\n\\ndist_matrix = [[INF]*N for _ in range(N)]\\n\\nfor A, B, C in ABC:\\n\\tdist_matrix[A-1][B-1] = C\\n\\tdist_matrix[B-1][A-1] = C\\n\\nfor i in range(N):\\n\\tdist_matrix[i][i] = 0\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\n#\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5  \\u6ce8\\u610f PyPy\\u975e\\u5bfe\\u5fdc!!!!\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\ngraph = csr_matrix(dist_matrix) #\\u96a3\\u63a5\\u884c\\u5217\\u304b\\u3089csr\\u884c\\u5217\\u3092\\u3064\\u304f\\u308b\\ndist_matrix = floyd_warshall(graph, directed=False) #\\u7121\\u5411\\u306e\\u5834\\u5408directed=False\\n#dist_matrix\\u306e\\u4e2d\\u8eab\\u306ffloat\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u306e\\u3067\\u6ce8\\u610f\\uff01\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\nans = INF\\nfor x in permutations(r):\\n\\ttmp = 0\\n\\tfor i in range(1, R):\\n\\t\\ttmp += dist_matrix[x[i-1]-1][x[i]-1]\\n\\tans = min(ans, tmp)\\n\\nprint((int(ans)))\\n\\t\\n\", \"from itertools import permutations\\nimport sys\\ninput = sys.stdin.readline\\n\\ndef main():\\n    N, M, RR = map(int, input().split())\\n    R = list(map(int, input().split()))\\n    INF = float(\\\"inf\\\")\\n    T = [[INF] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = tuple(map(int, input().split()))\\n        T[a-1][b-1] = c\\n        T[b-1][a-1] = c\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if T[i][j] > T[i][k]+T[k][j]:\\n                    T[i][j] = T[i][k]+T[k][j]\\n    ans = INF\\n    for rs in permutations(R):\\n        cost = 0\\n        for i in range(RR-1):\\n            cost += T[rs[i]-1][rs[i+1]-1]\\n        ans = min(ans, cost)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations as p\\n \\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\nn, m, r = map(int, input().split())\\nR = list(map(int, input().split()))\\nl = [[0]*n for _ in range(n)]\\nfor _ in range(m):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    l[a][b] = c\\n    l[b][a] = c\\nF = floyd_warshall(l)\\n \\nans = float(\\\"inf\\\")\\nfor v in p(R):\\n    temp = 0\\n    for i in range(r-1):\\n        temp += F[v[i]-1][v[i+1]-1]\\n    ans = min(ans, temp)\\nprint(int(ans))\", \"n,m,R = map(int,input().split())\\nr = list(map(int,input().split()))\\n\\nfrom itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\ncost = np.ones((n,n))*float('inf')\\nfor i in range(m):\\n    a,b,c = map(int,input().split())\\n    cost[a-1][b-1]=cost[b-1][a-1]=c\\nd = floyd_warshall(cost)\\ndef main():\\n    m = float('inf')\\n\\n    for j in permutations(r):\\n        l = 0\\n        for i in range(len(j)-1):\\n            s,t = j[i],j[i+1]\\n            l += d[s-1][t-1]\\n        if l<m:\\n            m = l\\n    print(int(m))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# https://atcoder.jp/contests/abc073/tasks/abc073_d\\n\\nimport sys\\nread = sys.stdin.readline\\n\\n\\ndef read_ints():\\n    return list(map(int, read().split()))\\n\\n\\n\\n# default import\\nfrom itertools import product, permutations, combinations\\n\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import dijkstra\\n\\n# \\u30c0\\u30a4\\u30af\\u30b9\\u30c8\\u30e9\\u304b\\uff1f\\n# \\u5168\\u70b9\\u9593\\u306e\\u6700\\u5c0f\\u8ddd\\u96e2\\u3092\\u53d6\\u5f97\\n# r\\u306b\\u3064\\u3044\\u3066permutation\\u3057\\u3066\\u3002\\u305d\\u306e\\u901a\\u308a\\u306b\\u8a2a\\u308c\\u305f\\u3068\\u304d\\u306e\\u8ddd\\u96e2\\u3092\\u30b7\\u30df\\u30e5\\u30ec\\u30fc\\u30b7\\u30e7\\u30f3\\n# \\u6700\\u5c0f\\u5024\\u3092\\u9078\\u3076\\n\\nN, M, r = read_ints()\\nR = [x - 1 for x in read_ints()]\\nadj_mat = [[0] * N for _ in range(N)]\\nfor _ in range(M):\\n    a, b, c = read_ints()\\n    a -= 1\\n    b -= 1\\n    adj_mat[a][b] = c\\n    adj_mat[b][a] = c\\n\\n# print(csr_matrix(adj_mat, dtype='int'))\\nD = dijkstra(csr_matrix(adj_mat, dtype='int'), directed=False)\\n# print(D)\\n# \\u5168\\u63a2\\u7d22\\u30d1\\u30fc\\u30c8\\nans = 2 ** 31\\n\\n\\ndef get_kyori(p):\\n    ret = 0\\n    for ps, pt in zip(p[:-1], p[1:]):\\n        ret += D[ps, pt]\\n    return ret\\n\\n\\nfor p in permutations(R):\\n    ans = min(ans, get_kyori(p))\\nprint((int(ans)))\\n\", \"import sys\\nimport heapq, math\\nfrom itertools import zip_longest, permutations, combinations, combinations_with_replacement\\nfrom itertools import accumulate, dropwhile, takewhile, groupby\\nfrom functools import lru_cache\\nfrom copy import deepcopy\\n\\nN, M, R = list(map(int, input().split()))\\n\\nRS = list(map(int, input().split()))\\n\\nG = [[] for _ in range(N + 1)]\\n\\nfor i in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    G[A].append((B, C))\\n    G[B].append((A, C))\\n\\n\\ndef dijkstra(s):\\n    d = [1 << 28] * (N + 1)\\n    d[s] = 0\\n    q = []\\n    heapq.heappush(q, (0, s))\\n\\n    while q:\\n        cur = heapq.heappop(q)\\n\\n        if d[cur[1]] != cur[0]:\\n            continue\\n\\n        for nx in G[cur[1]]:\\n            if d[nx[0]] > cur[0] + nx[1]:\\n                d[nx[0]] = cur[0] + nx[1]\\n                heapq.heappush(q, (cur[0] + nx[1], nx[0]))\\n\\n    return d\\n\\n\\nDIRS = [dijkstra(s) for s in RS]\\nMAT = [[d[r] for r in RS] for d in DIRS]\\n\\nmem = {}\\n\\n\\ndef solve(idx, st):\\n    if st == (1 << R) - 1:\\n        return 0\\n    if (idx, st) in mem:\\n        return mem[(idx, st)]\\n\\n    ret = 1 << 28\\n    for i in range(R):\\n        if ((st >> i) & 1) == 0:\\n            ret = min(ret, solve(i, st | (1 << i)) + MAT[idx][i])\\n\\n    mem[(idx, st)] = ret\\n    return ret\\n\\n\\nprint((min([solve(i, 1 << i) for i in range(R)])))\\n\", \"from scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list(map(int, input().split()))\\n    r = [i-1 for i in r]\\n    l = [[0] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        l[a][b] = c\\n        l[b][a] = c\\n    G = csr_matrix(l)\\n    dd = dijkstra(G, directed=False)\\n    ans = 10**100\\n    for i in permutations(r, len(r)):\\n        t = 0\\n        for j in range(len(r)-1):\\n            t += dd[i[j]][i[j+1]]\\n        ans = min(ans, t)\\n    return int(ans)\\nprint((main()))\\n\", \"import sys\\nfrom itertools import permutations\\n\\nsys.setrecursionlimit(10 ** 6)\\nINF = float(\\\"inf\\\")\\nMOD = 10 ** 9 + 7\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list([int(x) - 1 for x in input().split()])\\n    P = permutations(r)\\n\\n    dp = [[float(\\\"inf\\\")] * N for _ in range(N)]\\n    for i in range(N):\\n        dp[i][i] = 0\\n\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        dp[a][b] = c\\n        dp[b][a] = c\\n\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                tmp = dp[i][k] + dp[k][j]\\n                if tmp < dp[i][j]:\\n                    dp[i][j] = tmp\\n\\n    ans = float(\\\"inf\\\")\\n    for p in P:\\n        tmp = 0\\n        for i in range(R - 1):\\n            tmp += dp[p[i]][p[i + 1]]\\n\\n        if tmp < ans:\\n            ans = tmp\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import heapq\\nimport itertools\\nfrom collections import defaultdict\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\n\\nedges = [[] for i in range(n+1)]\\nfor _ in range(m):\\n    _from, to, distance = list(map(int, input().split()))\\n    edges[_from].append([to, distance])\\n    edges[to].append([_from, distance])\\n\\n\\ncomp_edges = defaultdict(dict)\\nfor _from in visiting_town:\\n    seen = [False] * (n+1)\\n    todo = []\\n    for to, dist in edges[_from]:\\n        heapq.heappush(todo, [dist, to])\\n\\n    while todo:\\n        dist, node = heapq.heappop(todo)\\n        if seen[node]:\\n            continue\\n\\n        seen[node] = dist\\n        for to, add_dist in edges[node]:\\n            if seen[to] or to == _from:\\n                continue\\n            new_dist = dist + add_dist\\n            heapq.heappush(todo, [new_dist, to])\\n\\n    for to in visiting_town:\\n        if to == _from:\\n            continue\\n        comp_edges[_from][to] = seen[to]\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        if to in comp_edges[_from]:\\n            dist = comp_edges[_from][to]\\n            result += dist\\n        else:\\n            break\\n    candidates.append(result)\\n\\nprint((min(candidates)))\\n\", \"import sys\\nimport numpy as np \\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nI = np.array(sys.stdin.read().split(), dtype=np.int64)\\nn, m, R = I[:3]\\nr = I[3:3+R] - 1\\na, b, c = I[3+R:].reshape(-1, 3).T\\ngraph = csr_matrix((c, (a-1, b-1)), shape=(n, n))\\n\\ndef main():\\n    dist = floyd_warshall(graph, directed=False).astype(np.int64)\\n    \\n    perms = np.array(list(permutations(r)))\\n    res = dist[perms[:, :-1], perms[:, 1:]]\\n    ans = np.amin(np.sum(res, axis=1))\\n    return ans\\n\\ndef __starting_point():\\n    ans = main()\\n    print(ans)\\n__starting_point()\", \"import itertools\\nimport sys\\n\\ndef main():\\n  input = sys.stdin.readline\\n  n, m, r = map(int, input().split())\\n  R = [int(x) for x in input().split()]\\n  inf = pow(10, 9)+7\\n\\n  roads = [[inf]*n for _ in range(n)]\\n  for i in range(m):\\n    a, b, c = map(int, input().split())\\n    roads[a-1][b-1] = c\\n    roads[b-1][a-1] = c\\n\\n  for k in range(n):\\n    for i in range(n):\\n      for j in range(n):\\n        if roads[i][j] > roads[i][k]+roads[k][j]:\\n          roads[i][j] = roads[i][k]+roads[k][j]\\n\\n  ans = inf\\n  for value in itertools.permutations(R):\\n    sub = 0\\n    for k in range(r-1):\\n      sub += roads[value[k]-1][value[k+1]-1]\\n    if ans > sub:\\n      ans = sub\\n\\n  print(ans)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from scipy.sparse.csgraph import csgraph_from_dense,dijkstra,floyd_warshall\\nN,M,R = map(int,input().split())\\n\\nmachi = list(map(int,input().split()))\\n\\nList = [[10**6]*(N) for i in range(N)]\\nfor i in range(M):\\n  a,b,c = map(int,input().split())\\n  \\n  if List[a-1][b-1] > c:\\n    List[a-1][b-1] = c\\n    List[b-1][a-1] = c\\nG = csgraph_from_dense(List, null_value=10**6)\\nd = floyd_warshall(G)\\n\\nimport itertools\\na = list(itertools.permutations(machi))\\narr = []\\nfor i in a:\\n  ans = 0\\n  for j in range(R):\\n    if j != R-1 :\\n      ans += d[i[j]-1][i[j+1]-1]\\n  arr.append(ans) \\n  \\nprint(int(min(arr)))\", \"from itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nN, M, R = map(int, input().split())\\nR = list(map(int, input().split()))\\nA, B, C = np.array([input().split() for _ in range(M)], dtype=int).T\\nG = csr_matrix((C, (A - 1, B - 1)), shape=(N, N))\\npath = dijkstra(G, directed=False)\\npath = (path + 0.5).astype(int)\\n\\nans = 10 ** 18\\nfor p in permutations(R):\\n    cnt = 0\\n    for x, y in zip(p[:-1], p[1:]):\\n        x -= 1\\n        y -= 1\\n        cnt += path[x][y]\\n    ans = min(ans, cnt)\\n\\nprint(ans)\", \"import numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nn,m,R = map(int, input().split())\\nr = list(map(int, input().split()))\\nr = [i-1 for i in r]\\nabc = [list(map(int, input().split())) for i in range(m)]\\nedge = [[10 ** 9] * (n) for i in range(n)]\\nfor a,b,c in abc:\\n  a-=1\\n  b-=1\\n  edge[a][b] = c\\n  edge[b][a] = c\\nd = floyd_warshall(edge)\\nans = 10 ** 18\\nfor i in permutations(r):\\n  tmp = 0\\n  for j in range(R-1):\\n    tmp += d[i[j]][i[j+1]]\\n  ans = min(ans,tmp)\\nprint(int(ans))\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nINF = 10 ** 18\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\ndp = [[INF for _ in range(N)] for _ in range(N)]\\nfor i in range(N):\\n    for j in range(N):\\n        if i == j:\\n            dp[i][j] = 0\\n\\nfor i in range(M):\\n    a, b, c = map(int, input().split())\\n    dp[a - 1][b - 1] = c\\n    dp[b - 1][a - 1] = c\\n\\nwf = floyd_warshall(dp)\\nans = INF\\nfor i in permutations(r):\\n    tmp = 0\\n    for j in range(1, len(i)):\\n        tmp += wf[i[j - 1] - 1][i[j] - 1]\\n    ans = min(ans, tmp)\\n\\nprint(int(ans))\", \"from itertools import permutations, combinations\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import dijkstra\\nN,M,R = map(int,input().split())\\nr = list(map(int,input().split()))\\nr = [r[i]-1 for i in range(R)]\\nvve = [list(map(int,input().split())) for i in range(M)]\\nv1,v2,edge = zip(*vve)\\nv1 = list(v1)\\nv2 = list(v2)\\nfor i in range(M):\\n  v1[i] -= 1\\n  v2[i] -= 1\\ncsr = csr_matrix((edge,(v1,v2)),shape = (N,N))\\ndist = [[0 for j in range(R)] for i in range(R)]\\nfor i,j in combinations(range(R),2):\\n  r1,r2=r[i],r[j]\\n  dist[i][j] = int(dijkstra(csr, directed = False,indices = r1)[r2])\\n  dist[j][i] = dist[i][j]\\nans = 10**18\\nfor x in permutations(range(R)):\\n  distsum = 0\\n  for i in range(1,R):\\n    distsum += dist[x[i]][x[i-1]]\\n  ans = min(ans,distsum)\\nprint(ans)\", \"def main():\\n    import sys\\n    input = sys.stdin.readline\\n    sys.setrecursionlimit(10**7)\\n    from collections import Counter, deque\\n    #from collections import defaultdict\\n    from itertools import combinations, permutations, accumulate, groupby\\n    #from itertools import product\\n    from bisect import bisect_left,bisect_right\\n    from heapq import heapify, heappop, heappush\\n    from math import floor, ceil\\n    #from operator import itemgetter\\n\\n    inf = 10**17\\n    #mod = 10**9 + 7\\n\\n    N,M,R = map(int, input().split())\\n    r = list(map(int, input().split()))\\n    for i in range(R):\\n        r[i] -= 1\\n\\n    def dijkstra_heap(start,edge):\\n        #\\u59cb\\u70b9\\u304b\\u3089\\u5404\\u9802\\u70b9\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2(\\u9802\\u70b9\\u756a\\u53f7:0~N-1)\\n        d = [inf]*N\\n        used = [False]*N\\n        d[start] = 0\\n        used[start] = True\\n        edgelist = []\\n        #a:\\u91cd\\u307f(//), b:\\u6b21\\u306e\\u9802\\u70b9(%)\\n        for a,b in edge[start]:\\n            heappush(edgelist,a*(10**6)+b)\\n\\n        while len(edgelist):\\n            #\\u307e\\u3060\\u6700\\u77ed\\u8ddd\\u96e2\\u304c\\u6c7a\\u307e\\u3063\\u3066\\u3044\\u306a\\u3044\\u9802\\u70b9\\u306e\\u4e2d\\u304b\\u3089\\u6700\\u5c0f\\u306e\\u8ddd\\u96e2\\u306e\\u3082\\u306e\\u3092\\u63a2\\u3059\\n            minedge = heappop(edgelist)\\n            if used[minedge%(10**6)]:\\n                continue\\n            node = minedge%(10**6)\\n            d[node] = minedge//(10**6)\\n            used[node] = True\\n\\n            for e in edge[node]:\\n                if not used[e[1]]:\\n                    heappush(edgelist,(e[0]+d[node])*(10**6)+e[1])\\n        return d\\n\\n    edge = [[] for i in range(N)]\\n    #edge[i] : i\\u304b\\u3089\\u51fa\\u308b\\u9053\\u306e[\\u91cd\\u307f,\\u884c\\u5148]\\u306e\\u914d\\u5217\\n    for _ in range(M):\\n        x,y,z = map(int,input().split())\\n        edge[x-1].append((z,y-1))\\n        edge[y-1].append((z,x-1))\\n\\n    kyori = []\\n    for i in r:\\n        kyori.append(dijkstra_heap(i,edge))\\n    \\n    res = inf\\n    for i in permutations(range(R), R):\\n        l = list(i)\\n        path = 0\\n        for j in range(R-1):\\n            path += kyori[l[j]][r[l[j+1]]]\\n        res = min(res, path)\\n\\n    print(res)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\n\\ndef warshal_floyd(d):\\n    # d[i][j] := i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n    nonlocal N\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                d[i][j] = min(d[i][j], d[i][k] + d[k][j])\\n    # \\\"\\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bi\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\\"\\u3068\\\"\\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bi\\u304b\\u3089k\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2 + \\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bk\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2 \\u306e\\u548c\\\"\\u3092\\u6bd4\\u8f03\\u3059\\u308b\\n    return d\\n\\n\\nN, M, R = list(map(int, input().split()))  # N\\u500b\\u306e\\u753a, M\\u672c\\u306e\\u9053, R\\u500b\\u306e\\u753a\\u3092\\u8a2a\\u308c\\u308b\\u3053\\u3068\\u306b\\u306a\\u3063\\u305f\\ntown = list([int(x)-1 for x in input().split()])\\ndistant = [[float('inf')]*N for _ in range(N)]\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    distant[a][b] = c\\n    distant[b][a] = c\\nfor i in range(N):\\n    distant[i][i] = 0\\ndistant = floyd_warshall(distant)\\n# warshal_floyd(distant)\\n# print(distant)  # R\\u306f\\u6700\\u59278\\u3060\\u304b\\u3089\\u5168\\u3066\\u3092\\u8a66\\u3057\\u3066\\u3082\\u5927\\u4e08\\u592b\\nans = 10**9\\nfor x in itertools.permutations(town):\\n    x = list(x)\\n    now = x[0]\\n    tmp = 0\\n    for i in range(1, len(x)):\\n        tmp += distant[now][x[i]]\\n        now = x[i]\\n    ans = min(ans, tmp)\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\nr = [ri-1 for ri in r]\\n\\ngraph = [[0]*N for i in range(N)]\\nfor i in range(M):\\n  a, b, c = map(int, input().split())\\n  a -= 1; b -= 1\\n  graph[a][b] = graph[b][a] = c\\n\\ndist = floyd_warshall(graph).astype(int)\\norders = list(itertools.permutations(r, R))\\n\\nans = float('inf')\\nfor order in orders:\\n  d = 0\\n  for u, v in zip(order, order[1:]):\\n    d += dist[u][v]\\n  ans = min(ans, d)\\n\\nprint(ans)\", \"import numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split(' ')))\\n    towns = list(map(lambda t: int(t) - 1, input().split(' ')))\\n    A = np.zeros((N, N), dtype=int)\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split(' ')))\\n        A[a-1][b-1] = c\\n        A[b-1][a-1] = c\\n    fw = floyd_warshall(A)\\n    d = fw[towns, :][:, towns]\\n    min_d = 10**12\\n    for trip in itertools.permutations(range(R)):\\n        min_d = min([min_d, sum([d[trip[i]][trip[i+1]] for i in range(len(trip) - 1)])])\\n    print(int(min_d))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nfrom itertools import permutations, combinations\\nn,m,R = map(int,input().split())\\nr=list(map(int,input().split()))\\nfor i in range(R):\\n    r[i]-=1\\nd = np.zeros((n,n)) \\n# \\u5165\\u529b\\nfor i in range(m):\\n    a,b,c = map(int,input().split())\\n    a,b = a-1, b-1\\n    d[a,b] = c\\ndist =floyd_warshall(d,directed=0).astype(int)\\nans=10**10\\nfor v in permutations(r):\\n    tmp=0\\n    for i in range(R-1):\\n        tmp+=dist[v[i],v[i+1]]\\n    ans=min(ans,tmp)\\nprint(ans)\", \"def main():\\n    from sys import stdin\\n    input = stdin.readline\\n    n, m, r = list(map(int, input().split()))\\n    root = list(map(int, input().split()))\\n    abc = [list(map(int, input().split())) for _ in [0]*m]\\n\\n    import numpy as np\\n    import scipy.sparse.csgraph as sp\\n    inf = float('inf')\\n    d = np.full((n, n), inf)\\n    for i in range(n):\\n        d[i][i] = 0\\n    for a, b, c in abc:\\n        d[a-1][b-1] = c\\n        d[b-1][a-1] = c\\n    # indices=x\\u3067x\\u304b\\u3089\\u306e\\u5358\\u4e00\\u59cb\\u70b9\\u306b,return_predecessors=True\\u3067\\u7d4c\\u8def\\u304c\\u51fa\\u308b\\n    s = sp.shortest_path(d)\\n\\n    from itertools import permutations\\n\\n    p = list(permutations(root))\\n\\n    D = 10**10\\n    for i in p:\\n        d = 0\\n        for j in range(len(i)-1):\\n            d += s[i[j]-1][i[j+1]-1]\\n        D = min(D, d)\\n    print((int(D)))\\n\\n\\nmain()\\n\", \"from scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\n\\ng = [[0] * (N + 1) for _ in range(N + 1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    g[A][B] = C\\n    g[B][A] = C\\ng = floyd_warshall(csgraph_from_dense(g))\\n\\nresult = 1000000000\\nfor p in permutations(r):\\n    t = 0\\n    for i in range(R - 1):\\n        t += g[p[i]][p[i + 1]]\\n    result = min(result, t)\\nprint((int(result)))\\n\", \"import sys\\nfrom itertools import permutations\\n\\nsys.setrecursionlimit(10 ** 6)\\nINF = float(\\\"inf\\\")\\nMOD = 10 ** 9 + 7\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list(map(int, input().split()))\\n    P = permutations(r)\\n\\n    dp = [[INF] * N for _ in range(N)]\\n    for i in range(N):\\n        dp[i][i] = 0\\n\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        dp[a][b] = c\\n        dp[b][a] = c\\n\\n    # \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                tmp = dp[i][k] + dp[k][j]\\n                if tmp < dp[i][j]:\\n                    dp[i][j] = tmp\\n\\n    ans = INF\\n    for p in P:\\n        tmp = 0\\n        for i in range(R - 1):\\n            tmp += dp[p[i] - 1][p[i + 1] - 1]\\n\\n        if tmp < ans:\\n            ans = tmp\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nint1 = lambda x: int(x) - 1\\n\\nN, M, R = map(int, input().split())\\nT = sorted(list(map(int1, input().split())))\\nA = np.array([tuple(map(int, input().split())) for _ in range(M)]).T\\n\\nmatr = csr_matrix((A[2], (A[0] - 1, A[1] - 1)), shape=(N, N))\\nway = [dijkstra(matr, indices=t, directed=False)[T].astype(int).tolist() for t in T]\\n\\nans = float('inf')\\nfor t in itertools.permutations(range(R)):\\n    tmp = 0\\n    for i in range(R - 1):\\n        tmp += way[t[i]][t[i + 1]]\\n    ans = min(ans, tmp)\\nprint(ans)\", \"import heapq\\nimport itertools\\n\\ndef dijkstra(s):\\n    inf = pow(10, 10)\\n    dist = [inf] * (n + 1)\\n    dist[s] = 0\\n    c = [0] * (n + 1)\\n    p = []\\n    heapq.heapify(p)\\n    heapq.heappush(p, (dist[s], s))\\n    while p:\\n        d, u = heapq.heappop(p)\\n        if dist[u] < d:\\n            continue\\n        c[u] = 1\\n        for g in G[u]:\\n            if c[g[0]] == 0 and dist[u] + g[1] < dist[g[0]]:\\n                dist[g[0]] = dist[u] + g[1]\\n                heapq.heappush(p, (dist[g[0]], g[0]))\\n    return dist\\n\\nn, m, r = map(int, input().split())\\nt = list(map(int, input().split()))\\nG = [[] for _ in range(n + 1)]\\nfor _ in range(m):\\n    a, b, c = map(int, input().split())\\n    G[a].append([b, c])\\n    G[b].append([a, c])\\nD = [[0] * r for _ in range(r)]\\nfor i in range(r):\\n    d = dijkstra(t[i])\\n    for j in range(r):\\n        D[i][j] = d[t[j]]\\n\\nx = [i for i in range(r)]\\nans = pow(10, 10)\\nfor y in itertools.permutations(x):\\n    dist = 0\\n    y = list(y)\\n    for i in range(r - 1):\\n        dist += D[y[i]][y[i + 1]]\\n    ans = min(ans, dist)\\nprint(ans)\", \"from itertools import permutations as perm\\n\\ndef warshall(d, n):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if d[i][k] + d[k][j] < d[i][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\nn,m,r = map(int, input().split())\\nrr = [i-1 for i in map(int, input().split())]\\n\\ninf = float('INF')\\nroute = [[inf for j in range(n)] for i in range(n)]\\nfor _ in range(m):\\n    a,b,c = map(int, input().split())\\n    route[a-1][b-1] = route[b-1][a-1] = c\\n\\nwarshall(route, n)\\n\\nans = inf\\nfor tmp in perm(rr):\\n    cost = 0\\n    for i in range(r-1):\\n        cost += route[tmp[i]][tmp[i+1]]\\n    if cost < ans:\\n        ans = cost\\nprint(int(ans))\", \"# coding: utf-8\\nfrom heapq import heappop, heappush\\nfrom itertools import permutations\\n\\nN,M,R=map(int,input().split())\\ntown=list(map(int,input().split()))\\nINF=10**9\\nG=[[INF for i in range(N+1)] for j in range(N+1)]\\n\\nfor i in range(M):\\n    A,B,C=map(int,input().split())\\n    G[A][B]=C\\n    G[B][A]=C\\n\\nd=[[INF*10 for i in range(N+1)] for j in range(R)]\\n\\nfor k in range(R):\\n    r=town[k]\\n    d[k][r]=0\\n    \\n    used=[False for i in range(N+1)]\\n    \\n    heap=[]\\n    heappush(heap,(d[k][r],r))\\n    \\n    while heap:\\n        d_u, u = heappop(heap)\\n\\n        used[u] = True\\n        \\n        if d[k][u] < d_u:\\n            continue\\n        \\n        for v in range(N+1):\\n            if not(used[v]) and d_u + G[u][v] < d[k][v]:\\n                d[k][v] = d_u + G[u][v]\\n                heappush(heap,(d[k][v],v))\\n\\nans=INF\\n\\nL=[i for i in range(R)]\\n\\nfor v in permutations(L,R):\\n    D=0\\n    for i in range(R-1):\\n        D+=d[v[i]][town[v[i+1]]]\\n    ans=min(ans,D)\\n    \\nprint(ans)\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import dijkstra\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = {}\\nfor node in visiting_town:\\n    dist_list = dijkstra(G, directed=False, indices=node-1)\\n    comp_dist[node-1] = dist_list\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\n\\nN,M,R = map(int, input().split())\\nr_list = np.array(list(map(int, input().split()))) -1 \\n\\nnodes = np.full((N,N), np.inf) \\nfor i in range(M):\\n    a,b,c = map(int, input().split())\\n    nodes[a-1,b-1] = nodes[b-1,a-1] = c\\nmin_ds = shortest_path(nodes)\\nres = 10**9\\nfor cmb in itertools.permutations(r_list):\\n    prev = cmb[0]\\n    tmpres = 0\\n    for to in cmb[1:]:\\n        tmpres += min_ds[prev,to]\\n        prev = to\\n    res = min(res, tmpres)\\n\\nprint(int(res))\", \"import numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\n\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int,input().split()))\\nfor i in range(R):\\n    r[i] -= 1\\n\\nconnections = np.zeros((N,N))\\nfor i in range(M):\\n    A,B,C = list(map(int,input().split()))\\n    A,B = A-1,B-1\\n    \\n    connections[A][B] = C\\n    connections[B][A] = C\\n\\nconnections = dijkstra(connections)\\n\\ndef explore(unvisited,total,start):\\n    if unvisited:\\n        ret = float('inf')\\n        for town in unvisited:\\n            ret = min(ret,explore(unvisited-set([town]),\\n                total+connections[start][town],town))\\n        return ret\\n    else:\\n        return total\\n\\nans = float('inf')\\ntargets = set(r)\\nfor s in r:\\n    ans = min(ans,explore(targets-set([s]),0,s))\\nprint(int(ans))\", \"def abc073_d():\\n    from scipy.sparse import lil_matrix\\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations\\n\\n    n, m, r = map(int, input().split())\\n    R = list(map(lambda x: int(x)-1, input().split()))\\n    graph = lil_matrix((n, n), dtype=int)\\n    for _ in range(m):\\n        a, b, c = map(int, input().split())\\n        a -= 1\\n        b -= 1\\n        graph[a, b] = c\\n        graph[b, a] = c\\n\\n    dist = floyd_warshall(csgraph=graph)\\n    #print(dist)\\n\\n    ans = 10**18\\n    for route in permutations(R, r):\\n        tmp = 0\\n        for i in range(r-1):\\n            s = route[i]\\n            t = route[i+1]\\n            tmp += dist[s][t]\\n            if tmp > ans: break\\n        ans = min(ans, tmp)\\n        #print(route, tmp)\\n\\n    print(int(ans))\\n\\ndef __starting_point():\\n    abc073_d()\\n__starting_point()\", \"from itertools import permutations\\nimport heapq\\n\\ndef dijkstra(N, G, s):\\n    INF = 10**40\\n    d = [INF] * N\\n    d[s] = 0\\n    q = [(0, s)]\\n    while q:\\n        c, v = heapq.heappop(q)\\n        if d[v] < c:\\n            continue\\n        for t, co in G[v]:\\n            if d[v] + co < d[t]:\\n                d[t] = d[v] + co\\n                heapq.heappush(q, (d[t], t))\\n    return d\\n\\n\\ndef main():\\n    N, M, _ = list(map(int, input().split()))\\n    R = list(map(int, input().split()))\\n    R = [r - 1 for r in R]\\n    G = [[] for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        G[a - 1].append((b - 1, c))\\n        G[b - 1].append((a - 1, c))\\n    RM = []\\n    for r in R:\\n        d = dijkstra(N, G, r)\\n        RM.append([d[rr] for rr in R])\\n    m = 10 ** 40\\n    for i in permutations(list(range(len(R)))):\\n        t = i[0]\\n        c = 0\\n        for r in i[1:]:\\n            c += RM[t][r]\\n            t = r\\n        m = min(m, c)\\n    return m\\n\\n\\nprint((main()))\\n\", \"import sys\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson, csgraph_from_dense\\nimport itertools\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, M, R = [int(x) for x in input().split()]\\n    r = [int(x) - 1 for x in input().split()]\\n    ABC = [[int(x) for x in input().split()] for _ in range(M)]\\n\\n    EDGE = [[10 ** 9] * N for j in range(N)]\\n\\n    for a, b, c in ABC:\\n        EDGE[a - 1][b - 1] = c\\n        EDGE[b - 1][a - 1] = c\\n\\n    G = csgraph_from_dense(EDGE, null_value=10 ** 9)\\n\\n    d = floyd_warshall(G)\\n\\n    ans = float(\\\"inf\\\")\\n    for a in itertools.permutations(r):\\n        tmp = 0\\n        c = a[0]\\n        for n in a:\\n            tmp += int(d[c][n])\\n            c = n\\n        ans = min(ans, tmp)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n  import numpy as np\\n  import itertools\\n  from scipy.sparse import csr_matrix\\n  from scipy.sparse.csgraph import dijkstra\\n  import sys\\n  readline = sys.stdin.readline\\n  readlines = sys.stdin.readlines\\n\\n  N, M, R= map(int, readline().split())\\n  r = list(map(int,input().split()))\\n  lines = readlines()\\n  edge = np.array([line.split() for line in lines], dtype = np.int64).T\\n  #print(r)\\n  #print(edge)\\n  graph = csr_matrix((edge[2], (edge[:2] - 1)), (N, N))\\n  ans = float(\\\"inf\\\")\\n  distance_mat = {}\\n  for i in r:\\n    distance_mat[i-1] = (dijkstra(graph, directed = False, indices = i-1))\\n\\n  for town in itertools.permutations(r,R):\\n    #print(town)\\n    tmp = 0\\n    for i in range(R-1):\\n      tmp += distance_mat[town[i]-1][town[i+1]-1]\\n    ans = min(ans,tmp)\\n  print(int(ans))\\nmain()\", \"import itertools\\ndef main():\\n    n,m,r=list(map(int,input().split()))\\n    rx=[int(i) for i in input().split()]\\n    inf=100000000\\n    wf=[[inf]*n for i in range(n)]\\n    for i in range(n):\\n        wf[i][i]=0\\n\\n    for i in range(m):\\n        a,b,c=list(map(int,input().split()))\\n        wf[a-1][b-1]=c\\n        wf[b-1][a-1]=c\\n\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if wf[i][j]>wf[i][k]+wf[k][j]:\\n                    wf[i][j]=wf[i][k]+wf[k][j]\\n    cnt=0\\n    l=list(itertools.permutations([i for i in range(r)]))\\n    cnt=inf\\n    for i in l:\\n        cnt_sub=0\\n        for j in range(r-1):\\n            cnt_sub+=wf[rx[i[j]]-1][rx[i[j+1]]-1]\\n        cnt=min(cnt,cnt_sub)\\n    print(cnt)\\nmain()\\n\", \"from collections import deque\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nimport itertools\\n\\nn, m, r = map(int,input().split())\\nrlist = list(map(int,input().split()))\\nfor i in range(r):\\n    rlist[i] -= 1\\ne = np.zeros((n,n), dtype=int)\\n\\nfor i in range(m):\\n    a, b, c = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    e[a][b] = c\\n    e[b][a] = c\\ncsr = csr_matrix(e)\\nf = floyd_warshall(csr)\\n\\nans = 10**18\\nfor v in itertools.permutations(rlist,r):\\n    d = 0\\n    for i in range(r-1):\\n        d += f[v[i]][v[i+1]]\\n    ans = min(ans, d)\\nprint(int(ans))\", \"def main():\\n    from sys import stdin\\n    input = stdin.readline\\n\\n    n, m, r = list(map(int, input().split()))\\n    l = [int(i) - 1 for i in input().split()]\\n    d = [[10**8] * n for _ in range(n)]\\n    for _ in range(m):\\n        i, j, k = list(map(int, input().split()))\\n        d[i-1][j-1] = k\\n        d[j-1][i-1] = k\\n\\n    # Warshall-Floyd algorithm\\n    for k in range(n):\\n        for i in range(n-1):\\n            for j in range(i+1, n):\\n                if d[i][j] > d[i][k] + d[k][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n                    d[j][i] = d[i][j]\\n\\n    # full search\\n    # 8! = 40320\\n    from itertools import permutations\\n\\n    answer = float('inf')\\n    for i in permutations(l):\\n        ans = 0\\n        for j in range(r-1):\\n            ans += d[i[j]][i[j+1]]\\n        if ans < answer:\\n            answer = ans\\n\\n    print(answer)\\n\\nmain()\\n\", \"from scipy.sparse.csgraph import shortest_path\\nimport numpy as np\\nimport itertools\\n\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int, input().split()))\\n\\nedge = np.zeros((N+1,N+1))\\nfor _ in range(M):\\n  A,B,C = list(map(int,input().split()))\\n  edge[A,B] = C  \\ndist = shortest_path(edge, directed = False).astype(int)\\n\\nanswer = 10**18\\nfor visit in itertools.permutations(r):\\n  p = 0\\n  prev = visit[0]\\n  for x in visit[1:]:\\n    if prev != None:\\n      p += dist[prev,x]\\n    prev = x\\n  answer = min(answer,p)\\n  \\nprint(answer)\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nfrom itertools import permutations\\n\\nn,m,r = map(int,input().split())\\nrr = np.array(list(map(int,input().split()))) - 1\\n\\nL = np.zeros((n,n),int)\\nfor _ in range(m):\\n    a,b,c = map(int,input().split())\\n    L[a-1,b-1] = c\\n\\nshortest = floyd_warshall(L, directed = False)\\n\\nans = float('inf')\\n\\nfor route in permutations(rr,len(rr)):\\n    ans = min(ans, shortest[route[:-1],route[1:]].sum())\\nprint(int(ans))\", \"# \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u3067\\u3001\\u5404\\u753a\\u9593\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\u306e\\u66f4\\u65b0\\u306f200^3 = 8,000,000\\n# \\u8a2a\\u308c\\u308b\\u3079\\u304d\\u753aR\\u306f\\u305f\\u304b\\u3060\\u304b8\\u500b\\u306a\\u306e\\u3067\\u3001\\u9806\\u756a\\u306e\\u5168\\u901a\\u308a\\u3092\\u8a66\\u3057\\u30668! = \\u7d0440000\\u901a\\u308a\\n\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall\\nfrom scipy.sparse import csr_matrix\\n\\nN,M,R = map(int,input().split())\\nr = list(map(int,input().split()))\\nr = list(map(lambda x:x-1,r))\\n\\nE = [[0 for j in range(N)] for i in range(N)]\\nfor i in range(M):\\n  a,b,c = map(int,input().split())\\n  E[a-1][b-1] = c\\n  E[b-1][a-1] = c\\n\\nE = np.array(E)\\nE = shortest_path(E,method = \\\"FW\\\")\\n\\n# DFS\\u3067\\u3059\\u3079\\u3066\\u306e\\u6570\\u3092\\u8a66\\u3059\\nstack = []\\nfor i in range(len(r)):\\n  stack.append([r[i],[],0])\\nans = 10 ** 18\\nwhile stack:\\n  v,visited,dist = stack.pop()\\n  if len(visited) != 0:\\n    dist += E[visited[-1]][v]\\n  visited2 = visited.copy()\\n  visited2.append(v)\\n  if len(visited2) == len(r):\\n    if dist < ans:\\n      ans = dist\\n    continue\\n  for i in range(len(r)):\\n    if r[i] not in visited2:\\n      stack.append([r[i],visited2,dist])\\n    \\nprint(int(ans))\", \"from sys import stdin\\nfrom itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\ndef main():\\n    #\\u5165\\u529b\\n    readline=stdin.readline\\n    inf=10**9\\n    n,m,r=map(int,readline().split())\\n    targets=list(map(lambda x:int(x)-1,readline().split()))\\n    G=[[inf]*n for _ in range(n)]\\n    for _ in range(m):\\n        a,b,c=map(int,readline().split())\\n        a-=1\\n        b-=1\\n        G[a][b]=min(G[a][b],c)\\n        G[b][a]=min(G[b][a],c)\\n\\n    #\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\n    G=csgraph_from_dense(G,null_value=10**9)\\n    li=floyd_warshall(G)\\n    li=[list(map(int,li[i])) for i in range(n)]\\n    \\n    #\\u9806\\u5217\\u5168\\u63a2\\u7d22\\n    ans=inf\\n    for p in permutations(targets,r):\\n        tmp=0\\n        for i in range(r-1):\\n            now=p[i]\\n            nex=p[i+1]\\n            tmp+=li[now][nex]\\n        ans=min(ans,tmp)\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import *\\nfrom scipy.sparse.csgraph import *\\nimport numpy as np\\nN,M,R = map(int,input().split())\\nT = list(map(int,input().split()))\\nE = [list(map(int,input().split())) for m in range(M)]\\nG = np.zeros((N,N))\\nA = []\\n\\nfor a,b,c in E:\\n  G[a-1][b-1] = c\\n  G[b-1][a-1] = c\\n\\nF = floyd_warshall(G)\\n\\nfor P in permutations(T):\\n  A+=[sum(F[P[r]-1][P[r+1]-1] for r in range(R-1))]\\n\\nprint(int(min(A)))\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\nfrom sys import stdin\\nnii=lambda:map(int,stdin.readline().split())\\n\\nn,m,r=nii()\\nr=list(nii())\\n\\ns=[[float('inf')]*n for i in range(n)]\\nfor i in range(n):\\n  for j in range(n):\\n    s[i][j]=0\\nfor i in range(m):\\n  a,b,c=nii()\\n  a-=1\\n  b-=1\\n  s[a][b]=c\\n  s[b][a]=c\\n\\nws=floyd_warshall(s)\\n\\nans=10**9\\nfor i in permutations(r):\\n  t_ans=0\\n  for j in range(1,len(i)):\\n    t_ans+=ws[i[j-1]-1][i[j]-1]\\n  ans=min(ans,t_ans)\\nprint(int(ans))\", \"import itertools\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nN, M, R = map(int, input().split())\\nr = tuple(map(int, input().split()))\\n\\nINF = 10**10\\n\\nd = [[INF] * N for _ in range(N)]\\n\\nfor i in range(N):\\n    d[i][i] = 0\\n\\nfor _ in range(M):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    if d[a][b] > c:\\n        d[a][b] = c\\n        d[b][a] = c\\n\\n\\ndef warshall(d):\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if d[i][j] > d[i][k] + d[k][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\n\\n# d = floyd_warshall(d)\\nwarshall(d)\\n\\n\\nans = INF\\nfor p in itertools.permutations(r):\\n    dist = 0\\n    for i in range(R-1):\\n        dist += d[p[i]-1][p[i+1]-1]\\n\\n    if ans > dist:\\n        ans = dist\\n\\nprint(int(ans))\", \"from itertools import permutations as perm\\nfrom scipy.sparse.csgraph import dijkstra as di\\n\\ndef warshall(d, n):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if d[i][k] + d[k][j] < d[i][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\nn,m,r = list(map(int, input().split()))\\nrr = [i-1 for i in map(int, input().split())]\\n\\ninf = float('INF')\\nroute = [[0 for j in range(n)] for i in range(n)]\\nfor _ in range(m):\\n    a,b,c = list(map(int, input().split()))\\n    route[a-1][b-1] = route[b-1][a-1] = c\\n\\nroute = di(route, n)\\n\\nans = inf\\nfor tmp in perm(rr):\\n    cost = 0\\n    for i in range(r-1):\\n        cost += route[tmp[i]][tmp[i+1]]\\n    if cost < ans:\\n        ans = cost\\nprint((int(ans)))\\n\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nN, M, R = list(map(int, input().split()))\\nr = tuple(map(int, input().split()))\\n\\ninf = 10**9\\ngraph = np.ones((N, N), dtype=int)*inf\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    graph[a][b] = c\\n    graph[b][a] = c\\nfy = floyd_warshall(graph)\\n\\nans = 10 ** 9\\nfor p in permutations(r):\\n    tmp = 0\\n    for x, y in zip(p[:-1], p[1:]):\\n        x -= 1\\n        y -= 1\\n        tmp += int(fy[x][y])\\n    ans = min(ans, tmp)\\nprint(ans)\\n\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nn,m,r=map(int,input().split())\\nR=list(map(int,input().split()))\\nl=[[float('inf')]*n for _ in range(n)]\\nfor _ in range(m):\\n    a,b,c,=map(int,input().split())\\n    a-=1\\n    b-=1\\n    l[a][b]=c\\n    l[b][a]=c\\nfor i in range(n):\\n    l[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\ndef warshall_floyd(d):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                d[i][j]=min(d[i][j],d[i][k]+d[k][j])\\n                \\n    return d\\n#F=warshall_floyd(l)\\nF1 = floyd_warshall(l)\\nans=float('inf')\\nfor v in permutations(R):\\n    temp=0\\n    for i in range(r-1):\\n        temp+=F1[v[i]-1][v[i+1]-1]\\n    ans=min(ans,temp)\\nprint(int(ans))\", \"# coding: utf-8\\n# Your code here!\\n\\n#\\u4fdd\\u5b58\\nimport heapq\\nimport sys\\nimport itertools\\n \\nsys.setrecursionlimit(10**7)\\n\\ndef bfs(cost,node):\\n    root[node]=min(root[node],cost)\\n    for next_c,next_n in way[node]:\\n        if root[next_n]==10**9:\\n            heapq.heappush(q,[cost+next_c,next_n])\\n    return\\n    \\nN,M,R=list(map(int,input().split()))\\nr=list(map(int,input().split()))\\n\\nway=[[] for i in range(N)]\\n\\nfor _ in range(M):\\n    A,B,C=list(map(int,input().split()))\\n    way[A-1].append([C,B-1])\\n    way[B-1].append([C,A-1])\\n\\nmade=[]\\n\\n#print(q)\\nfor start in r:\\n    root=[10**9]*N\\n    q=[[0,start-1]]\\n    heapq.heapify(q)#cost\\u3068node\\n    while q:\\n        temp=heapq.heappop(q)\\n        #print(temp)\\n        if root[temp[1]]==10**9:\\n            bfs(temp[0],temp[1])\\n    made.append(root)\\n#print(made)\\n\\nans=10**9\\nfor order in itertools.permutations([i for i in range(len(r))]):\\n    #print(order)\\n    temp=0\\n    for i in range(len(order)-1):\\n        fro=r[order[i]]\\n        aft=r[order[i+1]]\\n        #print(\\\"YES\\\")\\n        #print(order[i]-1,order[i+1]-1)\\n        #print(root[order[i]-1][0])\\n        temp+=abs(made[order[i]][aft-1]-made[order[i]][fro-1])\\n    ans=min(ans,temp)\\nprint(ans)\\n\\n\\n\", \"import time\\nimport itertools\\n\\ndef main():\\n    N,M,R = tuple([int(x) for x in input().split()])\\n\\n    r = [int(x)-1 for x in input().split()]\\n    \\n    w_e = [tuple([int(x)for x in input().split()]) for _ in [0]*M]\\n    w_e = [(a-1,b-1,w) for (a,b,w) in w_e]\\n\\n\\n    g = Graph(list(range(N)),w_e)\\n\\n    minimums = {}\\n    for r_i in r:\\n        minimums[r_i] = g.dijkstra(r_i)[0]\\n\\n    bf = itertools.permutations(r)\\n\\n    candidates = []\\n    for bf_i in bf:\\n        distance = 0\\n        for i in range(R-1):\\n            distance += minimums[bf_i[i]][bf_i[i+1]]\\n        candidates.append(distance)\\n\\n    print(min(candidates))\\n\\ndef speedtest(func,*args):\\n    b = time.perf_counter()\\n    res = func(*args)\\n    e = time.perf_counter()\\n    elapsed = e-b\\n    print(\\\"{} (sec)\\\".format(elapsed))\\n\\n    return res\\n\\nclass Graph:\\n    def __init__(self,w_v,w_e):\\n        super().__init__()\\n        self.size = len(w_v)\\n        self.w_v = [v_i for v_i in w_v]#value of vartex\\n        self.w_e = [{} for _ in [0]*self.size]\\n\\n        self.neighbor = [[] for _ in [0]*self.size]\\n        for a_i,b_i,w_i in w_e:\\n            self.w_e[a_i][b_i] = w_i#weight of edge\\n            self.w_e[b_i][a_i] = w_i#weight of edge\\n\\n            self.neighbor[a_i].append(b_i)\\n            self.neighbor[b_i].append(a_i)\\n\\n    def dijkstra(self,v_n):\\n        d = [-1]*self.size\\n        temp_d = [(10**9,i)for i in range(self.size)]\\n        temp_d[v_n] = (0,v_n)\\n        prev = [-1]*self.size\\n\\n        q = set(range(self.size))\\n        \\n        while len(q)>0:\\n            u = min([temp_d[q_i] for q_i in q])[1]\\n            d[u] = temp_d[u][0]\\n            q.discard(u)\\n            for v in self.neighbor[u]:\\n                temp = temp_d[u][0]+self.w_e[u][v]\\n                if temp_d[v][0]>temp:\\n                    temp_d[v] = temp,v\\n                    prev[v] = u\\n\\n        return d,prev\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations\\nimport sys\\ninput = sys.stdin.readline\\n\\ndef main():\\n    N, M, RR = map(int, input().split())\\n    R = list(map(int, input().split()))\\n    INF = float(\\\"inf\\\")\\n    T = [[INF] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = tuple(map(int, input().split()))\\n        T[a-1][b-1] = c\\n        T[b-1][a-1] = c\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if T[i][j] > T[i][k]+T[k][j]:\\n                    T[i][j] = T[i][k]+T[k][j]\\n    print(min(sum(T[rs[i]-1][rs[i+1]-1] for i in range(RR-1)) for rs in permutations(R)))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import dijkstra\\nimport numpy as np\\nfrom itertools import permutations\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int,input().split()))\\n\\n\\\"\\\"\\\"\\nhttps://juppy.hatenablog.com/entry/2018/11/01/%E8%9F%BB%E6%9C%AC_python_%E5%85%A8%E7%82%B9%E5%AF%BE%E6%9C%80%E7%9F%AD%E7%B5%8C%E8%B7%AF%E6%B3%95%EF%BC%88%E3%83%AF%E3%83%BC%E3%82%B7%E3%83%A3%E3%83%AB%E3%83%95%E3%83%AD%E3%82%A4%E3%83%89%E6%B3%95\\n\\\"\\\"\\\"\\n\\ndef warshall_floyd(d):\\n    n = len(d)\\n    #d[i][j]: i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                d[i][j] = min(d[i][j],d[i][k] + d[k][j])\\n    return d\\n\\ngraph = [[float(\\\"inf\\\") for i in range(N)] for j in range(N)] \\nfor i in range(M):\\n    A,B,C = list(map(int,input().split()))\\n    graph[A-1][B-1] = C\\n    graph[B-1][A-1] = C\\n\\ndist = dijkstra(graph)\\n\\nans = 1e10\\nfor i in permutations(r):\\n    flag = True\\n    way = list(i)\\n    tmp = 0\\n    for i in range(len(way)-1):\\n        if dist[way[i]-1][way[i+1]-1] < 0:\\n            flag = False\\n            break\\n        tmp += dist[way[i]-1][way[i+1]-1]\\n    if flag:\\n        ans = min(tmp,ans)\\n\\nprint((int(ans)))\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time, copy,bisect\\nfrom operator import itemgetter\\n#from heapq import heappush, heappop\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\n#from scipy.sparse import csr_matrix\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\nmod = 10**9 + 7\\n\\nstdin = sys.stdin\\n\\nni = lambda: int(ns())\\nnf = lambda: float(ns())\\nna = lambda: list(map(int, stdin.readline().split()))\\nnb = lambda: list(map(float, stdin.readline().split()))\\nns = lambda: stdin.readline().rstrip()  # ignore trailing spaces\\n\\nN, M, R = na()\\nr = na()\\ng = [[0] * N for _ in range(N)]\\nfor i in range(M):\\n    a, b, c = na()\\n    a -= 1\\n    b -= 1\\n    g[a][b] = c\\n    g[b][a] = c\\n\\ng = np.array(g)\\nd = shortest_path(g)\\n\\nR = len(r)\\nans = inf\\nfor x in itertools.permutations(r):\\n    tmp = 0\\n    for i in range(R-1):\\n        tmp += d[x[i] - 1][x[i+1] - 1]\\n    ans = min(ans, tmp)\\nprint((int(ans)))\\n\\n\\n\\n\\n\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import dijkstra\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = dijkstra(G, directed=False)\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"from itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\n\\nwith open(0) as f:\\n    N, M, R = map(int, f.readline().split())\\n    r = list(map(int, f.readline().split()))\\n    path = [tuple(map(int, line.split())) for line in f.readlines()]\\n\\ngraph = np.full((N,N), np.inf)\\nfor a, b, c in path:\\n    graph[a-1, b-1] = c\\n    graph[b-1, a-1] = c\\n\\ngraph = dijkstra(graph, directed=False)\\nans = np.inf\\npermutation = list(permutations(r))\\nfor p in permutation:\\n    way = 0\\n    for x,y in zip(p[:len(r)], p[1:]):\\n        way += graph[x-1, y-1]\\n    ans = min(ans, way)\\nprint(int(ans))\", \"n,m,r  = list(map(int,input().split()))\\nr_list = list(map(int,input().split()))\\nmatrix = [[float(\\\"inf\\\") for i in range(n)] for j in range(n)]\\nfor c in range(n):\\n  matrix[c][c] = 0\\nfor v in range(m):\\n  a,b,c = list(map(int,input().split()))\\n  if matrix[a-1][b-1] > c:\\n    matrix[a-1][b-1] = c\\n    matrix[b-1][a-1] = c\\nimport numpy as np\\nmatrix = np.array(matrix)\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nload = shortest_path(matrix, directed=False)\\nimport itertools\\nans_list = []\\nfor balls in itertools.permutations(r_list):\\n    balls = list(balls)\\n    ans = 0\\n    for i in range(1,len(r_list)):\\n      ans += load[balls[i-1]-1][balls[i]-1]\\n    ans_list.append(int(ans))\\n    \\nprint((min(ans_list)))\\n      \\n\\n\\n\", \"N, M, K = (int(x) for x in input().split())\\nR = [int(x)-1 for x in input().split()]\\n\\ndist = [[10e8 for _ in range(N)] for _ in range(N)]\\nfor i in range(N): dist[i][i] = 0\\nfor _ in range(M):\\n    a, b, c = (int(x) for x in input().split())\\n    dist[a-1][b-1] = dist[b-1][a-1] = c \\ndel a, b, c, K\\n\\nfor k in range(N):\\n    for i in range(N):\\n        for j in range(i,N):\\n            dist[i][j] = dist[j][i] = min(dist[i][j], dist[i][k]+dist[k][j])\\n\\ndef mindist(x, X):\\n    nonlocal dist\\n    Y = X.copy()\\n    Y.remove(x)\\n    if len(Y) == 0: return 0\\n    return min([dist[x][y] + mindist(y,Y) for y in Y])\\n\\nans = min([mindist(r, R) for r in R])\\nprint(ans)\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\ninf = 10 ** 10\\nn, m, r = map(int,input().split())\\nR = list(map(int,input().split()))\\n\\nedge = [[inf] * n for i in range(n)]\\nfor i in range(m):\\n    a, b, c = map(int,input().split())\\n    edge[a - 1][b - 1] = c\\n\\nG = csgraph_from_dense(edge, null_value = inf)\\nd = floyd_warshall(G, False)\\n\\nrec = inf\\nfor i in permutations(R, r):\\n    temp = 0\\n    #print(i)\\n    for j in range(r):\\n        if j == r - 1:continue\\n        temp += d[i[j] - 1][i[j + 1] - 1]\\n    rec = min(rec, temp)\\n\\nprint(int(rec))\"]",
        "difficulty": "interview",
        "input": "3 3 2\n1 3\n2 3 2\n1 3 6\n1 2 2\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc073/tasks/abc073_d"
    },
    {
        "id": 1068,
        "task_id": 2517,
        "test_case_id": 3,
        "question": "There are N towns in the State of Atcoder, connected by M bidirectional roads.\nThe i-th road connects Town A_i and B_i and has a length of C_i.\nJoisino is visiting R towns in the state, r_1,r_2,..,r_R (not necessarily in this order).\nShe will fly to the first town she visits, and fly back from the last town she visits, but for the rest of the trip she will have to travel by road.\nIf she visits the towns in the order that minimizes the distance traveled by road, what will that distance be?\n\n-----Constraints-----\n - 2≤N≤200\n - 1≤M≤N×(N-1)/2\n - 2≤R≤min(8,N) (min(8,N) is the smaller of 8 and N.)\n - r_i≠r_j (i≠j)\n - 1≤A_i,B_i≤N, A_i≠B_i\n - (A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)\n - 1≤C_i≤100000\n - Every town can be reached from every town by road.\n - All input values are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN M R\nr_1 ... r_R\nA_1 B_1 C_1\n:\nA_M B_M C_M\n\n-----Output-----\nPrint the distance traveled by road if Joisino visits the towns in the order that minimizes it.\n\n-----Sample Input-----\n3 3 3\n1 2 3\n1 2 1\n2 3 1\n3 1 4\n\n-----Sample Output-----\n2\n\nFor example, if she visits the towns in the order of 1, 2, 3, the distance traveled will be 2, which is the minimum possible.",
        "solutions": "[\"from itertools import permutations as p\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nn, m, r = map(int, input().split())\\nR = list(map(int, input().split()))\\nl = [[0]*n for _ in range(n)]\\nfor _ in range(m):\\n  a, b, c = map(int, input().split())\\n  a -= 1\\n  b -= 1\\n  l[a][b] = c\\n  l[b][a] = c\\nF = floyd_warshall(l)\\n\\nans = float(\\\"inf\\\")\\nfor v in p(R):\\n  temp = 0\\n  for i in range(r-1):\\n    temp += F[v[i]-1][v[i+1]-1]\\n  ans = min(ans, temp)\\n  \\nprint(int(ans))\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import floyd_warshall\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = floyd_warshall(G, directed=False)\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"def main():\\n    import itertools\\n    n,m,r=list(map(int,input().split()))\\n    rx=[int(i) for i in input().split()]\\n    inf=100000000\\n    wf=[[inf]*n for i in range(n)]\\n    for i in range(n):\\n        wf[i][i]=0\\n\\n    for i in range(m):\\n        a,b,c=list(map(int,input().split()))\\n        wf[a-1][b-1]=c\\n        wf[b-1][a-1]=c\\n\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if wf[i][j]>wf[i][k]+wf[k][j]:\\n                    wf[i][j]=wf[i][k]+wf[k][j]\\n    cnt=0\\n    l=list(itertools.permutations([i for i in range(r)]))\\n    cnt=inf\\n    for i in l:\\n        cnt_sub=0\\n        for j in range(r-1):\\n            cnt_sub+=wf[rx[i[j]]-1][rx[i[j+1]]-1]\\n        cnt=min(cnt,cnt_sub)\\n    print(cnt)\\nmain()\\n\", \"import itertools as it\\nn,m,r=map(int,input().split())\\nll=list(map(int,input().split()))\\nd=[[float(\\\"inf\\\") for _ in range(n)] for _ in range(n)] #\\u5404\\u9802\\u70b9\\u304b\\u3089\\u5404\\u9802\\u70b9\\u3078\\u306e\\u6700\\u5c0f\\u30b3\\u30b9\\u30c8\\nfor i in range(m): #\\u91cd\\u8907\\u306a\\u3057ver\\n    a,b,t=map(int,input().split())\\n    d[a-1][b-1]=t\\n    d[b-1][a-1]=t #\\u6709\\u5411\\u30b0\\u30e9\\u30d5\\u306e\\u5834\\u5408\\u6d88\\u3059\\nfor i in range(n):\\n    d[i][i]=0\\nfor k in range(n):\\n    for i in range(n):\\n        for j in range(n):\\n            if d[i][j]>d[i][k]+d[k][j]:\\n                d[i][j]=d[i][k]+d[k][j]\\nans=10**9\\nfor p in it.permutations(ll):\\n    tmp=0\\n    for j in range(r-1):\\n        s,g=p[j]-1,p[j+1]-1\\n        tmp+=d[s][g]\\n    ans=min(ans,tmp)\\nprint(ans)\", \"import sys, re\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees#, log2\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left\\nfrom fractions import gcd\\nfrom heapq import heappush, heappop, heapify\\nfrom functools import reduce\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nr = LIST()\\nABC = [LIST() for _ in range(M)]\\n\\ndist_matrix = [[0]*N for _ in range(N)]\\nfor A, B, C in ABC:\\n\\tdist_matrix[A-1][B-1] = C\\n\\tdist_matrix[B-1][A-1] = C\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\n#\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5  \\u6ce8\\u610f PyPy\\u975e\\u5bfe\\u5fdc!!!!\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\ngraph = csr_matrix(dist_matrix) #\\u96a3\\u63a5\\u884c\\u5217\\u304b\\u3089csr\\u884c\\u5217\\u3092\\u3064\\u304f\\u308b\\ndist_matrix = floyd_warshall(graph, directed=False) #\\u7121\\u5411\\u306e\\u5834\\u5408directed=False\\n#dist_matrix\\u306e\\u4e2d\\u8eab\\u306ffloat\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u306e\\u3067\\u6ce8\\u610f\\uff01\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\nans = INF\\nfor x in permutations(r):\\n\\tdis = 0\\n\\tfor i in range(1, R):\\n\\t\\tdis += dist_matrix[x[i-1]-1][x[i]-1]\\n\\tans = min(ans, dis)\\n\\nprint((int(ans)))\\n\\n\\n\\t\\n\", \"# ABC073 D - joisino's travel\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\n\\ninf = 10**18\\ndist = [[inf]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = map(int, input().split())\\n    dist[A-1][B-1] = dist[B-1][A-1] = C\\n\\ndist = floyd_warshall(dist)\\n\\nans = inf\\nfor rr in permutations(r):\\n    cost = 0\\n    for i in range(R-1):\\n        cost += dist[rr[i]-1][rr[i+1]-1]\\n    if cost < ans:\\n        ans = cost\\nprint(int(ans))\", \"import itertools\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nN, M, R = list(map(int, input().split()))\\nr = tuple(map(int, input().split()))\\n\\nINF = 10**10\\n\\nd = [[INF] * N for _ in range(N)]\\n\\nfor i in range(N):\\n    d[i][i] = 0\\n\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    if d[a][b] > c:\\n        d[a][b] = c\\n        d[b][a] = c\\n\\n# for k in range(N):\\n#     for i in range(N):\\n#         for j in range(N):\\n#             d[i][j] = min(d[i][j], d[i][k] + d[k][j])\\n\\nd = floyd_warshall(d)\\n\\n\\nans = INF\\nfor p in itertools.permutations(r):\\n    dist = 0\\n    for i in range(R-1):\\n        dist += d[p[i]-1][p[i+1]-1]\\n\\n    ans = min(ans, dist)\\n\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\n\\nd = [[0] * (N + 1) for _ in range(N + 1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    d[A][B] = C\\n    d[B][A] = C\\ng = csgraph_from_dense(d)\\ng = floyd_warshall(g)\\n\\nresult = 1000000000\\nfor p in permutations(r):\\n    t = 0\\n    for i in range(R - 1):\\n        t += g[p[i]][p[i + 1]]\\n    result = min(result, t)\\nprint((int(result)))\\n\", \"import sys\\nfrom collections import defaultdict\\nfrom heapq import heapify,heappush,heappop\\nreadline=sys.stdin.readline\\n\\ndef main():\\n    N,M,R=list(map(int,readline().split()))\\n    r=list(map(int,readline().split()))\\n    ew=[tuple(map(int,readline().split())) for _ in range(M)]\\n    edges=defaultdict(list)\\n    wt=defaultdict(int)\\n    for e in ew:\\n        edges[e[0]].append(e[1])\\n        edges[e[1]].append(e[0])\\n        wt[(e[0],e[1])]=e[2]\\n        wt[(e[1],e[0])]=e[2]\\n    inf=float('inf')\\n    def dijkstra(v):\\n        dist=[inf]*(N+1)\\n        dist[v]=0\\n        vw=[(0,v)]\\n        heapify(vw)\\n        while vw:\\n            cvw=heappop(vw)\\n            if cvw[0]>dist[cvw[1]]:\\n                continue\\n            for w in edges[cvw[1]]:\\n                cand=cvw[0]+wt[(cvw[1],w)]\\n                if dist[w]>cand:\\n                    dist[w]=cand\\n                    heappush(vw,(cand,w))\\n        return dist\\n    distdict=dict((v,dijkstra(v)) for v in r)\\n    mindist=inf\\n    def permlist(lst):\\n        tmp=[]\\n        if not lst:\\n            return [[]]\\n        for i,x in enumerate(lst):\\n            lstx=lst[:i]+lst[i+1:]\\n            ret=permlist(lstx)\\n            for e in ret:\\n                e.append(x)\\n            tmp.extend(ret)\\n        return tmp\\n    #print(distdict,permlist(r))\\n    pathlist=[sum([distdict[v][w] for v,w in zip(lst,lst[1:])]) for lst in permlist(r)]\\n    print((min(pathlist)))\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**6)\\ninput = sys.stdin.readline\\n\\ndef warshallFloyd():\\n    for k in range(N):\\n        for i in range(N-1):\\n            for j in range(i, N):\\n                tmp = min(E[i][j], E[i][k] + E[k][j])\\n                E[i][j] = E[j][i] = tmp\\n\\ndef dfs(c, p, d):\\n    if c == R:\\n        ans[0] = min(ans[0], d)\\n        return\\n    for i in range(R):\\n        if used[i]:\\n            continue\\n        used[i] = True\\n        if p == -1:\\n            dfs(c+1, i, 0)\\n        else:\\n            dfs(c+1, i, d + E[S[p]-1][S[i]-1])\\n        used[i] = False\\n\\nN, M, R = map(int, input().split())\\nS = list(map(int, input().split()))\\nE = [[float('inf')] * N for _ in range(N)]\\nfor i in range(N):\\n    E[i][i] = 0\\nfor _ in range(M):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    E[a][b] = E[b][a] = c\\nwarshallFloyd()\\nans = [float('inf')]\\nused = [False] * R\\ndfs(0, -1, 0)\\nprint(*ans)\", \"import sys\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\ninput = sys.stdin.readline\\nans = -1\\n\\nn, m, r = list(map(int, input().split()))\\nr  = list(map(int, input().split()))\\nA, B, C = [], [], []\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    A.append(a-1)\\n    B.append(b-1)\\n    C.append(c)\\n\\nwf = floyd_warshall(csr_matrix((C, (A, B)), shape = (n, n)), directed=False)\\n\\nfor pi in permutations(r):\\n    cand = 0\\n    for r1, r2 in zip(pi, pi[1:]):\\n        cand += int(wf[r1-1][r2-1])\\n    if ans < 0 or ans > cand:\\n        ans = cand\\n\\nprint(ans)\\n\", \"#!/usr/bin/env python3\\nimport sys\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    import numpy as np    \\n    from scipy.sparse import coo_matrix    \\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations, islice\\n    from functools import reduce\\n    # coo_matrix((data, (i, j)), [shape=(M, N)])\\n    mat = floyd_warshall(coo_matrix((C, (A, B)), shape=(N+1, N+1), dtype=np.int32).tocsr(), directed=False)\\n    f = lambda a, b: (a[0]+mat[a[1]][b], b)\\n    return int(min(reduce(f, q, (0, s))[0] for s, *q in permutations(r)))\\n\\n# Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    print((solve(N, M, R, r, A, B, C)))\\n\\ndef test():\\n    import doctest\\n    doctest.testmod()\\n\\ndef __starting_point():\\n    #test()\\n    main()\\n\\n__starting_point()\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nn,m,r=map(int,input().split())\\nvis=list(map(int,input().split()))\\nINF=10**18\\npath=[[INF]*n for i in range(n)]\\nfor i in range(m):\\n  a,b,c=map(int,input().split())\\n  path[a-1][b-1]=c\\n  path[b-1][a-1]=c\\npath=floyd_warshall(path)\\nans=10**18\\nfor root in permutations(vis):\\n  cnt=0\\n  for j in range(1,r):\\n    cnt+=path[root[j-1]-1][root[j]-1]\\n  ans=min(ans,cnt)\\nprint(int(ans))\", \"import sys\\nfrom itertools import permutations\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n\\n    def warshall_floyd(d):\\n        # d[i][j]: i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n        for k in range(N):\\n            for i in range(N):\\n                for j in range(N):\\n                    tmp = d[i][k] + d[k][j]\\n                    if d[i][j] > tmp:\\n                        d[i][j] = tmp\\n        return d\\n\\n    r = list(map(int, input().split()))\\n    d = [[float(\\\"inf\\\")] * N for i in range(N)]\\n    for i in range(M):\\n        x, y, z = list(map(int, input().split()))\\n        d[x - 1][y - 1] = z\\n        d[y - 1][x - 1] = z\\n    for i in range(N):\\n        d[i][i] = 0  # \\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\n    D = warshall_floyd(d)\\n    P = list(permutations(r))\\n    cost = float(\\\"inf\\\")\\n    for p in P:\\n        c = 0\\n        for i in range(R - 1):\\n            c += D[p[i] - 1][p[i + 1] - 1]\\n        if c < cost:\\n            cost = c\\n    print(cost)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"#!/usr/bin/env python3\\nimport sys\\nimport itertools\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    from scipy.sparse import csr_matrix\\n    from scipy.sparse.csgraph import floyd_warshall\\n    matrix = [[0]*N for _ in range(N)]\\n    for i in range(M):\\n        matrix[A[i]-1][B[i]-1] = C[i]\\n        matrix[B[i]-1][A[i]-1] = C[i]\\n    \\n    dist_matrix = floyd_warshall(csgraph=matrix, directed=False)\\n\\n    perm = list(itertools.permutations(r)) \\n\\n    answer = float('inf')\\n    for p in perm:\\n        a = 0\\n        for index in range(len(p)-1):\\n            a += dist_matrix[p[index]-1][p[index+1]-1]\\n        answer = min(answer,a)\\n    print((int(answer)))\\n    return\\n\\n\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    solve(N, M, R, r, A, B, C)\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"from scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\nn,m,r,*L=map(int,open(0).read().split())\\nf=floyd_warshall(csr_matrix((L[r+2::3],(L[r::3],L[r+1::3])),(n+1,n+1)),0)\\nprint(int(min(sum(f[s,t]for s,t in zip(o,o[1:]))for o in permutations(L[:r]))))\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path\\n\\nN,M,R = map(int, input().split())\\nrs = np.array(list(map(int, input().split()))) -1\\n\\nnodes = np.full((N,N), np.inf) \\nfor i in range(M):\\n  a,b,c = map(int, input().split())\\n  nodes[a-1,b-1] = nodes[b-1,a-1] = c\\nmin_ds = shortest_path(nodes)\\nres = float('inf')\\nfor p in itertools.permutations(rs):\\n  prev = p[0]\\n  tmp = 0\\n  for next in p[1:]:\\n    tmp += min_ds[prev,next]\\n    prev = next\\n  res = min(res, tmp)\\n\\nprint(int(res))\", \"def main():\\n  import numpy as np\\n  import itertools\\n  from scipy.sparse import csr_matrix\\n  from scipy.sparse.csgraph import floyd_warshall\\n  import sys\\n  readline = sys.stdin.readline\\n  readlines = sys.stdin.readlines\\n\\n  N, M, R= map(int, readline().split())\\n  r = list(map(int,input().split()))\\n  lines = readlines()\\n  edge = np.array([line.split() for line in lines], dtype = np.int64).T\\n  graph = csr_matrix((edge[2], (edge[:2] - 1)), (N, N))\\n  distance_mat = floyd_warshall(graph,directed = False)\\n  #print(distance_mat)\\n  ans = float(\\\"inf\\\")\\n  for town in itertools.permutations(r,R):\\n    #print(town)\\n    tmp = 0\\n    for i in range(R-1):\\n      tmp += distance_mat[town[i]-1][town[i+1]-1]\\n    ans = min(ans,tmp)\\n  print(int(ans))\\nmain()\", \"import sys\\nimport numpy as np \\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nI = np.array(sys.stdin.read().split(), dtype=np.int64)\\nn, m, R = I[:3]\\nr = I[3:3+R] - 1\\na, b, c = I[3+R:].reshape(-1, 3).T\\na -= 1; b -= 1\\ngraph = csr_matrix((c, (a, b)), shape=(n, n))\\n\\ndef main():\\n    dist = floyd_warshall(graph, directed=False).astype(np.int64)\\n    *perms, = permutations(r)\\n    perms = np.array(perms)\\n    res = dist[perms[:, :-1], perms[:, 1:]]\\n    ans = np.amin(np.sum(res, axis=1))\\n    return ans\\n\\ndef __starting_point():\\n    ans = main()\\n    print(ans)\\n__starting_point()\", \"import itertools\\nfrom scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nN, M, R = list(map(int, input().split()))\\nr = [int(i) for i in input().split()]\\n\\nG = [[10**8]*(N+1) for _ in range(N+1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    G[A][B] = C\\n    G[B][A] = C\\n\\nDG = csgraph_from_dense(G, null_value=10**8)\\nd = floyd_warshall(DG)\\n\\nans = 10**8\\nfor route in itertools.permutations(r):\\n    c = 0\\n    for i in range(len(route)-1):\\n        c += d[route[i]][route[i+1]]\\n    ans = min(ans, c)\\nprint((int(ans)))\\n\", \"import heapq\\nfrom itertools import permutations\\nn, m, r = list(map(int, input().split()))\\nR = list([int(x)-1 for x in input().split()])\\nedges = [[]for _ in range(n)]\\nfor _ in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    edges[a].append((c, b))\\n    edges[b].append((c, a))\\n\\n\\ndef dijkstra(init_v):\\n    next_v = [(0, init_v)]\\n    INF = 10**18\\n    dist = [INF]*n\\n    dist[init_v] = 0\\n    while next_v:\\n        d, v = heapq.heappop(next_v)\\n        if dist[v] < d:\\n            continue\\n        for d, v2 in edges[v]:\\n            if dist[v2] <= dist[v]+d:\\n                continue\\n            dist[v2] = dist[v]+d\\n            heapq.heappush(next_v, (dist[v2], v2))\\n    return dist\\n\\n\\ndists = []\\nfor x in R:\\n    dist = dijkstra(x)\\n    dists.append(dist)\\n\\nINF = 10**18\\nans = INF\\nfor subset in permutations(list(range(r))):\\n    d = 0\\n    for v, v2 in zip(subset, subset[1:]):\\n        d += dists[v][R[v2]]\\n    if d < ans:\\n        ans = d\\n\\nprint(ans)\\n\", \"# ABC073 D - joisino's travel\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\n\\ninf = 10**18\\ndist = [[inf]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = map(int, input().split())\\n    dist[A-1][B-1] = dist[B-1][A-1] = C\\n\\ndist = floyd_warshall(dist)\\n\\nans = inf\\nfor rr in permutations(r):\\n    cost = 0\\n    for i in range(R-1):\\n        cost += dist[rr[i]-1][rr[i+1]-1]\\n    ans = min(ans, cost)\\nprint(int(ans))\", \"from itertools import permutations\\n\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph._shortest_path import floyd_warshall\\n\\n\\ndef solve():\\n    N, M, R = list(map(int, input().split()))\\n    r = list([int(x)-1 for x in input().split()])\\n    dist = [[0 for _ in range(N)] for _ in range(N)]\\n    for _ in range(M):\\n        A, B, C = list(map(int, input().split()))\\n        dist[A-1][B-1] = C\\n        dist[B-1][A-1] = C\\n    fw = floyd_warshall(csr_matrix(dist))\\n    \\n    ans = 10**12\\n    for visit in permutations(r, R):\\n        m_sum = 0\\n        for i in range(R-1):\\n            m_sum += fw[visit[i]][visit[i+1]]\\n        ans = min(ans, m_sum)\\n    print((int(ans)))\\n\\n\\ndef __starting_point():\\n    solve()\\n\\n__starting_point()\", \"import numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nn, m, r = map(int,input().split()) #n:\\u9802\\u70b9\\u6570\\u3000m:\\u8fba\\u306e\\u6570 r:\\u8a2a\\u308c\\u308b\\u753a\\u306e\\u6570\\nR = list(map(int,input().split()))\\n\\nd = np.array([[float(\\\"inf\\\") for i in range(n)] for i in range(n)])\\n#d[u][v] : \\u8fbauv\\u306e\\u30b3\\u30b9\\u30c8(\\u5b58\\u5728\\u3057\\u306a\\u3044\\u3068\\u304d\\u306finf)\\nfor i in range(m):\\n    x, y, z = map(int,input().split())\\n    d[x-1][y-1] = z\\n    d[y-1][x-1] = z\\nfor i in range(n):\\n    d[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\nwa = floyd_warshall(d)\\n\\n# \\u8a2a\\u308c\\u308b\\u753a\\u306e\\u9806\\u756a\\u3092\\u9806\\u5217\\u3067\\u7528\\u610f\\nfrom itertools import permutations\\nptn = permutations(R)\\n\\nans = 10**19\\nfor p in ptn:\\n  cost = 0\\n  roots = [(j-1, i-1) for i, j in zip(p, p[1:])]\\n  for i, j in roots:\\n    cost += wa[i][j]\\n  ans = min(ans, cost)\\n  \\nprint(int(ans))\", \"import itertools\\n\\n\\ndef dijkstra(v, G):\\n    import heapq\\n\\n    ret = [10 ** 10] * len(G)\\n    ret[v] = 0\\n    q = [(ret[i], i) for i in range(len(G))]\\n    heapq.heapify(q)\\n\\n    while len(q):\\n        tmpr, u = heapq.heappop(q)\\n        if tmpr == ret[u]:\\n            for w, l in G[u]:\\n                if ret[w] > ret[u] + l:\\n                    ret[w] = ret[u] + l\\n                    heapq.heappush(q, (ret[w], w))\\n    return ret\\n\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\nNE = [[] for _ in range(N)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    NE[A - 1].append((B - 1, C))\\n    NE[B - 1].append((A - 1, C))\\n\\nRE = [[0] * R for _ in range(R)]\\nfor iR in range(R):\\n    for jR in range(iR + 1, R):\\n        lR = dijkstra(r[iR] - 1, NE)[r[jR] - 1]\\n        RE[iR][jR] = lR\\n        RE[jR][iR] = lR\\n\\nans = 10 ** 10\\nfor p in itertools.permutations(list(range(R))):\\n    ans = min(ans, sum([RE[p[i]][p[i + 1]] for i in range(R - 1)]))\\n\\nprint(ans)\\n\", \"# solution\\n\\nimport itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nint1 = lambda x: int(x) - 1\\n\\nN, M, R = map(int, input().split())\\nT = sorted(list(map(int1, input().split())))\\nA = np.array([tuple(map(int, input().split())) for _ in range(M)]).T\\n\\nmatr = csr_matrix((A[2], (A[0] - 1, A[1] - 1)), shape=(N, N))\\nway = [dijkstra(matr, indices=t, directed=False)[T].astype(int).tolist() for t in T]\\n\\nresult = float('inf')\\nfor t in itertools.permutations(range(R)):\\n    tmp = 0\\n    for i in range(R - 1):\\n        tmp += way[t[i]][t[i + 1]]\\n    result = min(result, tmp)\\nprint(result)\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nsys.setrecursionlimit(10 ** 9)\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nA = [a-1 for a in LIST()]\\nG = list2d(N, N, INF)\\nfor i in range(M):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    G[a][b] = c\\n    G[b][a] = c\\n\\n# \\u5168\\u4f53\\u30b0\\u30e9\\u30d5\\u306e\\u6700\\u77ed\\u8ddd\\u96e2(\\u3053\\u3053\\u304b\\u3089\\u5fc5\\u8981\\u306a\\u9802\\u70b9\\u9593\\u3060\\u3051\\u4f7f\\u3046)\\nwf = floyd_warshall(G)\\n\\nans = INF\\nfor r in range(R):\\n    # TSP(\\u5de1\\u56de\\u30bb\\u30fc\\u30eb\\u30b9\\u30de\\u30f3)\\n    dp = list2d(1<<R, R, INF)\\n    dp[1<<r][r] = 0\\n    for bit in range(1, (1<<R)-1):\\n        for i in range(R):\\n            if not (bit >> i & 1):\\n                continue\\n            for j in range(R):\\n                if bit >> j & 1:\\n                    continue\\n                a, b = A[i], A[j]\\n                dp[bit|1<<j][j] = min(dp[bit|1<<j][j], dp[bit][i] + int(wf[a,b]))\\n    ans = min(ans, min(dp[-1]))\\nprint(ans)\\n\", \"import numpy as np\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\nI = [int(_) for _ in open(0).read().split()]\\nN, M, R = I[:3]\\nr = np.array(I[3:3 + R])\\nABC = I[3 + R:]\\nF = floyd_warshall(csr_matrix((ABC[2::3], (ABC[::3], ABC[1::3])), (N + 1, N + 1)), 0).astype(np.int64)\\nans = float('inf')\\nfor root in itertools.permutations(r, R):\\n    ans = min(ans, sum(F[i,j] for i, j in zip(root, root[1:])))\\nprint(ans)\\n\", \"#!/usr/bin python3\\n# -*- coding: utf-8 -*-\\n\\n# \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5\\n# \\u5168\\u9802\\u70b9\\u9593\\u6700\\u77ed\\u8def\\n# d[i][j]\\u306f2\\u9802\\u70b9\\u9593i, j\\u9593\\u306e\\u79fb\\u52d5\\u30b3\\u30b9\\u30c8\\u3092\\u683c\\u7d0d, M\\u306f\\u9802\\u70b9\\u6570\\n\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\n\\ndef main():\\n    N, M, R = map(int,input().split()) #N:\\u9802\\u70b9\\u6570\\u3000M:\\u8fba\\u306e\\u6570\\n    d = [[float(\\\"inf\\\")]*N for i in range(N)]\\n    #d[u][v] : \\u8fbauv\\u306e\\u30b3\\u30b9\\u30c8(\\u5b58\\u5728\\u3057\\u306a\\u3044\\u3068\\u304d\\u306finf)\\n    r = list(map(int,input().split()))\\n    for i in range(M):\\n        u, v, w = map(int,input().split())\\n        d[u-1][v-1] = w\\n        d[v-1][u-1] = w\\n    for i in range(N):\\n        d[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\n    cost = floyd_warshall(d)\\n    L = list(permutations(r, R))\\n    ret = 10**9\\n    for rt in L:\\n        cos = 0\\n        for i in range(1,R):\\n            cos += cost[rt[i]-1][rt[i-1]-1]\\n        ret = min(ret, cos)\\n    print(int(ret))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nINF = 1001001001\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\ngraph = [[INF]*N for _ in range(N)]\\nfor i in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    graph[A-1][B-1] = C\\n    graph[B-1][A-1] = C\\ndist_matrix = floyd_warshall(graph)\\n\\nans = INF\\nfor root in permutations(r):\\n    cnt = 0\\n    for j in range(R-1):\\n        cnt += dist_matrix[root[j]-1][root[j+1]-1]\\n    ans = min(ans, cnt)\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\nabc = [list(map(int, input().split())) for _ in range(M)]\\n\\nr = [e - 1 for e in r]\\n\\ng = [[0] * N for _ in range(N)]\\nfor a, b, c in abc:\\n\\ta -= 1\\n\\tb -= 1\\n\\tg[a][b] = c\\n\\tg[b][a] = c\\n\\ndist = floyd_warshall(csr_matrix(g))\\n\\nans = float(\\\"inf\\\")\\nfor pat in permutations(r, len(r)):\\n\\tsm = 0\\n\\tfor e1, e2 in zip(pat, pat[1:]):\\n\\t\\tsm += dist[e1][e2]\\n\\n\\tans = min(ans, sm)\\n\\nans = int(ans)\\nprint(ans)\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = map(int, input().split())\\n\\ns = [[10**9]*N for i in range(N)]\\nt = list(map(int, input().split()))\\nfor i in range(M):\\n    a, b, c = map(int, input().split())\\n    s[a-1][b-1] = c\\n    s[b-1][a-1] = c\\n    \\ns = floyd_warshall(s)\\nans = float('inf')\\nfor i in permutations(t):\\n    A_n = 0\\n    for j in range(R-1):\\n        A_n += s[i[j] - 1][i[j+1] - 1]\\n    ans = min(ans, A_n)\\n    \\nprint(int(ans))\", \"import sys\\nfrom itertools import permutations\\n\\nread = sys.stdin.read\\nreadline = sys.stdin.readline\\nreadlines = sys.stdin.readlines\\nsys.setrecursionlimit(10 ** 9)\\nINF = 1 << 60\\nMOD = 1000000007\\n\\n\\ndef main():\\n    N, M, R = list(map(int, readline().split()))\\n    city = [int(s) - 1 for s in readline().split()]\\n    ABC = list(map(int, read().split()))\\n\\n    G = [[INF] * N for _ in range(N)]\\n    for a, b, c in zip(*[iter(ABC)] * 3):\\n        G[a - 1][b - 1] = G[b - 1][a - 1] = c\\n\\n    for i in range(N):\\n        G[i][i] = 0\\n\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if G[i][j] > G[i][k] + G[k][j]:\\n                    G[i][j] = G[i][k] + G[k][j]\\n\\n    ans = INF\\n    for plan in permutations(city):\\n        res = 0\\n        for i in range(R - 1):\\n            res += G[plan[i]][plan[i + 1]]\\n        if ans > res:\\n            ans = res\\n\\n    print(ans)\\n    return\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"#!/usr/bin/env python3\\nimport sys\\n\\ndef solve(N: int, M: int, R: int, r: \\\"List[int]\\\", A: \\\"List[int]\\\", B: \\\"List[int]\\\", C: \\\"List[int]\\\"):\\n    from scipy.sparse import coo_matrix    \\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations\\n    from functools import reduce\\n    # coo_matrix((data, (i, j)), [shape=(M, N)])\\n    mat = floyd_warshall(coo_matrix((C, (A, B)), shape=(N+1, N+1)).tocsr(), directed=False)\\n    f = lambda a, b: (a[0]+mat[a[1]][b], b)\\n    return int(min(reduce(f, q, (0, s))[0] for s, *q in permutations(r)))\\n\\n# Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools\\ndef main():\\n    def iterate_tokens():\\n        for line in sys.stdin:\\n            for word in line.split():\\n                yield word\\n    tokens = iterate_tokens()\\n    N = int(next(tokens))  # type: int\\n    M = int(next(tokens))  # type: int\\n    R = int(next(tokens))  # type: int\\n    r = [int(next(tokens)) for _ in range(R)]  # type: \\\"List[int]\\\"\\n    A = [int()] * (M)  # type: \\\"List[int]\\\"\\n    B = [int()] * (M)  # type: \\\"List[int]\\\"\\n    C = [int()] * (M)  # type: \\\"List[int]\\\"\\n    for i in range(M):\\n        A[i] = int(next(tokens))\\n        B[i] = int(next(tokens))\\n        C[i] = int(next(tokens))\\n    print(solve(N, M, R, r, A, B, C))\\n\\ndef test():\\n    import doctest\\n    doctest.testmod()\\n\\ndef __starting_point():\\n    #test()\\n    main()\\n__starting_point()\", \"n,m,r_=list(map(int,input().split()))\\nr=list(map(int,input().split()))\\ng=[[]for _ in range(n)]\\nfor _ in range(m):\\n  a,b,c=list(map(int,input().split()))\\n  a-=1\\n  b-=1\\n  g[a].append([b,c])\\n  g[b].append([a,c])\\nfrom heapq import heapify,heappush,heappop\\ndef dks(t0,t1):\\n  kyori=[-1]*n\\n  todo=[[0,t0]]\\n  heapify(todo)\\n  while todo:\\n    d,t=heappop(todo)\\n    kyori[t]=d\\n    if t==t1:break\\n    l=g[t]\\n    for li,c in l:\\n      if kyori[li]==-1:\\n        heappush(todo,[c+d,li])\\n  return kyori[t1]\\nans=pow(10,9)\\nallr=[]\\nimport sys\\nsys.setrecursionlimit(10**7)\\ndef dfs(s,k): #s:list, k:set\\n  if len(k)==1:\\n    s.append(k.pop())\\n    return [s]\\n  ret=[]\\n  for ki in k:\\n    ret.extend(dfs(s+[ki],k-{ki}))\\n  return ret\\nallr=dfs([],set(r))\\nd={}\\nfor i in range(r_-1):\\n  for j in range(i+1,r_):\\n    k=dks(r[i]-1,r[j]-1)\\n    d[r[i]-1,r[j]-1]=k\\n    d[r[j]-1,r[i]-1]=k\\n\\nfor j in range(len(allr)):\\n  r1=allr[j]\\n  ansi=0\\n  for i in range(r_-1):\\n    ansi+=d[(r1[i]-1,r1[i+1]-1)]\\n  ans=min(ans,ansi)\\nprint(ans)\\n\", \"#!/usr/bin/env python3\\n\\nfrom itertools import permutations\\nimport numpy as np\\n\\nHUGE = 10 ** 18\\n\\ndef main():\\n    n, m, r = list(map(int, input().split()))\\n    togo = list(map(int, input().split()))\\n    adj_mat = [[HUGE for j in range(n)] for i in range(n)]\\n    for i in range(n):\\n        adj_mat[i][i] = HUGE\\n    for i in range(m):\\n        a, b, c = list(map(int, input().split()))\\n        a0 = a - 1\\n        b0 = b - 1\\n        adj_mat[a0][b0] = c\\n        adj_mat[b0][a0] = c\\n    adj_mat = np.array(adj_mat)\\n\\n    wf(adj_mat, n)\\n\\n    res = HUGE\\n    for way in permutations(togo):\\n        assert len(way) > 1\\n        x = 0\\n        for i in range(len(way) - 1):\\n            x += adj_mat[way[i] - 1][way[i + 1] - 1]\\n        res = min(res, x)\\n\\n    print(res)\\n\\ndef wf(adj_mat, n):\\n    for k in range(n):\\n        for i in range(n):\\n            adj_mat[i, :] = np.minimum(adj_mat[i, :], adj_mat[i][k] + adj_mat[k, :])\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys, re\\nfrom math import ceil, sqrt, hypot, factorial, pi, sin, cos, tan, asin, acos, atan, radians, degrees#, log2\\nfrom collections import deque, defaultdict, Counter\\nfrom itertools import accumulate, permutations, combinations, combinations_with_replacement, product, groupby\\nfrom operator import itemgetter, mul\\nfrom copy import deepcopy\\nfrom string import ascii_lowercase, ascii_uppercase, digits\\nfrom bisect import bisect, bisect_left\\nfrom fractions import gcd\\nfrom heapq import heappush, heappop, heapify\\nfrom functools import reduce\\ndef input(): return sys.stdin.readline().strip()\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(): return list(map(int, input().split()))\\ndef ZIP(n): return list(zip(*(MAP() for _ in range(n))))\\nsys.setrecursionlimit(10 ** 9)\\nINF = float('inf')\\nmod = 10 ** 9 + 7\\n\\nN, M, R = MAP()\\nr = LIST()\\nABC = [LIST() for _ in range(M)]\\n\\ndist_matrix = [[INF]*N for _ in range(N)]\\n\\nfor A, B, C in ABC:\\n\\tdist_matrix[A-1][B-1] = C\\n\\tdist_matrix[B-1][A-1] = C\\n\\nfor i in range(N):\\n\\tdist_matrix[i][i] = 0\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\n#\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u6cd5  \\u6ce8\\u610f PyPy\\u975e\\u5bfe\\u5fdc!!!!\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\ngraph = csr_matrix(dist_matrix) #\\u96a3\\u63a5\\u884c\\u5217\\u304b\\u3089csr\\u884c\\u5217\\u3092\\u3064\\u304f\\u308b\\ndist_matrix = floyd_warshall(graph, directed=False) #\\u7121\\u5411\\u306e\\u5834\\u5408directed=False\\n#dist_matrix\\u306e\\u4e2d\\u8eab\\u306ffloat\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u306e\\u3067\\u6ce8\\u610f\\uff01\\n\\n#for i in range(N):\\n#\\tprint(dist_matrix[i])\\n\\nans = INF\\nfor x in permutations(r):\\n\\ttmp = 0\\n\\tfor i in range(1, R):\\n\\t\\ttmp += dist_matrix[x[i-1]-1][x[i]-1]\\n\\tans = min(ans, tmp)\\n\\nprint((int(ans)))\\n\\t\\n\", \"from itertools import permutations\\nimport sys\\ninput = sys.stdin.readline\\n\\ndef main():\\n    N, M, RR = map(int, input().split())\\n    R = list(map(int, input().split()))\\n    INF = float(\\\"inf\\\")\\n    T = [[INF] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = tuple(map(int, input().split()))\\n        T[a-1][b-1] = c\\n        T[b-1][a-1] = c\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if T[i][j] > T[i][k]+T[k][j]:\\n                    T[i][j] = T[i][k]+T[k][j]\\n    ans = INF\\n    for rs in permutations(R):\\n        cost = 0\\n        for i in range(RR-1):\\n            cost += T[rs[i]-1][rs[i+1]-1]\\n        ans = min(ans, cost)\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations as p\\n \\nfrom scipy.sparse.csgraph import floyd_warshall\\n \\nn, m, r = map(int, input().split())\\nR = list(map(int, input().split()))\\nl = [[0]*n for _ in range(n)]\\nfor _ in range(m):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    l[a][b] = c\\n    l[b][a] = c\\nF = floyd_warshall(l)\\n \\nans = float(\\\"inf\\\")\\nfor v in p(R):\\n    temp = 0\\n    for i in range(r-1):\\n        temp += F[v[i]-1][v[i+1]-1]\\n    ans = min(ans, temp)\\nprint(int(ans))\", \"n,m,R = map(int,input().split())\\nr = list(map(int,input().split()))\\n\\nfrom itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\ncost = np.ones((n,n))*float('inf')\\nfor i in range(m):\\n    a,b,c = map(int,input().split())\\n    cost[a-1][b-1]=cost[b-1][a-1]=c\\nd = floyd_warshall(cost)\\ndef main():\\n    m = float('inf')\\n\\n    for j in permutations(r):\\n        l = 0\\n        for i in range(len(j)-1):\\n            s,t = j[i],j[i+1]\\n            l += d[s-1][t-1]\\n        if l<m:\\n            m = l\\n    print(int(m))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"# https://atcoder.jp/contests/abc073/tasks/abc073_d\\n\\nimport sys\\nread = sys.stdin.readline\\n\\n\\ndef read_ints():\\n    return list(map(int, read().split()))\\n\\n\\n\\n# default import\\nfrom itertools import product, permutations, combinations\\n\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import dijkstra\\n\\n# \\u30c0\\u30a4\\u30af\\u30b9\\u30c8\\u30e9\\u304b\\uff1f\\n# \\u5168\\u70b9\\u9593\\u306e\\u6700\\u5c0f\\u8ddd\\u96e2\\u3092\\u53d6\\u5f97\\n# r\\u306b\\u3064\\u3044\\u3066permutation\\u3057\\u3066\\u3002\\u305d\\u306e\\u901a\\u308a\\u306b\\u8a2a\\u308c\\u305f\\u3068\\u304d\\u306e\\u8ddd\\u96e2\\u3092\\u30b7\\u30df\\u30e5\\u30ec\\u30fc\\u30b7\\u30e7\\u30f3\\n# \\u6700\\u5c0f\\u5024\\u3092\\u9078\\u3076\\n\\nN, M, r = read_ints()\\nR = [x - 1 for x in read_ints()]\\nadj_mat = [[0] * N for _ in range(N)]\\nfor _ in range(M):\\n    a, b, c = read_ints()\\n    a -= 1\\n    b -= 1\\n    adj_mat[a][b] = c\\n    adj_mat[b][a] = c\\n\\n# print(csr_matrix(adj_mat, dtype='int'))\\nD = dijkstra(csr_matrix(adj_mat, dtype='int'), directed=False)\\n# print(D)\\n# \\u5168\\u63a2\\u7d22\\u30d1\\u30fc\\u30c8\\nans = 2 ** 31\\n\\n\\ndef get_kyori(p):\\n    ret = 0\\n    for ps, pt in zip(p[:-1], p[1:]):\\n        ret += D[ps, pt]\\n    return ret\\n\\n\\nfor p in permutations(R):\\n    ans = min(ans, get_kyori(p))\\nprint((int(ans)))\\n\", \"import sys\\nimport heapq, math\\nfrom itertools import zip_longest, permutations, combinations, combinations_with_replacement\\nfrom itertools import accumulate, dropwhile, takewhile, groupby\\nfrom functools import lru_cache\\nfrom copy import deepcopy\\n\\nN, M, R = list(map(int, input().split()))\\n\\nRS = list(map(int, input().split()))\\n\\nG = [[] for _ in range(N + 1)]\\n\\nfor i in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    G[A].append((B, C))\\n    G[B].append((A, C))\\n\\n\\ndef dijkstra(s):\\n    d = [1 << 28] * (N + 1)\\n    d[s] = 0\\n    q = []\\n    heapq.heappush(q, (0, s))\\n\\n    while q:\\n        cur = heapq.heappop(q)\\n\\n        if d[cur[1]] != cur[0]:\\n            continue\\n\\n        for nx in G[cur[1]]:\\n            if d[nx[0]] > cur[0] + nx[1]:\\n                d[nx[0]] = cur[0] + nx[1]\\n                heapq.heappush(q, (cur[0] + nx[1], nx[0]))\\n\\n    return d\\n\\n\\nDIRS = [dijkstra(s) for s in RS]\\nMAT = [[d[r] for r in RS] for d in DIRS]\\n\\nmem = {}\\n\\n\\ndef solve(idx, st):\\n    if st == (1 << R) - 1:\\n        return 0\\n    if (idx, st) in mem:\\n        return mem[(idx, st)]\\n\\n    ret = 1 << 28\\n    for i in range(R):\\n        if ((st >> i) & 1) == 0:\\n            ret = min(ret, solve(i, st | (1 << i)) + MAT[idx][i])\\n\\n    mem[(idx, st)] = ret\\n    return ret\\n\\n\\nprint((min([solve(i, 1 << i) for i in range(R)])))\\n\", \"from scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list(map(int, input().split()))\\n    r = [i-1 for i in r]\\n    l = [[0] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        l[a][b] = c\\n        l[b][a] = c\\n    G = csr_matrix(l)\\n    dd = dijkstra(G, directed=False)\\n    ans = 10**100\\n    for i in permutations(r, len(r)):\\n        t = 0\\n        for j in range(len(r)-1):\\n            t += dd[i[j]][i[j+1]]\\n        ans = min(ans, t)\\n    return int(ans)\\nprint((main()))\\n\", \"import sys\\nfrom itertools import permutations\\n\\nsys.setrecursionlimit(10 ** 6)\\nINF = float(\\\"inf\\\")\\nMOD = 10 ** 9 + 7\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list([int(x) - 1 for x in input().split()])\\n    P = permutations(r)\\n\\n    dp = [[float(\\\"inf\\\")] * N for _ in range(N)]\\n    for i in range(N):\\n        dp[i][i] = 0\\n\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        dp[a][b] = c\\n        dp[b][a] = c\\n\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                tmp = dp[i][k] + dp[k][j]\\n                if tmp < dp[i][j]:\\n                    dp[i][j] = tmp\\n\\n    ans = float(\\\"inf\\\")\\n    for p in P:\\n        tmp = 0\\n        for i in range(R - 1):\\n            tmp += dp[p[i]][p[i + 1]]\\n\\n        if tmp < ans:\\n            ans = tmp\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import heapq\\nimport itertools\\nfrom collections import defaultdict\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\n\\nedges = [[] for i in range(n+1)]\\nfor _ in range(m):\\n    _from, to, distance = list(map(int, input().split()))\\n    edges[_from].append([to, distance])\\n    edges[to].append([_from, distance])\\n\\n\\ncomp_edges = defaultdict(dict)\\nfor _from in visiting_town:\\n    seen = [False] * (n+1)\\n    todo = []\\n    for to, dist in edges[_from]:\\n        heapq.heappush(todo, [dist, to])\\n\\n    while todo:\\n        dist, node = heapq.heappop(todo)\\n        if seen[node]:\\n            continue\\n\\n        seen[node] = dist\\n        for to, add_dist in edges[node]:\\n            if seen[to] or to == _from:\\n                continue\\n            new_dist = dist + add_dist\\n            heapq.heappush(todo, [new_dist, to])\\n\\n    for to in visiting_town:\\n        if to == _from:\\n            continue\\n        comp_edges[_from][to] = seen[to]\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        if to in comp_edges[_from]:\\n            dist = comp_edges[_from][to]\\n            result += dist\\n        else:\\n            break\\n    candidates.append(result)\\n\\nprint((min(candidates)))\\n\", \"import sys\\nimport numpy as np \\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nI = np.array(sys.stdin.read().split(), dtype=np.int64)\\nn, m, R = I[:3]\\nr = I[3:3+R] - 1\\na, b, c = I[3+R:].reshape(-1, 3).T\\ngraph = csr_matrix((c, (a-1, b-1)), shape=(n, n))\\n\\ndef main():\\n    dist = floyd_warshall(graph, directed=False).astype(np.int64)\\n    \\n    perms = np.array(list(permutations(r)))\\n    res = dist[perms[:, :-1], perms[:, 1:]]\\n    ans = np.amin(np.sum(res, axis=1))\\n    return ans\\n\\ndef __starting_point():\\n    ans = main()\\n    print(ans)\\n__starting_point()\", \"import itertools\\nimport sys\\n\\ndef main():\\n  input = sys.stdin.readline\\n  n, m, r = map(int, input().split())\\n  R = [int(x) for x in input().split()]\\n  inf = pow(10, 9)+7\\n\\n  roads = [[inf]*n for _ in range(n)]\\n  for i in range(m):\\n    a, b, c = map(int, input().split())\\n    roads[a-1][b-1] = c\\n    roads[b-1][a-1] = c\\n\\n  for k in range(n):\\n    for i in range(n):\\n      for j in range(n):\\n        if roads[i][j] > roads[i][k]+roads[k][j]:\\n          roads[i][j] = roads[i][k]+roads[k][j]\\n\\n  ans = inf\\n  for value in itertools.permutations(R):\\n    sub = 0\\n    for k in range(r-1):\\n      sub += roads[value[k]-1][value[k+1]-1]\\n    if ans > sub:\\n      ans = sub\\n\\n  print(ans)\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"from scipy.sparse.csgraph import csgraph_from_dense,dijkstra,floyd_warshall\\nN,M,R = map(int,input().split())\\n\\nmachi = list(map(int,input().split()))\\n\\nList = [[10**6]*(N) for i in range(N)]\\nfor i in range(M):\\n  a,b,c = map(int,input().split())\\n  \\n  if List[a-1][b-1] > c:\\n    List[a-1][b-1] = c\\n    List[b-1][a-1] = c\\nG = csgraph_from_dense(List, null_value=10**6)\\nd = floyd_warshall(G)\\n\\nimport itertools\\na = list(itertools.permutations(machi))\\narr = []\\nfor i in a:\\n  ans = 0\\n  for j in range(R):\\n    if j != R-1 :\\n      ans += d[i[j]-1][i[j+1]-1]\\n  arr.append(ans) \\n  \\nprint(int(min(arr)))\", \"from itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nN, M, R = map(int, input().split())\\nR = list(map(int, input().split()))\\nA, B, C = np.array([input().split() for _ in range(M)], dtype=int).T\\nG = csr_matrix((C, (A - 1, B - 1)), shape=(N, N))\\npath = dijkstra(G, directed=False)\\npath = (path + 0.5).astype(int)\\n\\nans = 10 ** 18\\nfor p in permutations(R):\\n    cnt = 0\\n    for x, y in zip(p[:-1], p[1:]):\\n        x -= 1\\n        y -= 1\\n        cnt += path[x][y]\\n    ans = min(ans, cnt)\\n\\nprint(ans)\", \"import numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nfrom itertools import permutations\\n\\nn,m,R = map(int, input().split())\\nr = list(map(int, input().split()))\\nr = [i-1 for i in r]\\nabc = [list(map(int, input().split())) for i in range(m)]\\nedge = [[10 ** 9] * (n) for i in range(n)]\\nfor a,b,c in abc:\\n  a-=1\\n  b-=1\\n  edge[a][b] = c\\n  edge[b][a] = c\\nd = floyd_warshall(edge)\\nans = 10 ** 18\\nfor i in permutations(r):\\n  tmp = 0\\n  for j in range(R-1):\\n    tmp += d[i[j]][i[j+1]]\\n  ans = min(ans,tmp)\\nprint(int(ans))\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nINF = 10 ** 18\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\ndp = [[INF for _ in range(N)] for _ in range(N)]\\nfor i in range(N):\\n    for j in range(N):\\n        if i == j:\\n            dp[i][j] = 0\\n\\nfor i in range(M):\\n    a, b, c = map(int, input().split())\\n    dp[a - 1][b - 1] = c\\n    dp[b - 1][a - 1] = c\\n\\nwf = floyd_warshall(dp)\\nans = INF\\nfor i in permutations(r):\\n    tmp = 0\\n    for j in range(1, len(i)):\\n        tmp += wf[i[j - 1] - 1][i[j] - 1]\\n    ans = min(ans, tmp)\\n\\nprint(int(ans))\", \"from itertools import permutations, combinations\\nfrom scipy.sparse import csr_matrix\\nfrom scipy.sparse.csgraph import dijkstra\\nN,M,R = map(int,input().split())\\nr = list(map(int,input().split()))\\nr = [r[i]-1 for i in range(R)]\\nvve = [list(map(int,input().split())) for i in range(M)]\\nv1,v2,edge = zip(*vve)\\nv1 = list(v1)\\nv2 = list(v2)\\nfor i in range(M):\\n  v1[i] -= 1\\n  v2[i] -= 1\\ncsr = csr_matrix((edge,(v1,v2)),shape = (N,N))\\ndist = [[0 for j in range(R)] for i in range(R)]\\nfor i,j in combinations(range(R),2):\\n  r1,r2=r[i],r[j]\\n  dist[i][j] = int(dijkstra(csr, directed = False,indices = r1)[r2])\\n  dist[j][i] = dist[i][j]\\nans = 10**18\\nfor x in permutations(range(R)):\\n  distsum = 0\\n  for i in range(1,R):\\n    distsum += dist[x[i]][x[i-1]]\\n  ans = min(ans,distsum)\\nprint(ans)\", \"def main():\\n    import sys\\n    input = sys.stdin.readline\\n    sys.setrecursionlimit(10**7)\\n    from collections import Counter, deque\\n    #from collections import defaultdict\\n    from itertools import combinations, permutations, accumulate, groupby\\n    #from itertools import product\\n    from bisect import bisect_left,bisect_right\\n    from heapq import heapify, heappop, heappush\\n    from math import floor, ceil\\n    #from operator import itemgetter\\n\\n    inf = 10**17\\n    #mod = 10**9 + 7\\n\\n    N,M,R = map(int, input().split())\\n    r = list(map(int, input().split()))\\n    for i in range(R):\\n        r[i] -= 1\\n\\n    def dijkstra_heap(start,edge):\\n        #\\u59cb\\u70b9\\u304b\\u3089\\u5404\\u9802\\u70b9\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2(\\u9802\\u70b9\\u756a\\u53f7:0~N-1)\\n        d = [inf]*N\\n        used = [False]*N\\n        d[start] = 0\\n        used[start] = True\\n        edgelist = []\\n        #a:\\u91cd\\u307f(//), b:\\u6b21\\u306e\\u9802\\u70b9(%)\\n        for a,b in edge[start]:\\n            heappush(edgelist,a*(10**6)+b)\\n\\n        while len(edgelist):\\n            #\\u307e\\u3060\\u6700\\u77ed\\u8ddd\\u96e2\\u304c\\u6c7a\\u307e\\u3063\\u3066\\u3044\\u306a\\u3044\\u9802\\u70b9\\u306e\\u4e2d\\u304b\\u3089\\u6700\\u5c0f\\u306e\\u8ddd\\u96e2\\u306e\\u3082\\u306e\\u3092\\u63a2\\u3059\\n            minedge = heappop(edgelist)\\n            if used[minedge%(10**6)]:\\n                continue\\n            node = minedge%(10**6)\\n            d[node] = minedge//(10**6)\\n            used[node] = True\\n\\n            for e in edge[node]:\\n                if not used[e[1]]:\\n                    heappush(edgelist,(e[0]+d[node])*(10**6)+e[1])\\n        return d\\n\\n    edge = [[] for i in range(N)]\\n    #edge[i] : i\\u304b\\u3089\\u51fa\\u308b\\u9053\\u306e[\\u91cd\\u307f,\\u884c\\u5148]\\u306e\\u914d\\u5217\\n    for _ in range(M):\\n        x,y,z = map(int,input().split())\\n        edge[x-1].append((z,y-1))\\n        edge[y-1].append((z,x-1))\\n\\n    kyori = []\\n    for i in r:\\n        kyori.append(dijkstra_heap(i,edge))\\n    \\n    res = inf\\n    for i in permutations(range(R), R):\\n        l = list(i)\\n        path = 0\\n        for j in range(R-1):\\n            path += kyori[l[j]][r[l[j+1]]]\\n        res = min(res, path)\\n\\n    print(res)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\n\\ndef warshal_floyd(d):\\n    # d[i][j] := i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n    nonlocal N\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                d[i][j] = min(d[i][j], d[i][k] + d[k][j])\\n    # \\\"\\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bi\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\\"\\u3068\\\"\\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bi\\u304b\\u3089k\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2 + \\u4eca\\u73fe\\u5728\\u6c42\\u307e\\u3063\\u3066\\u3044\\u308bk\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2 \\u306e\\u548c\\\"\\u3092\\u6bd4\\u8f03\\u3059\\u308b\\n    return d\\n\\n\\nN, M, R = list(map(int, input().split()))  # N\\u500b\\u306e\\u753a, M\\u672c\\u306e\\u9053, R\\u500b\\u306e\\u753a\\u3092\\u8a2a\\u308c\\u308b\\u3053\\u3068\\u306b\\u306a\\u3063\\u305f\\ntown = list([int(x)-1 for x in input().split()])\\ndistant = [[float('inf')]*N for _ in range(N)]\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    distant[a][b] = c\\n    distant[b][a] = c\\nfor i in range(N):\\n    distant[i][i] = 0\\ndistant = floyd_warshall(distant)\\n# warshal_floyd(distant)\\n# print(distant)  # R\\u306f\\u6700\\u59278\\u3060\\u304b\\u3089\\u5168\\u3066\\u3092\\u8a66\\u3057\\u3066\\u3082\\u5927\\u4e08\\u592b\\nans = 10**9\\nfor x in itertools.permutations(town):\\n    x = list(x)\\n    now = x[0]\\n    tmp = 0\\n    for i in range(1, len(x)):\\n        tmp += distant[now][x[i]]\\n        now = x[i]\\n    ans = min(ans, tmp)\\nprint((int(ans)))\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\nN, M, R = map(int, input().split())\\nr = list(map(int, input().split()))\\nr = [ri-1 for ri in r]\\n\\ngraph = [[0]*N for i in range(N)]\\nfor i in range(M):\\n  a, b, c = map(int, input().split())\\n  a -= 1; b -= 1\\n  graph[a][b] = graph[b][a] = c\\n\\ndist = floyd_warshall(graph).astype(int)\\norders = list(itertools.permutations(r, R))\\n\\nans = float('inf')\\nfor order in orders:\\n  d = 0\\n  for u, v in zip(order, order[1:]):\\n    d += dist[u][v]\\n  ans = min(ans, d)\\n\\nprint(ans)\", \"import numpy as np\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport itertools\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split(' ')))\\n    towns = list(map(lambda t: int(t) - 1, input().split(' ')))\\n    A = np.zeros((N, N), dtype=int)\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split(' ')))\\n        A[a-1][b-1] = c\\n        A[b-1][a-1] = c\\n    fw = floyd_warshall(A)\\n    d = fw[towns, :][:, towns]\\n    min_d = 10**12\\n    for trip in itertools.permutations(range(R)):\\n        min_d = min([min_d, sum([d[trip[i]][trip[i+1]] for i in range(len(trip) - 1)])])\\n    print(int(min_d))\\n\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nfrom itertools import permutations, combinations\\nn,m,R = map(int,input().split())\\nr=list(map(int,input().split()))\\nfor i in range(R):\\n    r[i]-=1\\nd = np.zeros((n,n)) \\n# \\u5165\\u529b\\nfor i in range(m):\\n    a,b,c = map(int,input().split())\\n    a,b = a-1, b-1\\n    d[a,b] = c\\ndist =floyd_warshall(d,directed=0).astype(int)\\nans=10**10\\nfor v in permutations(r):\\n    tmp=0\\n    for i in range(R-1):\\n        tmp+=dist[v[i],v[i+1]]\\n    ans=min(ans,tmp)\\nprint(ans)\", \"def main():\\n    from sys import stdin\\n    input = stdin.readline\\n    n, m, r = list(map(int, input().split()))\\n    root = list(map(int, input().split()))\\n    abc = [list(map(int, input().split())) for _ in [0]*m]\\n\\n    import numpy as np\\n    import scipy.sparse.csgraph as sp\\n    inf = float('inf')\\n    d = np.full((n, n), inf)\\n    for i in range(n):\\n        d[i][i] = 0\\n    for a, b, c in abc:\\n        d[a-1][b-1] = c\\n        d[b-1][a-1] = c\\n    # indices=x\\u3067x\\u304b\\u3089\\u306e\\u5358\\u4e00\\u59cb\\u70b9\\u306b,return_predecessors=True\\u3067\\u7d4c\\u8def\\u304c\\u51fa\\u308b\\n    s = sp.shortest_path(d)\\n\\n    from itertools import permutations\\n\\n    p = list(permutations(root))\\n\\n    D = 10**10\\n    for i in p:\\n        d = 0\\n        for j in range(len(i)-1):\\n            d += s[i[j]-1][i[j+1]-1]\\n        D = min(D, d)\\n    print((int(D)))\\n\\n\\nmain()\\n\", \"from scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\nfrom itertools import permutations\\n\\nN, M, R = list(map(int, input().split()))\\nr = list(map(int, input().split()))\\n\\ng = [[0] * (N + 1) for _ in range(N + 1)]\\nfor _ in range(M):\\n    A, B, C = list(map(int, input().split()))\\n    g[A][B] = C\\n    g[B][A] = C\\ng = floyd_warshall(csgraph_from_dense(g))\\n\\nresult = 1000000000\\nfor p in permutations(r):\\n    t = 0\\n    for i in range(R - 1):\\n        t += g[p[i]][p[i + 1]]\\n    result = min(result, t)\\nprint((int(result)))\\n\", \"import sys\\nfrom itertools import permutations\\n\\nsys.setrecursionlimit(10 ** 6)\\nINF = float(\\\"inf\\\")\\nMOD = 10 ** 9 + 7\\n\\n\\ndef input():\\n    return sys.stdin.readline().strip()\\n\\n\\ndef main():\\n    N, M, R = list(map(int, input().split()))\\n    r = list(map(int, input().split()))\\n    P = permutations(r)\\n\\n    dp = [[INF] * N for _ in range(N)]\\n    for i in range(N):\\n        dp[i][i] = 0\\n\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        a -= 1\\n        b -= 1\\n        dp[a][b] = c\\n        dp[b][a] = c\\n\\n    # \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                tmp = dp[i][k] + dp[k][j]\\n                if tmp < dp[i][j]:\\n                    dp[i][j] = tmp\\n\\n    ans = INF\\n    for p in P:\\n        tmp = 0\\n        for i in range(R - 1):\\n            tmp += dp[p[i] - 1][p[i + 1] - 1]\\n\\n        if tmp < ans:\\n            ans = tmp\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\nfrom scipy.sparse import csr_matrix\\n\\nint1 = lambda x: int(x) - 1\\n\\nN, M, R = map(int, input().split())\\nT = sorted(list(map(int1, input().split())))\\nA = np.array([tuple(map(int, input().split())) for _ in range(M)]).T\\n\\nmatr = csr_matrix((A[2], (A[0] - 1, A[1] - 1)), shape=(N, N))\\nway = [dijkstra(matr, indices=t, directed=False)[T].astype(int).tolist() for t in T]\\n\\nans = float('inf')\\nfor t in itertools.permutations(range(R)):\\n    tmp = 0\\n    for i in range(R - 1):\\n        tmp += way[t[i]][t[i + 1]]\\n    ans = min(ans, tmp)\\nprint(ans)\", \"import heapq\\nimport itertools\\n\\ndef dijkstra(s):\\n    inf = pow(10, 10)\\n    dist = [inf] * (n + 1)\\n    dist[s] = 0\\n    c = [0] * (n + 1)\\n    p = []\\n    heapq.heapify(p)\\n    heapq.heappush(p, (dist[s], s))\\n    while p:\\n        d, u = heapq.heappop(p)\\n        if dist[u] < d:\\n            continue\\n        c[u] = 1\\n        for g in G[u]:\\n            if c[g[0]] == 0 and dist[u] + g[1] < dist[g[0]]:\\n                dist[g[0]] = dist[u] + g[1]\\n                heapq.heappush(p, (dist[g[0]], g[0]))\\n    return dist\\n\\nn, m, r = map(int, input().split())\\nt = list(map(int, input().split()))\\nG = [[] for _ in range(n + 1)]\\nfor _ in range(m):\\n    a, b, c = map(int, input().split())\\n    G[a].append([b, c])\\n    G[b].append([a, c])\\nD = [[0] * r for _ in range(r)]\\nfor i in range(r):\\n    d = dijkstra(t[i])\\n    for j in range(r):\\n        D[i][j] = d[t[j]]\\n\\nx = [i for i in range(r)]\\nans = pow(10, 10)\\nfor y in itertools.permutations(x):\\n    dist = 0\\n    y = list(y)\\n    for i in range(r - 1):\\n        dist += D[y[i]][y[i + 1]]\\n    ans = min(ans, dist)\\nprint(ans)\", \"from itertools import permutations as perm\\n\\ndef warshall(d, n):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if d[i][k] + d[k][j] < d[i][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\nn,m,r = map(int, input().split())\\nrr = [i-1 for i in map(int, input().split())]\\n\\ninf = float('INF')\\nroute = [[inf for j in range(n)] for i in range(n)]\\nfor _ in range(m):\\n    a,b,c = map(int, input().split())\\n    route[a-1][b-1] = route[b-1][a-1] = c\\n\\nwarshall(route, n)\\n\\nans = inf\\nfor tmp in perm(rr):\\n    cost = 0\\n    for i in range(r-1):\\n        cost += route[tmp[i]][tmp[i+1]]\\n    if cost < ans:\\n        ans = cost\\nprint(int(ans))\", \"# coding: utf-8\\nfrom heapq import heappop, heappush\\nfrom itertools import permutations\\n\\nN,M,R=map(int,input().split())\\ntown=list(map(int,input().split()))\\nINF=10**9\\nG=[[INF for i in range(N+1)] for j in range(N+1)]\\n\\nfor i in range(M):\\n    A,B,C=map(int,input().split())\\n    G[A][B]=C\\n    G[B][A]=C\\n\\nd=[[INF*10 for i in range(N+1)] for j in range(R)]\\n\\nfor k in range(R):\\n    r=town[k]\\n    d[k][r]=0\\n    \\n    used=[False for i in range(N+1)]\\n    \\n    heap=[]\\n    heappush(heap,(d[k][r],r))\\n    \\n    while heap:\\n        d_u, u = heappop(heap)\\n\\n        used[u] = True\\n        \\n        if d[k][u] < d_u:\\n            continue\\n        \\n        for v in range(N+1):\\n            if not(used[v]) and d_u + G[u][v] < d[k][v]:\\n                d[k][v] = d_u + G[u][v]\\n                heappush(heap,(d[k][v],v))\\n\\nans=INF\\n\\nL=[i for i in range(R)]\\n\\nfor v in permutations(L,R):\\n    D=0\\n    for i in range(R-1):\\n        D+=d[v[i]][town[v[i+1]]]\\n    ans=min(ans,D)\\n    \\nprint(ans)\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import dijkstra\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = {}\\nfor node in visiting_town:\\n    dist_list = dijkstra(G, directed=False, indices=node-1)\\n    comp_dist[node-1] = dist_list\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"import itertools\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\n\\nN,M,R = map(int, input().split())\\nr_list = np.array(list(map(int, input().split()))) -1 \\n\\nnodes = np.full((N,N), np.inf) \\nfor i in range(M):\\n    a,b,c = map(int, input().split())\\n    nodes[a-1,b-1] = nodes[b-1,a-1] = c\\nmin_ds = shortest_path(nodes)\\nres = 10**9\\nfor cmb in itertools.permutations(r_list):\\n    prev = cmb[0]\\n    tmpres = 0\\n    for to in cmb[1:]:\\n        tmpres += min_ds[prev,to]\\n        prev = to\\n    res = min(res, tmpres)\\n\\nprint(int(res))\", \"import numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\n\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int,input().split()))\\nfor i in range(R):\\n    r[i] -= 1\\n\\nconnections = np.zeros((N,N))\\nfor i in range(M):\\n    A,B,C = list(map(int,input().split()))\\n    A,B = A-1,B-1\\n    \\n    connections[A][B] = C\\n    connections[B][A] = C\\n\\nconnections = dijkstra(connections)\\n\\ndef explore(unvisited,total,start):\\n    if unvisited:\\n        ret = float('inf')\\n        for town in unvisited:\\n            ret = min(ret,explore(unvisited-set([town]),\\n                total+connections[start][town],town))\\n        return ret\\n    else:\\n        return total\\n\\nans = float('inf')\\ntargets = set(r)\\nfor s in r:\\n    ans = min(ans,explore(targets-set([s]),0,s))\\nprint(int(ans))\", \"def abc073_d():\\n    from scipy.sparse import lil_matrix\\n    from scipy.sparse.csgraph import floyd_warshall\\n    from itertools import permutations\\n\\n    n, m, r = map(int, input().split())\\n    R = list(map(lambda x: int(x)-1, input().split()))\\n    graph = lil_matrix((n, n), dtype=int)\\n    for _ in range(m):\\n        a, b, c = map(int, input().split())\\n        a -= 1\\n        b -= 1\\n        graph[a, b] = c\\n        graph[b, a] = c\\n\\n    dist = floyd_warshall(csgraph=graph)\\n    #print(dist)\\n\\n    ans = 10**18\\n    for route in permutations(R, r):\\n        tmp = 0\\n        for i in range(r-1):\\n            s = route[i]\\n            t = route[i+1]\\n            tmp += dist[s][t]\\n            if tmp > ans: break\\n        ans = min(ans, tmp)\\n        #print(route, tmp)\\n\\n    print(int(ans))\\n\\ndef __starting_point():\\n    abc073_d()\\n__starting_point()\", \"from itertools import permutations\\nimport heapq\\n\\ndef dijkstra(N, G, s):\\n    INF = 10**40\\n    d = [INF] * N\\n    d[s] = 0\\n    q = [(0, s)]\\n    while q:\\n        c, v = heapq.heappop(q)\\n        if d[v] < c:\\n            continue\\n        for t, co in G[v]:\\n            if d[v] + co < d[t]:\\n                d[t] = d[v] + co\\n                heapq.heappush(q, (d[t], t))\\n    return d\\n\\n\\ndef main():\\n    N, M, _ = list(map(int, input().split()))\\n    R = list(map(int, input().split()))\\n    R = [r - 1 for r in R]\\n    G = [[] for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = list(map(int, input().split()))\\n        G[a - 1].append((b - 1, c))\\n        G[b - 1].append((a - 1, c))\\n    RM = []\\n    for r in R:\\n        d = dijkstra(N, G, r)\\n        RM.append([d[rr] for rr in R])\\n    m = 10 ** 40\\n    for i in permutations(list(range(len(R)))):\\n        t = i[0]\\n        c = 0\\n        for r in i[1:]:\\n            c += RM[t][r]\\n            t = r\\n        m = min(m, c)\\n    return m\\n\\n\\nprint((main()))\\n\", \"import sys\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson, csgraph_from_dense\\nimport itertools\\n\\ninput = sys.stdin.readline\\n\\n\\ndef main():\\n    N, M, R = [int(x) for x in input().split()]\\n    r = [int(x) - 1 for x in input().split()]\\n    ABC = [[int(x) for x in input().split()] for _ in range(M)]\\n\\n    EDGE = [[10 ** 9] * N for j in range(N)]\\n\\n    for a, b, c in ABC:\\n        EDGE[a - 1][b - 1] = c\\n        EDGE[b - 1][a - 1] = c\\n\\n    G = csgraph_from_dense(EDGE, null_value=10 ** 9)\\n\\n    d = floyd_warshall(G)\\n\\n    ans = float(\\\"inf\\\")\\n    for a in itertools.permutations(r):\\n        tmp = 0\\n        c = a[0]\\n        for n in a:\\n            tmp += int(d[c][n])\\n            c = n\\n        ans = min(ans, tmp)\\n\\n    print(ans)\\n\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"def main():\\n  import numpy as np\\n  import itertools\\n  from scipy.sparse import csr_matrix\\n  from scipy.sparse.csgraph import dijkstra\\n  import sys\\n  readline = sys.stdin.readline\\n  readlines = sys.stdin.readlines\\n\\n  N, M, R= map(int, readline().split())\\n  r = list(map(int,input().split()))\\n  lines = readlines()\\n  edge = np.array([line.split() for line in lines], dtype = np.int64).T\\n  #print(r)\\n  #print(edge)\\n  graph = csr_matrix((edge[2], (edge[:2] - 1)), (N, N))\\n  ans = float(\\\"inf\\\")\\n  distance_mat = {}\\n  for i in r:\\n    distance_mat[i-1] = (dijkstra(graph, directed = False, indices = i-1))\\n\\n  for town in itertools.permutations(r,R):\\n    #print(town)\\n    tmp = 0\\n    for i in range(R-1):\\n      tmp += distance_mat[town[i]-1][town[i+1]-1]\\n    ans = min(ans,tmp)\\n  print(int(ans))\\nmain()\", \"import itertools\\ndef main():\\n    n,m,r=list(map(int,input().split()))\\n    rx=[int(i) for i in input().split()]\\n    inf=100000000\\n    wf=[[inf]*n for i in range(n)]\\n    for i in range(n):\\n        wf[i][i]=0\\n\\n    for i in range(m):\\n        a,b,c=list(map(int,input().split()))\\n        wf[a-1][b-1]=c\\n        wf[b-1][a-1]=c\\n\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if wf[i][j]>wf[i][k]+wf[k][j]:\\n                    wf[i][j]=wf[i][k]+wf[k][j]\\n    cnt=0\\n    l=list(itertools.permutations([i for i in range(r)]))\\n    cnt=inf\\n    for i in l:\\n        cnt_sub=0\\n        for j in range(r-1):\\n            cnt_sub+=wf[rx[i[j]]-1][rx[i[j+1]]-1]\\n        cnt=min(cnt,cnt_sub)\\n    print(cnt)\\nmain()\\n\", \"from collections import deque\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nimport itertools\\n\\nn, m, r = map(int,input().split())\\nrlist = list(map(int,input().split()))\\nfor i in range(r):\\n    rlist[i] -= 1\\ne = np.zeros((n,n), dtype=int)\\n\\nfor i in range(m):\\n    a, b, c = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    e[a][b] = c\\n    e[b][a] = c\\ncsr = csr_matrix(e)\\nf = floyd_warshall(csr)\\n\\nans = 10**18\\nfor v in itertools.permutations(rlist,r):\\n    d = 0\\n    for i in range(r-1):\\n        d += f[v[i]][v[i+1]]\\n    ans = min(ans, d)\\nprint(int(ans))\", \"def main():\\n    from sys import stdin\\n    input = stdin.readline\\n\\n    n, m, r = list(map(int, input().split()))\\n    l = [int(i) - 1 for i in input().split()]\\n    d = [[10**8] * n for _ in range(n)]\\n    for _ in range(m):\\n        i, j, k = list(map(int, input().split()))\\n        d[i-1][j-1] = k\\n        d[j-1][i-1] = k\\n\\n    # Warshall-Floyd algorithm\\n    for k in range(n):\\n        for i in range(n-1):\\n            for j in range(i+1, n):\\n                if d[i][j] > d[i][k] + d[k][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n                    d[j][i] = d[i][j]\\n\\n    # full search\\n    # 8! = 40320\\n    from itertools import permutations\\n\\n    answer = float('inf')\\n    for i in permutations(l):\\n        ans = 0\\n        for j in range(r-1):\\n            ans += d[i[j]][i[j+1]]\\n        if ans < answer:\\n            answer = ans\\n\\n    print(answer)\\n\\nmain()\\n\", \"from scipy.sparse.csgraph import shortest_path\\nimport numpy as np\\nimport itertools\\n\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int, input().split()))\\n\\nedge = np.zeros((N+1,N+1))\\nfor _ in range(M):\\n  A,B,C = list(map(int,input().split()))\\n  edge[A,B] = C  \\ndist = shortest_path(edge, directed = False).astype(int)\\n\\nanswer = 10**18\\nfor visit in itertools.permutations(r):\\n  p = 0\\n  prev = visit[0]\\n  for x in visit[1:]:\\n    if prev != None:\\n      p += dist[prev,x]\\n    prev = x\\n  answer = min(answer,p)\\n  \\nprint(answer)\\n\", \"from scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nfrom itertools import permutations\\n\\nn,m,r = map(int,input().split())\\nrr = np.array(list(map(int,input().split()))) - 1\\n\\nL = np.zeros((n,n),int)\\nfor _ in range(m):\\n    a,b,c = map(int,input().split())\\n    L[a-1,b-1] = c\\n\\nshortest = floyd_warshall(L, directed = False)\\n\\nans = float('inf')\\n\\nfor route in permutations(rr,len(rr)):\\n    ans = min(ans, shortest[route[:-1],route[1:]].sum())\\nprint(int(ans))\", \"# \\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\u3067\\u3001\\u5404\\u753a\\u9593\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\u306e\\u66f4\\u65b0\\u306f200^3 = 8,000,000\\n# \\u8a2a\\u308c\\u308b\\u3079\\u304d\\u753aR\\u306f\\u305f\\u304b\\u3060\\u304b8\\u500b\\u306a\\u306e\\u3067\\u3001\\u9806\\u756a\\u306e\\u5168\\u901a\\u308a\\u3092\\u8a66\\u3057\\u30668! = \\u7d0440000\\u901a\\u308a\\n\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall\\nfrom scipy.sparse import csr_matrix\\n\\nN,M,R = map(int,input().split())\\nr = list(map(int,input().split()))\\nr = list(map(lambda x:x-1,r))\\n\\nE = [[0 for j in range(N)] for i in range(N)]\\nfor i in range(M):\\n  a,b,c = map(int,input().split())\\n  E[a-1][b-1] = c\\n  E[b-1][a-1] = c\\n\\nE = np.array(E)\\nE = shortest_path(E,method = \\\"FW\\\")\\n\\n# DFS\\u3067\\u3059\\u3079\\u3066\\u306e\\u6570\\u3092\\u8a66\\u3059\\nstack = []\\nfor i in range(len(r)):\\n  stack.append([r[i],[],0])\\nans = 10 ** 18\\nwhile stack:\\n  v,visited,dist = stack.pop()\\n  if len(visited) != 0:\\n    dist += E[visited[-1]][v]\\n  visited2 = visited.copy()\\n  visited2.append(v)\\n  if len(visited2) == len(r):\\n    if dist < ans:\\n      ans = dist\\n    continue\\n  for i in range(len(r)):\\n    if r[i] not in visited2:\\n      stack.append([r[i],visited2,dist])\\n    \\nprint(int(ans))\", \"from sys import stdin\\nfrom itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\ndef main():\\n    #\\u5165\\u529b\\n    readline=stdin.readline\\n    inf=10**9\\n    n,m,r=map(int,readline().split())\\n    targets=list(map(lambda x:int(x)-1,readline().split()))\\n    G=[[inf]*n for _ in range(n)]\\n    for _ in range(m):\\n        a,b,c=map(int,readline().split())\\n        a-=1\\n        b-=1\\n        G[a][b]=min(G[a][b],c)\\n        G[b][a]=min(G[b][a],c)\\n\\n    #\\u30ef\\u30fc\\u30b7\\u30e3\\u30eb\\u30d5\\u30ed\\u30a4\\u30c9\\n    G=csgraph_from_dense(G,null_value=10**9)\\n    li=floyd_warshall(G)\\n    li=[list(map(int,li[i])) for i in range(n)]\\n    \\n    #\\u9806\\u5217\\u5168\\u63a2\\u7d22\\n    ans=inf\\n    for p in permutations(targets,r):\\n        tmp=0\\n        for i in range(r-1):\\n            now=p[i]\\n            nex=p[i+1]\\n            tmp+=li[now][nex]\\n        ans=min(ans,tmp)\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import *\\nfrom scipy.sparse.csgraph import *\\nimport numpy as np\\nN,M,R = map(int,input().split())\\nT = list(map(int,input().split()))\\nE = [list(map(int,input().split())) for m in range(M)]\\nG = np.zeros((N,N))\\nA = []\\n\\nfor a,b,c in E:\\n  G[a-1][b-1] = c\\n  G[b-1][a-1] = c\\n\\nF = floyd_warshall(G)\\n\\nfor P in permutations(T):\\n  A+=[sum(F[P[r]-1][P[r+1]-1] for r in range(R-1))]\\n\\nprint(int(min(A)))\", \"from scipy.sparse.csgraph import floyd_warshall\\nfrom itertools import permutations\\nfrom sys import stdin\\nnii=lambda:map(int,stdin.readline().split())\\n\\nn,m,r=nii()\\nr=list(nii())\\n\\ns=[[float('inf')]*n for i in range(n)]\\nfor i in range(n):\\n  for j in range(n):\\n    s[i][j]=0\\nfor i in range(m):\\n  a,b,c=nii()\\n  a-=1\\n  b-=1\\n  s[a][b]=c\\n  s[b][a]=c\\n\\nws=floyd_warshall(s)\\n\\nans=10**9\\nfor i in permutations(r):\\n  t_ans=0\\n  for j in range(1,len(i)):\\n    t_ans+=ws[i[j-1]-1][i[j]-1]\\n  ans=min(ans,t_ans)\\nprint(int(ans))\", \"import itertools\\nfrom scipy.sparse.csgraph import floyd_warshall\\n\\nN, M, R = map(int, input().split())\\nr = tuple(map(int, input().split()))\\n\\nINF = 10**10\\n\\nd = [[INF] * N for _ in range(N)]\\n\\nfor i in range(N):\\n    d[i][i] = 0\\n\\nfor _ in range(M):\\n    a, b, c = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    if d[a][b] > c:\\n        d[a][b] = c\\n        d[b][a] = c\\n\\n\\ndef warshall(d):\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if d[i][j] > d[i][k] + d[k][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\n\\n# d = floyd_warshall(d)\\nwarshall(d)\\n\\n\\nans = INF\\nfor p in itertools.permutations(r):\\n    dist = 0\\n    for i in range(R-1):\\n        dist += d[p[i]-1][p[i+1]-1]\\n\\n    if ans > dist:\\n        ans = dist\\n\\nprint(int(ans))\", \"from itertools import permutations as perm\\nfrom scipy.sparse.csgraph import dijkstra as di\\n\\ndef warshall(d, n):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                if d[i][k] + d[k][j] < d[i][j]:\\n                    d[i][j] = d[i][k] + d[k][j]\\n\\nn,m,r = list(map(int, input().split()))\\nrr = [i-1 for i in map(int, input().split())]\\n\\ninf = float('INF')\\nroute = [[0 for j in range(n)] for i in range(n)]\\nfor _ in range(m):\\n    a,b,c = list(map(int, input().split()))\\n    route[a-1][b-1] = route[b-1][a-1] = c\\n\\nroute = di(route, n)\\n\\nans = inf\\nfor tmp in perm(rr):\\n    cost = 0\\n    for i in range(r-1):\\n        cost += route[tmp[i]][tmp[i+1]]\\n    if cost < ans:\\n        ans = cost\\nprint((int(ans)))\\n\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nimport numpy as np\\nN, M, R = list(map(int, input().split()))\\nr = tuple(map(int, input().split()))\\n\\ninf = 10**9\\ngraph = np.ones((N, N), dtype=int)*inf\\nfor _ in range(M):\\n    a, b, c = list(map(int, input().split()))\\n    a -= 1\\n    b -= 1\\n    graph[a][b] = c\\n    graph[b][a] = c\\nfy = floyd_warshall(graph)\\n\\nans = 10 ** 9\\nfor p in permutations(r):\\n    tmp = 0\\n    for x, y in zip(p[:-1], p[1:]):\\n        x -= 1\\n        y -= 1\\n        tmp += int(fy[x][y])\\n    ans = min(ans, tmp)\\nprint(ans)\\n\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import floyd_warshall\\nn,m,r=map(int,input().split())\\nR=list(map(int,input().split()))\\nl=[[float('inf')]*n for _ in range(n)]\\nfor _ in range(m):\\n    a,b,c,=map(int,input().split())\\n    a-=1\\n    b-=1\\n    l[a][b]=c\\n    l[b][a]=c\\nfor i in range(n):\\n    l[i][i] = 0 #\\u81ea\\u8eab\\u306e\\u3068\\u3053\\u308d\\u306b\\u884c\\u304f\\u30b3\\u30b9\\u30c8\\u306f\\uff10\\ndef warshall_floyd(d):\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                d[i][j]=min(d[i][j],d[i][k]+d[k][j])\\n                \\n    return d\\n#F=warshall_floyd(l)\\nF1 = floyd_warshall(l)\\nans=float('inf')\\nfor v in permutations(R):\\n    temp=0\\n    for i in range(r-1):\\n        temp+=F1[v[i]-1][v[i+1]-1]\\n    ans=min(ans,temp)\\nprint(int(ans))\", \"# coding: utf-8\\n# Your code here!\\n\\n#\\u4fdd\\u5b58\\nimport heapq\\nimport sys\\nimport itertools\\n \\nsys.setrecursionlimit(10**7)\\n\\ndef bfs(cost,node):\\n    root[node]=min(root[node],cost)\\n    for next_c,next_n in way[node]:\\n        if root[next_n]==10**9:\\n            heapq.heappush(q,[cost+next_c,next_n])\\n    return\\n    \\nN,M,R=list(map(int,input().split()))\\nr=list(map(int,input().split()))\\n\\nway=[[] for i in range(N)]\\n\\nfor _ in range(M):\\n    A,B,C=list(map(int,input().split()))\\n    way[A-1].append([C,B-1])\\n    way[B-1].append([C,A-1])\\n\\nmade=[]\\n\\n#print(q)\\nfor start in r:\\n    root=[10**9]*N\\n    q=[[0,start-1]]\\n    heapq.heapify(q)#cost\\u3068node\\n    while q:\\n        temp=heapq.heappop(q)\\n        #print(temp)\\n        if root[temp[1]]==10**9:\\n            bfs(temp[0],temp[1])\\n    made.append(root)\\n#print(made)\\n\\nans=10**9\\nfor order in itertools.permutations([i for i in range(len(r))]):\\n    #print(order)\\n    temp=0\\n    for i in range(len(order)-1):\\n        fro=r[order[i]]\\n        aft=r[order[i+1]]\\n        #print(\\\"YES\\\")\\n        #print(order[i]-1,order[i+1]-1)\\n        #print(root[order[i]-1][0])\\n        temp+=abs(made[order[i]][aft-1]-made[order[i]][fro-1])\\n    ans=min(ans,temp)\\nprint(ans)\\n\\n\\n\", \"import time\\nimport itertools\\n\\ndef main():\\n    N,M,R = tuple([int(x) for x in input().split()])\\n\\n    r = [int(x)-1 for x in input().split()]\\n    \\n    w_e = [tuple([int(x)for x in input().split()]) for _ in [0]*M]\\n    w_e = [(a-1,b-1,w) for (a,b,w) in w_e]\\n\\n\\n    g = Graph(list(range(N)),w_e)\\n\\n    minimums = {}\\n    for r_i in r:\\n        minimums[r_i] = g.dijkstra(r_i)[0]\\n\\n    bf = itertools.permutations(r)\\n\\n    candidates = []\\n    for bf_i in bf:\\n        distance = 0\\n        for i in range(R-1):\\n            distance += minimums[bf_i[i]][bf_i[i+1]]\\n        candidates.append(distance)\\n\\n    print(min(candidates))\\n\\ndef speedtest(func,*args):\\n    b = time.perf_counter()\\n    res = func(*args)\\n    e = time.perf_counter()\\n    elapsed = e-b\\n    print(\\\"{} (sec)\\\".format(elapsed))\\n\\n    return res\\n\\nclass Graph:\\n    def __init__(self,w_v,w_e):\\n        super().__init__()\\n        self.size = len(w_v)\\n        self.w_v = [v_i for v_i in w_v]#value of vartex\\n        self.w_e = [{} for _ in [0]*self.size]\\n\\n        self.neighbor = [[] for _ in [0]*self.size]\\n        for a_i,b_i,w_i in w_e:\\n            self.w_e[a_i][b_i] = w_i#weight of edge\\n            self.w_e[b_i][a_i] = w_i#weight of edge\\n\\n            self.neighbor[a_i].append(b_i)\\n            self.neighbor[b_i].append(a_i)\\n\\n    def dijkstra(self,v_n):\\n        d = [-1]*self.size\\n        temp_d = [(10**9,i)for i in range(self.size)]\\n        temp_d[v_n] = (0,v_n)\\n        prev = [-1]*self.size\\n\\n        q = set(range(self.size))\\n        \\n        while len(q)>0:\\n            u = min([temp_d[q_i] for q_i in q])[1]\\n            d[u] = temp_d[u][0]\\n            q.discard(u)\\n            for v in self.neighbor[u]:\\n                temp = temp_d[u][0]+self.w_e[u][v]\\n                if temp_d[v][0]>temp:\\n                    temp_d[v] = temp,v\\n                    prev[v] = u\\n\\n        return d,prev\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from itertools import permutations\\nimport sys\\ninput = sys.stdin.readline\\n\\ndef main():\\n    N, M, RR = map(int, input().split())\\n    R = list(map(int, input().split()))\\n    INF = float(\\\"inf\\\")\\n    T = [[INF] * N for _ in range(N)]\\n    for _ in range(M):\\n        a, b, c = tuple(map(int, input().split()))\\n        T[a-1][b-1] = c\\n        T[b-1][a-1] = c\\n    for k in range(N):\\n        for i in range(N):\\n            for j in range(N):\\n                if T[i][j] > T[i][k]+T[k][j]:\\n                    T[i][j] = T[i][k]+T[k][j]\\n    print(min(sum(T[rs[i]-1][rs[i+1]-1] for i in range(RR-1)) for rs in permutations(R)))\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from scipy.sparse.csgraph import dijkstra\\nimport numpy as np\\nfrom itertools import permutations\\nN,M,R = list(map(int,input().split()))\\nr = list(map(int,input().split()))\\n\\n\\\"\\\"\\\"\\nhttps://juppy.hatenablog.com/entry/2018/11/01/%E8%9F%BB%E6%9C%AC_python_%E5%85%A8%E7%82%B9%E5%AF%BE%E6%9C%80%E7%9F%AD%E7%B5%8C%E8%B7%AF%E6%B3%95%EF%BC%88%E3%83%AF%E3%83%BC%E3%82%B7%E3%83%A3%E3%83%AB%E3%83%95%E3%83%AD%E3%82%A4%E3%83%89%E6%B3%95\\n\\\"\\\"\\\"\\n\\ndef warshall_floyd(d):\\n    n = len(d)\\n    #d[i][j]: i\\u304b\\u3089j\\u3078\\u306e\\u6700\\u77ed\\u8ddd\\u96e2\\n    for k in range(n):\\n        for i in range(n):\\n            for j in range(n):\\n                d[i][j] = min(d[i][j],d[i][k] + d[k][j])\\n    return d\\n\\ngraph = [[float(\\\"inf\\\") for i in range(N)] for j in range(N)] \\nfor i in range(M):\\n    A,B,C = list(map(int,input().split()))\\n    graph[A-1][B-1] = C\\n    graph[B-1][A-1] = C\\n\\ndist = dijkstra(graph)\\n\\nans = 1e10\\nfor i in permutations(r):\\n    flag = True\\n    way = list(i)\\n    tmp = 0\\n    for i in range(len(way)-1):\\n        if dist[way[i]-1][way[i+1]-1] < 0:\\n            flag = False\\n            break\\n        tmp += dist[way[i]-1][way[i+1]-1]\\n    if flag:\\n        ans = min(tmp,ans)\\n\\nprint((int(ans)))\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time, copy,bisect\\nfrom operator import itemgetter\\n#from heapq import heappush, heappop\\nimport numpy as np\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\n#from scipy.sparse import csr_matrix\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\nmod = 10**9 + 7\\n\\nstdin = sys.stdin\\n\\nni = lambda: int(ns())\\nnf = lambda: float(ns())\\nna = lambda: list(map(int, stdin.readline().split()))\\nnb = lambda: list(map(float, stdin.readline().split()))\\nns = lambda: stdin.readline().rstrip()  # ignore trailing spaces\\n\\nN, M, R = na()\\nr = na()\\ng = [[0] * N for _ in range(N)]\\nfor i in range(M):\\n    a, b, c = na()\\n    a -= 1\\n    b -= 1\\n    g[a][b] = c\\n    g[b][a] = c\\n\\ng = np.array(g)\\nd = shortest_path(g)\\n\\nR = len(r)\\nans = inf\\nfor x in itertools.permutations(r):\\n    tmp = 0\\n    for i in range(R-1):\\n        tmp += d[x[i] - 1][x[i+1] - 1]\\n    ans = min(ans, tmp)\\nprint((int(ans)))\\n\\n\\n\\n\\n\", \"import itertools\\nimport sys\\nfrom scipy.sparse.csgraph import csgraph_from_dense\\nfrom scipy.sparse.csgraph import dijkstra\\ninput = sys.stdin.readline\\n\\n\\nn, m, r = list(map(int, input().split()))\\nvisiting_town = list(map(int, input().split()))\\nedges = [[float('INF')]*n for _ in range(n)]\\n\\nfor i in range(m):\\n    a, b, c = list(map(int, input().split()))\\n    edges[a-1][b-1] = c\\n\\nG = csgraph_from_dense(edges, null_value=float('INF'))\\ncomp_dist = dijkstra(G, directed=False)\\n\\ncandidates = []\\nfor route in itertools.permutations(visiting_town):\\n    result = 0\\n    for _from, to in zip(route[:-1], route[1:]):\\n        dist = comp_dist[_from-1][to-1]\\n        result += dist\\n    candidates.append(result)\\n\\nprint((int(min(candidates))))\\n\", \"from itertools import permutations\\nimport numpy as np\\nfrom scipy.sparse.csgraph import dijkstra\\n\\nwith open(0) as f:\\n    N, M, R = map(int, f.readline().split())\\n    r = list(map(int, f.readline().split()))\\n    path = [tuple(map(int, line.split())) for line in f.readlines()]\\n\\ngraph = np.full((N,N), np.inf)\\nfor a, b, c in path:\\n    graph[a-1, b-1] = c\\n    graph[b-1, a-1] = c\\n\\ngraph = dijkstra(graph, directed=False)\\nans = np.inf\\npermutation = list(permutations(r))\\nfor p in permutation:\\n    way = 0\\n    for x,y in zip(p[:len(r)], p[1:]):\\n        way += graph[x-1, y-1]\\n    ans = min(ans, way)\\nprint(int(ans))\", \"n,m,r  = list(map(int,input().split()))\\nr_list = list(map(int,input().split()))\\nmatrix = [[float(\\\"inf\\\") for i in range(n)] for j in range(n)]\\nfor c in range(n):\\n  matrix[c][c] = 0\\nfor v in range(m):\\n  a,b,c = list(map(int,input().split()))\\n  if matrix[a-1][b-1] > c:\\n    matrix[a-1][b-1] = c\\n    matrix[b-1][a-1] = c\\nimport numpy as np\\nmatrix = np.array(matrix)\\nfrom scipy.sparse.csgraph import shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson\\nfrom scipy.sparse import csr_matrix\\nload = shortest_path(matrix, directed=False)\\nimport itertools\\nans_list = []\\nfor balls in itertools.permutations(r_list):\\n    balls = list(balls)\\n    ans = 0\\n    for i in range(1,len(r_list)):\\n      ans += load[balls[i-1]-1][balls[i]-1]\\n    ans_list.append(int(ans))\\n    \\nprint((min(ans_list)))\\n      \\n\\n\\n\", \"N, M, K = (int(x) for x in input().split())\\nR = [int(x)-1 for x in input().split()]\\n\\ndist = [[10e8 for _ in range(N)] for _ in range(N)]\\nfor i in range(N): dist[i][i] = 0\\nfor _ in range(M):\\n    a, b, c = (int(x) for x in input().split())\\n    dist[a-1][b-1] = dist[b-1][a-1] = c \\ndel a, b, c, K\\n\\nfor k in range(N):\\n    for i in range(N):\\n        for j in range(i,N):\\n            dist[i][j] = dist[j][i] = min(dist[i][j], dist[i][k]+dist[k][j])\\n\\ndef mindist(x, X):\\n    nonlocal dist\\n    Y = X.copy()\\n    Y.remove(x)\\n    if len(Y) == 0: return 0\\n    return min([dist[x][y] + mindist(y,Y) for y in Y])\\n\\nans = min([mindist(r, R) for r in R])\\nprint(ans)\", \"from itertools import permutations\\nfrom scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall\\ninf = 10 ** 10\\nn, m, r = map(int,input().split())\\nR = list(map(int,input().split()))\\n\\nedge = [[inf] * n for i in range(n)]\\nfor i in range(m):\\n    a, b, c = map(int,input().split())\\n    edge[a - 1][b - 1] = c\\n\\nG = csgraph_from_dense(edge, null_value = inf)\\nd = floyd_warshall(G, False)\\n\\nrec = inf\\nfor i in permutations(R, r):\\n    temp = 0\\n    #print(i)\\n    for j in range(r):\\n        if j == r - 1:continue\\n        temp += d[i[j] - 1][i[j + 1] - 1]\\n    rec = min(rec, temp)\\n\\nprint(int(rec))\"]",
        "difficulty": "interview",
        "input": "4 6 3\n2 3 4\n1 2 4\n2 3 3\n4 3 1\n1 4 1\n4 2 2\n3 1 6\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc073/tasks/abc073_d"
    },
    {
        "id": 1069,
        "task_id": 2653,
        "test_case_id": 1,
        "question": "Given is a rooted tree with N vertices numbered 1 to N.\nThe root is Vertex 1, and the i-th edge (1 \\leq i \\leq N - 1) connects Vertex a_i and b_i.\nEach of the vertices has a counter installed. Initially, the counters on all the vertices have the value 0.\nNow, the following Q operations will be performed:\n - Operation j (1 \\leq j \\leq Q): Increment by x_j the counter on every vertex contained in the subtree rooted at Vertex p_j.\nFind the value of the counter on each vertex after all operations.\n\n-----Constraints-----\n - 2 \\leq N \\leq 2 \\times 10^5\n - 1 \\leq Q \\leq 2 \\times 10^5\n - 1 \\leq a_i < b_i \\leq N\n - 1 \\leq p_j \\leq N\n - 1 \\leq x_j \\leq 10^4\n - The given graph is a tree.\n - All values in input are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN Q\na_1 b_1\n:\na_{N-1} b_{N-1}\np_1 x_1\n:\np_Q x_Q\n\n-----Output-----\nPrint the values of the counters on Vertex 1, 2, \\ldots, N after all operations, in this order, with spaces in between.\n\n-----Sample Input-----\n4 3\n1 2\n2 3\n2 4\n2 10\n1 100\n3 1\n\n-----Sample Output-----\n100 110 111 110\n\nThe tree in this input is as follows:\n\nEach operation changes the values of the counters on the vertices as follows:\n - Operation 1: Increment by 10 the counter on every vertex contained in the subtree rooted at Vertex 2, that is, Vertex 2, 3, 4. The values of the counters on Vertex 1, 2, 3, 4 are now 0, 10, 10, 10, respectively.\n - Operation 2: Increment by 100 the counter on every vertex contained in the subtree rooted at Vertex 1, that is, Vertex 1, 2, 3, 4. The values of the counters on Vertex 1, 2, 3, 4 are now 100, 110, 110, 110, respectively.\n - Operation 3: Increment by 1 the counter on every vertex contained in the subtree rooted at Vertex 3, that is, Vertex 3. The values of the counters on Vertex 1, 2, 3, 4 are now 100, 110, 111, 110, respectively.",
        "solutions": "[\"import sys\\nsys.setrecursionlimit(10**7)\\n \\nN, Q = list(map(int, input().split()))\\nroot = [[] for _ in range(N)]\\nans = [0]*N\\n \\nfor _ in range(N-1):\\n  a,b=list(map(int, input().split()))\\n  a,b=a-1,b-1\\n  root[a].append(b)\\n  root[b].append(a)\\n\\nfor _ in range(Q):\\n  p,x=list(map(int, input().split()))\\n  ans[p-1]+=x\\n  \\nvisited=[False]*N\\ndef dfs(v):\\n  visited[v] = True\\n  for go in root[v]:\\n    if visited[go]:\\n      continue\\n    ans[go] += ans[v]\\n    dfs(go)\\n\\ndfs(0)\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\n\\ndef solve():\\n    N, Q = list(map(int, input().split()))\\n    graph = [[] for _ in range(N)]\\n    for i in range(N - 1):\\n        a, b = [int(x) - 1 for x in input().split()]\\n        graph[a].append(b)\\n        graph[b].append(a)\\n    ans = [0] * N\\n    for i in range(Q):\\n        p, x = list(map(int, input().split()))\\n        p -= 1\\n        ans[p] += x\\n    def dfs(cur, pre):\\n        for to in graph[cur]:\\n            if to == pre:\\n                continue\\n            ans[to] += ans[cur]\\n            dfs(to, cur)\\n    dfs(0, -1)\\n    print((*ans))\\n    return\\n\\nsolve()\\n\", \"def main():\\n    n,q=map(int,input().split())\\n    ab=[list(map(int,input().split())) for _ in range(n-1)]\\n    tree=[list() for _ in range(n)]\\n    score=[0]*n\\n    for a,b in ab:\\n        a,b=a-1,b-1\\n        tree[a].append(b)\\n        tree[b].append(a)\\n    e = [0]\\n    while len(e) > 0:\\n        i = e.pop()\\n        for j in tree[i]:\\n            tree[j].remove(i)\\n            e.append(j)\\n    px=[list(map(int,input().split())) for _ in range(q)]\\n    for p,x in px:\\n        p-=1\\n        score[p]+=x\\n    add(tree,score)\\n    print(*score)\\n\\ndef add(tree,score):\\n    s=tree[0][:]\\n    for i in s:\\n        score[i] += score[0]\\n    while len(s)>0:\\n        t = s.pop()\\n        for i in tree[t]:\\n            s.append(i)\\n            score[i] += score[t]\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections,sys\\ndef LI(): return list(map(int,sys.stdin.readline().rstrip().split()))\\nN,Q = LI()\\nab = [LI() for _ in [None]*(N-1)]\\npx = [LI() for _ in [None]*(Q)]\\nans = [0]*(N+1) #1_indexed\\ngraph = {i:collections.deque() for i in range(1,N+1)} #1_indexed\\nfor a,b in ab:\\n        graph[a].append(b)\\n        graph[b].append(a)\\nfor p,x in px:\\n    ans[p] += x\\nseen = [0]*(N+1) #1_indexed\\nstack = []\\ndef dfs():\\n    seen[1] = 1\\n    stack.append(1)\\n    while stack:\\n        s = stack.pop()\\n        if not graph[s]:\\n            continue\\n        for j in range(len(graph[s])):\\n            g_NO = graph[s].popleft()\\n            if seen[g_NO]:\\n                continue\\n            seen[g_NO] = 1\\n            stack.append(g_NO)\\n            ans[g_NO] += ans[s]\\ndfs()\\nprint(*ans[1:])\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN,Q=map(int,input().split())\\nT=[[] for _ in range(N)]\\nfor _ in range(N-1):\\n  a,b=map(int,input().split())\\n  a-=1;b-=1\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nV=[0]*N\\nfor _ in range(Q):\\n  p,x=map(int,input().split())\\n  V[p-1]+=x\\n  \\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next!=prev:\\n      V[next]+=V[now]\\n      dfs(next,now)\\n\\ndfs(0)\\nprint(*V)\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10 ** 6)\\n\\nn, q = list(map(int, input().split()))\\nG = [[]*n for i in range(n)]\\nfor _ in range(n-1):\\n    a, b = list(map(int, input().split()))\\n    G[a-1].append(b-1)\\n    G[b-1].append(a-1)\\n\\n\\ndone = [False]*n\\npoints = [0]*n\\nd = defaultdict(lambda: 0)\\n\\nfor _ in range(q):\\n    a, b = list(map(int, input().split()))\\n    d[a-1] += b\\n\\n\\ndef dfs(i, cnt):\\n    cnt += d[i]\\n    points[i] += cnt\\n    done[i] = True\\n    for j in G[i]:\\n        if not done[j]:\\n            dfs(j, cnt)\\n\\n\\ndfs(0, 0)\\nprint((*points))\\n\", \"N,Q = map(int,input().split())\\nf = [[] for i in range(N)]\\nfor i in range(N-1):\\n    a,b = map(int,input().split())\\n    f[a-1].append(b-1)\\n    f[b-1].append(a-1)\\nans = [0] * N\\nfor i in range(Q):\\n    p,x = map(int,input().split())\\n    ans[p-1] += x\\nfrom collections import deque\\nd = deque()\\nd.append(0)\\nroot = [-1]*N\\nroot[0] = 0\\nwhile len(d) > 0:\\n    z = d.popleft()\\n    for i in f[z]:\\n        if root[i] == -1:\\n            root[i] = z\\n            ans[i] += ans[z]\\n            d.append(i)\\nprint(*ans)\", \"import sys\\nsys.setrecursionlimit(1<<30)\\ndef dfs(x):\\n    for y in edges[x]:\\n        if y != parent[x]:\\n            parent[y] = x\\n            count[y] += count[x]\\n            dfs(y)\\n\\nN, Q = map(int, input().split())\\nedges = [[] for _ in range(N+1)]\\ncount = [0]*(N+1)\\nfor i in range(N-1):\\n    a, b = map(int, input().split())\\n    edges[a].append(b)\\n    edges[b].append(a)\\nfor i in range(Q):\\n    p, x = map(int, input().split())\\n    count[p] += x\\n\\nparent = [-1]*(N+1)\\ndfs(1)\\nprint(*count[1:])\", \"def abc138d_ki():\\n    import heapq\\n    n, q = map(int, input().split())\\n    cnt = [0] * n\\n    graph = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        a, b = map(int, input().split())\\n        graph[a - 1].append(b - 1)\\n        graph[b - 1].append(a - 1)\\n    for _ in range(q):\\n        p, x = map(int, input().split())\\n        cnt[p - 1] += x\\n    qu = [(0, 0)]\\n    heapq.heapify(qu)\\n    check = [False] * n\\n    while len(qu) != 0:\\n        h, no = heapq.heappop(qu)\\n        check[no] = True\\n        for nxt in graph[no]:\\n            if not check[nxt]:\\n                cnt[nxt] += cnt[no]\\n                heapq.heappush(qu, (h + 1, nxt))\\n    for c in cnt:\\n        print(c, end=' ')\\n\\n\\nabc138d_ki()\", \"import sys\\nsys.setrecursionlimit(10**9) #\\u518d\\u5e30\\u306e\\u4e0a\\u9650\\u6307\\u5b9a\\n# \\u6728\\u69cb\\u9020\\u306e\\u6a19\\u6e96\\u5165\\u529b\\nN,Q = list(map(int,input().split()))\\nT=[[] for _ in range(N)]\\nfor _ in range(N-1):\\n  a,b=list(map(int,input().split()))\\n  a-=1;b-=1\\n  T[a].append(b)\\n  T[b].append(a)\\n\\n# \\u5165\\u529b\\u4f8b1\\u306e\\u3068\\u304d\\u3001\\u7e4b\\u304c\\u308b\\u9802\\u70b9\\u306e\\u30ea\\u30b9\\u30c8 [[1], [0, 2, 3], [1], [1]]\\n# \\u8db3\\u7b97\\u90e8\\u5206\\u306e\\u6a19\\u6e96\\u5165\\u529b\\nV=[0]*N\\nfor _ in range(Q):\\n  p,x=list(map(int,input().split()))\\n  V[p-1]+=x\\n# \\u5165\\u529b\\u4f8b1\\u306e\\u3068\\u304d\\u3001\\u5404\\u30ce\\u30fc\\u30c9\\u4ee5\\u4e0b\\u306b\\u8db3\\u3059\\u5024 [100, 10, 1, 0]\\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next!=prev:# \\u89aa\\u3058\\u3083\\u306a\\u3051\\u308c\\u3070\\u3001(\\u5b50\\u3067\\u3042\\u308c\\u3070)\\u8db3\\u3057\\u7b97\\u3059\\u308b\\n      V[next]+=V[now]\\n      dfs(next,now)\\n\\ndfs(0)\\n\\\"\\\"\\\"\\n\\u5165\\u529b\\u4f8b1\\u306e\\u3068\\u304d\\ndfs(0,-1)\\u2192dfs(1,0)\\u2192dfs(2,1)\\n                  \\u2192dfs(3,1)\\n\\\"\\\"\\\"\\nprint((*V))\\n\", \"from collections import deque\\nN, Q = list(map(int, input().split()))\\ntree = [[] for _ in range(N+1)]\\ncounter = [0]*(N+1)\\nfor i in range(N-1):\\n    a, b = list(map(int, input().split()))\\n    tree[a].append(b)\\n    tree[b].append(a)\\nfor i in range(Q):\\n    p, x = list(map(int, input().split()))\\n    counter[p] += x\\nparent = [-1]*(N+1)\\nd = deque([1])\\n\\nwhile d:\\n    a = d.popleft()\\n    for b in tree[a]:\\n        if parent[a] != b:\\n            parent[b] = a\\n            counter[b] += counter[a]\\n            d.append(b)\\n\\nprint((*counter[1:]))\\n\", \"from collections import deque\\n\\n\\ndef main():\\n  n,q=map(int,input().split())\\n  node=[[]*n for i in range(n)]\\n  for i in range(n-1):\\n    a,b=map(lambda x:int(x)-1,input().split())\\n    node[b].append(a)\\n    node[a].append(b)\\n  \\n  ans=[0]*n\\n  for i in range(q):\\n    p,x=map(int,input().split())\\n    p-=1\\n    ans[p]+=x\\n\\n  root=[0]*n\\n  edge=[[]*n for i in range(n)]\\n  root[0]=0\\n  q=deque([0])\\n  visited=[False]*n\\n  visited[0]=True\\n  \\n  while q:\\n    r=q.popleft()\\n    for e in node[r]:\\n      if not visited[e]:\\n        visited[e]=True\\n        q.append(e)\\n        root[e]=r\\n        edge[r].append(e)\\n  \\n  \\n  p=[0]*n\\n  q=deque(edge[0])\\n  visited=[False]*n\\n  while q:\\n    e=q.popleft()\\n    p[e]=p[root[e]]+1\\n    for e2 in edge[e]:\\n      if not visited[e2]:\\n        visited[e2]=True\\n        q.append(e2)\\n  \\n  p=sorted(enumerate(p), key=lambda x:x[1])\\n  for i in range(1,n):\\n    e=p[i][0]\\n    r=root[e]\\n    ans[e]+=ans[r]\\n    \\n  print(*ans,sep=' ')\\n\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"import sys\\nimport math\\n\\n#https://atcoder.jp/contests/agc008/submissions/15248942\\nsys.setrecursionlimit(10 ** 8)\\nini = lambda: int(sys.stdin.readline())\\ninm = lambda: map(int, sys.stdin.readline().split())\\ninl = lambda: list(inm())\\nins = lambda: sys.stdin.readline().rstrip()\\ndebug = lambda *a, **kw: print(\\\"\\\\033[33m\\\", *a, \\\"\\\\033[0m\\\", **dict(file=sys.stderr, **kw))\\n\\ncounts = []\\n\\nclass Node:\\n    def __init__(self):\\n        self.edge = []\\n        self.count = 0\\n\\ndef dfs(tree,node,p_node,p_count):\\n    count = tree[node].count + p_count\\n    counts[node] = count\\n    for x in tree[node].edge:\\n        if x == p_node:\\n            continue\\n        dfs(tree,x,node,count)\\n\\nN,Q = inm()\\n\\ntree = []\\nfor _ in range(N):\\n    tree.append(Node())\\n\\n#node = Node()\\n#tree = [node]*N\\n\\nfor _ in range(N-1):\\n    a,b = inm()\\n    tree[a-1].edge.append(b-1)\\n    tree[b-1].edge.append(a-1)\\n\\nfor _ in range(Q):\\n    p,x = inm()\\n    tree[p-1].count += x\\n\\nfor _ in range(N):\\n    counts.append(0)\\n\\n#for i in range(N):\\n#    print(tree[i].edge)\\n#    print(tree[i].count)\\n\\n\\ndfs(tree,0,-1,0)\\n\\ncounts_str = [str(n) for n in counts]\\ns = \\\" \\\".join(counts_str)\\nprint(s)\\n\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\nN, Q = map(int, input().split())\\n\\nG = [[] for i in range(N)]\\n\\nfor _ in range(N - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    G[b].append(a)\\n    G[a].append(b)\\n\\nans = [0] * N\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    p -= 1\\n    ans[p] += x\\n\\ndef dfs(v, p):\\n    for to in G[v]:\\n        if to == p: continue\\n        ans[to] += ans[v]\\n        dfs(to, v)\\n\\ndfs(0, -1)\\nfor i in range(N):\\n    print(ans[i],end=' ')\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef dfs(tree, n, ans):\\n    for i in tree[n]:\\n        ans[i] += ans[n]\\n        tree[i].remove(n)\\n        dfs(tree, i, ans)\\n\\nn, q = map(int,input().split())\\nab = [list(map(int,input().split())) for _ in range(n-1)]\\npx = [list(map(int,input().split())) for _ in range(q)]\\n\\ng = [[] for _ in range(n)]\\nfor i in range(n-1):\\n    g[ab[i][0]-1].append(ab[i][1]-1)\\n    g[ab[i][1]-1].append(ab[i][0]-1)\\n\\nans = [0]*n\\nfor i in range(q):\\n    ans[px[i-1][0]-1] += px[i-1][1]\\ndfs(g, 0, ans)\\n\\nprint(*ans)\", \"import sys\\nread = sys.stdin.read\\nsys.setrecursionlimit(10**7)\\n#readlines = sys.stdin.readlines\\ndef main():\\n    def dfs(v, s):\\n        if added[v]:\\n            pass\\n        else:\\n            scores[v] += s\\n            added[v] = 1\\n            for nextv in gg[v]:\\n                dfs(nextv, scores[v])\\n\\n    data = tuple(map(int, read().split()))\\n    n, q = data[0], data[1]\\n    gg = {i: set() for i in range(1, n + 1)}\\n    for i1, v1 in zip(data[2:n*2:2], data[3:n*2:2]):\\n        gg[v1].add(i1)\\n        gg[i1].add(v1)\\n    scores = [0] * (n + 1)\\n    for j1, v1 in zip(data[n*2::2], data[n*2+1::2]):\\n        scores[j1] += v1\\n\\n    added = [0] * (n + 1)\\n    dfs(1, 0)\\n    print(*scores[1:], sep=' ')\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"n,q = map(int,input().split())\\ngraph = [[] for i in range(n)]\\nfor i in range(n-1):\\n  a,b = map(int,input().split())\\n  a -= 1\\n  b -= 1\\n  graph[a].append(b)\\n  graph[b].append(a)\\n\\nweight = [0 for i in range(n)]\\nfor i in range(q):\\n  p,x = map(int,input().split())\\n  weight[p-1] += x\\n\\nst = [0]\\nvisit = [False for i in range(n)]\\nwhile not len(st) == 0:\\n  now = st.pop()\\n  visit[now] = True\\n  for e in graph[now]:\\n    if visit[e]:\\n      continue\\n    weight[e] += weight[now]\\n    st.append(e)\\n    \\nprint(' '.join(map(str,weight)))\", \"\\nimport sys\\nsys.setrecursionlimit(10 ** 6)\\ndef main2():\\n    n, q = list(map(int, input().split()))\\n    tree = [[] for i in range(n)]\\n    point = [0] * n\\n\\n    for _ in range(n - 1):\\n        a, b = list(map(int, input().split()))\\n        tree[b - 1].append(a - 1)\\n        tree[a - 1].append(b - 1)\\n    for i in range(q):\\n        p, x = list(map(int, input().split()))\\n        point[p - 1] += x\\n\\n    def dfs(v, parent):\\n        nonlocal point, tree\\n        for next_val in tree[v]:\\n            if next_val == parent:\\n                continue\\n            point[next_val] += point[v]\\n            dfs(next_val, v)\\n    dfs(0, 0)\\n    print((*point))\\n\\n\\ndef __starting_point():\\n    main2()\\n\\n__starting_point()\", \"# abc138 D ki\\n\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\n# def input():\\n#     return sys.stdin.readline()[:-1]\\n\\nN,Q = map(int,input().split())\\ng=[[] for _ in range(N)]\\n\\npoint=[0]*N\\n\\nfor _ in range(N-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    g[a].append(b)\\n    g[b].append(a)\\n    \\nfor _ in range(Q):\\n    a,b=map(int,input().split())\\n    a-=1\\n    point[a]+=b\\n\\ndef dfs(n,pre=-1):\\n    for ne in g[n]:\\n        if ne == pre:\\n            continue\\n        \\n        point[ne]+=point[n]\\n        \\n        dfs(ne,n)\\n        \\ndfs(0)\\nprint(*point)\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(v,prev = -1):\\n    for u in graph[v]:\\n        if u == prev:\\n            continue\\n        point[u] += point[v]\\n        dfs(u,v)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [[] for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"from collections import defaultdict\\nN,Q = list(map(int,input().split()))\\nd = defaultdict(list)\\n\\nfor _ in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    d[a].append(b)\\n    d[b].append(a)\\n\\nc = defaultdict(int)\\n\\nfor _ in range(Q):\\n    p,x = list(map(int,input().split()))\\n    c[p] += x\\n\\nans = [0 for i in range(N)]\\nque = [(1,0,-1)]\\nwhile len(que)>0:\\n    tmp = []\\n    for v,x,par in que:\\n        x += c[v]\\n        ans[v-1] = str(x)\\n        for w in d[v]:\\n            if w != par:\\n                tmp.append((w,x,v))\\n    que = tmp\\n\\nprint((' '.join(ans)))\\n\", \"import sys\\nsys.setrecursionlimit(500000)\\n\\nn, q = list(map(int, input().split()))\\nab = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a, b = list(map(int, input().split()))\\n    ab[a-1].append(b-1)\\n    ab[b-1].append(a-1)\\nadd = [0]*n\\nfor _ in range(q):\\n    pp, xx = list(map(int, input().split()))\\n    add[pp-1] += xx\\n\\n\\ndef dfs(v, p, value):\\n    value += add[v]\\n    ans[v] = value\\n    for c in ab[v]:\\n        if c == p:\\n            continue\\n        dfs(c, v, value)\\n\\n\\nans = [0]*n\\ndfs(0, -1, 0)\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)# \\u5404\\u9802\\u70b9\\u3054\\u3068\\u306b\\u6240\\u5c5e\\u3059\\u308b\\u6728\\u306e\\u5927\\u304d\\u3055\\u3092\\u8a08\\u7b97\\u3059\\u308b\\nN,Q = list(map(int,input().split()))\\n\\nE = [[] for _ in range(N+1)]\\nvisited = [False for _ in range(N+1)]\\nNode_len = [1 for _ in range(N+1)]\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nAS = [0 for i in range(N+1)]\\ndef dfs(node,parent=-1):\\n    for child in E[node]:\\n        if child == parent:\\n            continue\\n        AS[child] += AS[node]\\n        dfs(child,node)\\n\\nfor q in range(Q):\\n    p,x = list(map(int,input().split()))\\n    AS[p] += x\\n\\ndfs(1)\\nprint((*AS[1:]))\\n\\n\\n\", \"import sys\\nfrom collections import deque\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(s):\\n    stack = [s]\\n    visited = [False]*N\\n    while stack:\\n        v = stack.pop()\\n        if visited[v]:\\n            continue\\n        visited[v] = True\\n        for u in graph[v]:\\n            if not visited[u]:\\n                point[u] += point[v]\\n                stack.append(u)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [deque([]) for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"from collections import defaultdict\\n\\nN, Q = list(map(int, input().split()))\\nd = defaultdict(list)\\n\\nfor _ in range(N - 1):\\n    a, b = list(map(int, input().split()))\\n    d[a].append(b)\\n    d[b].append(a)\\n\\nc = defaultdict(int)\\n\\nfor _ in range(Q):\\n    p, x = list(map(int, input().split()))\\n    c[p] += x\\n\\nans = [0 for i in range(N)]\\nque = [(1, 0, -1)]\\nwhile len(que) > 0:\\n    tmp = []\\n    for v, x, par in que:\\n        x += c[v]\\n        ans[v - 1] = str(x)\\n        for w in d[v]:\\n            if w != par:\\n                tmp.append((w, x, v))\\n    que = tmp\\n\\nprint((*ans))\\n\", \"n,q=map(int,input().split())\\ne=[[] for i in range(n)]\\nfor i in range(n-1):\\n  a,b=map(lambda x:int(x)-1,input().split())\\n  e[a].append(b)\\n  e[b].append(a)\\nps=[0 for i in range(n)]\\nfor i in range(q):\\n  p,x=map(int,input().split())\\n  ps[p-1]+=x\\n\\ns=[-1 for i in range(n)]\\nsc=[0]\\ns[0]=0\\nwhile sc:\\n  nsc=[]\\n  for i in sc:\\n    s[i]+=ps[i]\\n    for j in e[i]:\\n      if s[j]==-1:\\n        nsc.append(j)\\n        s[j]=s[i]\\n  sc=nsc\\n\\nr=\\\"\\\"\\nfor i in range(n):\\n  r+=str(s[i])+\\\" \\\"\\nprint(r[:-1])\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(v,prev = -1):\\n    for u in graph[v]:\\n        if u == prev:\\n            continue\\n        point[u] += point[v]\\n        dfs(u,v)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [[] for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"from collections import deque\\n\\n\\ndef nearlist(N, LIST):\\n    NEAR = [set() for _ in range(N)]\\n    for a, b in LIST:\\n        NEAR[a - 1].add(b - 1)\\n        NEAR[b - 1].add(a - 1)\\n    return NEAR\\n\\n\\ndef bfs(NEAR):\\n    que, frag = deque([0]), set([0])\\n    while que:\\n        q = que.popleft()\\n        for i in NEAR[q]:\\n            if i in frag:\\n                continue\\n            ans[i] += ans[q]\\n            que.append(i), frag.add(i)\\n    return\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n - 1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nans = [0] * n\\nfor p, x in px:\\n    ans[p - 1] += x\\n\\nnear = nearlist(n, ab)\\nbfs(near)\\nprint(*ans)\", \"#!/usr/bin/env python3\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 8)\\ninput = sys.stdin.readline\\n\\n\\ndef dfs(now):\\n    seen[now] = True\\n    for next in branch[now]:\\n        if seen[next] is False:\\n            score[next] += score[now]\\n            dfs(next)\\n\\n\\nN, Q = list(map(int, input().split()))\\nbranch = [set() for _ in range(N)]\\nfor _ in range(N - 1):\\n    a, b = list(map(int, input().split()))\\n    branch[a - 1].add(b - 1)\\n    branch[b - 1].add(a - 1)\\nscore = [0] * N\\nfor _ in range(Q):\\n    p, x = list(map(int, input().split()))\\n    score[p - 1] += x\\n\\nseen = [False] * N\\nfor i in range(N):\\n    if seen[i] is False:\\n        dfs(i)\\n\\nprint((*score))\\n\", \"from collections import deque\\nn,q = map(int,input().split())\\nz = [[] for i in range(n+1)]\\nfor i in range(n-1):\\n    a,b = map(int,input().split())\\n    z[a].append(b)\\n    z[b].append(a)\\nl = [list(map(int,input().split())) for i in range(q)]\\np,x = [list(i) for i in zip(*l)]\\nc = [0] * n\\nfor i in range(q):\\n    c[p[i]-1] += x[i]\\nqueue = deque([1])\\ncheck = [1] * (n+1)\\nwhile queue:\\n    p = queue.popleft()\\n    if check[p]:\\n        for i in z[p]:\\n            if check[i]:\\n                c[i-1] += c[p-1]\\n                queue.append(i)\\n    check[p] = 0\\nprint(' '.join(map(str,c)))\", \"#\\n# abc138 d\\n#\\nimport sys\\nfrom io import StringIO\\nimport unittest\\nfrom collections import deque\\n\\n\\nclass TestClass(unittest.TestCase):\\n    def assertIO(self, input, output):\\n        stdout, stdin = sys.stdout, sys.stdin\\n        sys.stdout, sys.stdin = StringIO(), StringIO(input)\\n        resolve()\\n        sys.stdout.seek(0)\\n        out = sys.stdout.read()[:-1]\\n        sys.stdout, sys.stdin = stdout, stdin\\n        self.assertEqual(out, output)\\n\\n    def test_\\u5165\\u529b\\u4f8b_1(self):\\n        input = \\\"\\\"\\\"4 3\\n1 2\\n2 3\\n2 4\\n2 10\\n1 100\\n3 1\\\"\\\"\\\"\\n        output = \\\"\\\"\\\"100 110 111 110\\\"\\\"\\\"\\n        self.assertIO(input, output)\\n\\n    def test_\\u5165\\u529b\\u4f8b_2(self):\\n        input = \\\"\\\"\\\"6 2\\n1 2\\n1 3\\n2 4\\n3 6\\n2 5\\n1 10\\n1 10\\\"\\\"\\\"\\n        output = \\\"\\\"\\\"20 20 20 20 20 20\\\"\\\"\\\"\\n        self.assertIO(input, output)\\n\\n\\ndef resolve():\\n    N, Q = list(map(int, input().split()))\\n    AB = [list(map(int, input().split())) for _ in range(N-1)]\\n    PX = [list(map(int, input().split())) for _ in range(Q)]\\n\\n    G = [[i+1, 0] for i in range(N)]\\n    for ab in AB:\\n        a, b = ab\\n        G[a-1][1] += 1\\n        G[b-1][1] += 1\\n        G[a-1].append(b)\\n        G[b-1].append(a)\\n\\n    ans = [0] * N\\n    for px in PX:\\n        p, x = px\\n        ans[p-1] += x\\n\\n    S = deque()\\n\\n    F = [False] * N\\n    S.append(1)\\n    F[0] = True\\n\\n    while S:\\n        p = S.pop()\\n        if G[p-1][1] == 0:\\n            continue\\n\\n        for np in G[p-1][2:]:\\n            if F[np-1]:\\n                continue\\n            S.append(np)\\n            F[np-1] = True\\n            ans[np-1] += ans[p-1]\\n\\n    print((*ans))\\n\\n\\ndef __starting_point():\\n    # unittest.main()\\n    resolve()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\nN, Q = map(int, input().split())\\n\\nAB_TREE = [[] for i in range(N)]\\n\\nfor _ in range(N - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    AB_TREE[b].append(a)\\n    AB_TREE[a].append(b)\\n\\nans = [0] * (N + 1)\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    p -= 1\\n    ans[p] += x\\n\\n\\ndef dfs(v, p):\\n    for to in AB_TREE[v]:\\n        if to == p: continue\\n        ans[to] += ans[v]\\n        dfs(to, v)\\n\\n\\ndfs(0, -1)\\nfor i in range(N):\\n    print(ans[i],end=' ')\\n\", \"import collections\\nN,Q = map(int,input().split())\\nlski = [[] for i in range(N+1)]\\nfor i in range(N-1):\\n    a,b = map(int,input().split())\\n    lski[a].append(b)\\n    lski[b].append(a)\\ncountercost = collections.Counter()\\nfor i in range(Q):\\n    p,x = map(int,input().split())\\n    countercost[p] += x\\n\\ndist = [-1] * (N+1)\\nparent = [-1] * (N+1)\\ndist[0] = 0\\ndist[1] = countercost[1]\\n\\nd = collections.deque()\\nd.append(1)\\n\\nwhile d:\\n    v = d.popleft()\\n    for i in lski[v]:\\n        if parent[a] == b:\\n            continue\\n        if dist[i] != -1:\\n            continue\\n        parent[b] = a\\n        dist[i] = dist[v] + countercost[i]\\n        d.append(i)\\ndist.pop(0)\\nansls = [str(i) for i in dist]\\nprint(' '.join(ansls))\", \"N,Q = map(int,input().split())\\nG = [[] for n in range(N)]\\nans = N*[0]\\n\\nfor n in range(N-1):\\n  a,b = map(int,input().split())\\n  G[a-1].append(b-1)\\n  G[b-1].append(a-1)\\n\\nfor q in range(Q):\\n  p,x = map(int,input().split())\\n  ans[p-1]+=x\\n\\nf = N*[1]\\nt = [0]\\nwhile t:\\n  v = t.pop()\\n  f[v] = 0\\n  for k in G[v]:\\n    if f[k]:\\n      ans[k]+=ans[v]\\n      t.append(k)\\n\\nprint(*ans)\", \"import sys\\nfrom collections import deque\\nsys.setrecursionlimit(10**7)\\ndef I(): return int(sys.stdin.readline().rstrip())\\ndef MI(): return list(map(int,sys.stdin.readline().rstrip().split()))\\ndef LI(): return list(map(int,sys.stdin.readline().rstrip().split()))  #\\u7a7a\\u767d\\u3042\\u308a\\ndef LI2(): return list(map(int,sys.stdin.readline().rstrip()))  #\\u7a7a\\u767d\\u306a\\u3057\\ndef S(): return sys.stdin.readline().rstrip()\\ndef LS(): return list(sys.stdin.readline().rstrip().split())  #\\u7a7a\\u767d\\u3042\\u308a\\ndef LS2(): return list(sys.stdin.readline().rstrip())  #\\u7a7a\\u767d\\u306a\\u3057\\n\\n\\nN,Q = MI()\\nGraph = [[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n    a,b = MI()\\n    Graph[a].append(b)\\n    Graph[b].append(a)\\n\\ncount = [0]*(N+1)\\nadd_count = [0]*(N+1)\\nfor i in range(Q):\\n    p,x = MI()\\n    add_count[p] += x\\n\\nflag = [-1]*(N+1)\\ncount[1] = add_count[1]\\ndeq = deque([(1,add_count[1])])\\nflag[1] = 0\\n\\nwhile deq:\\n    n,r = deq.pop()\\n    for d in Graph[n]:\\n        if flag[d] == -1:\\n            flag[d] = 0\\n            count[d] = r+add_count[d]\\n            deq.appendleft((d,count[d]))\\n\\nprint((*count[1:]))\\n\", \"# D - Ki TLE\\nimport sys\\nsys.setrecursionlimit(10**7)\\nN,Q = map(int,input().split())\\n\\n# \\u6709\\u5411\\u30b0\\u30e9\\u30d5\\nG = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    a -= 1; b -= 1\\n    G[a].append(b)\\n    G[b].append(a)\\n    \\nlst = []\\nfor _ in range(Q):\\n    P,X = map(int,input().split())\\n    P-=1\\n    lst.append((P,X))\\n\\ncnt = [0]*N\\nfor p,x in lst:\\n    cnt[p] += x \\n    \\nseen = [False]*N\\ndef dfs(v):\\n    seen[v] = True# v \\u3092\\u8a2a\\u554f\\u6e08\\u307f\\u306b\\u3059\\u308b\\n    # v \\u304b\\u3089\\u884c\\u3051\\u308b\\u5404\\u9802\\u70b9 next_v \\u306b\\u3064\\u3044\\u3066\\n    for next_v in G[v]:\\n        # next_v \\u304c\\u63a2\\u7d22\\u6e08\\u307f\\u306a\\u3089\\u30b9\\u30eb\\u30fc\\n        if seen[next_v] == True:\\n             continue\\n        cnt[next_v] += cnt[v] \\n        dfs(next_v)\\ndfs(0)\\nprint(*cnt)\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\n\\ndef dfs(v,prev = -1):\\n    for u in graph[v]:\\n        if u == prev:\\n            continue\\n        point[u] += point[v]\\n        dfs(u,v)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [[] for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"import sys\\nfrom typing import List, Tuple\\nimport collections\\n\\nGraph = List[List[int]]\\n\\n\\ndef main():\\n    def input(): return sys.stdin.readline()[:-1]\\n    N, Q = list(map(int, input().split()))\\n\\n    graph = [[] for _ in range(N+1)]\\n    for _ in range(1, N):\\n        a, b = list(map(int, input().split()))\\n        graph[a].append(b)\\n        graph[b].append(a)\\n\\n    px = [0] * (N + 1)\\n    for _ in range(Q):\\n        p, x = list(map(int, input().split()))\\n        px[p] += x\\n\\n    queue = collections.deque()\\n    queue.append(1)\\n    checked = [0] * (N + 1)\\n    while queue:\\n        v = queue.pop()\\n        checked[v] = 1\\n        for next_v in graph[v]:\\n            if checked[next_v] == 1: continue\\n            px[next_v] += px[v]\\n            queue.append(next_v)\\n    print((*px[1:]))\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**7)\\n\\nn,q=map(int,input().split())\\nans=[0 for _ in range(n)]\\ngra=[[] for _ in range(n)]\\nvisited=[0 for _ in range(n)]\\n\\nfor _ in range(n-1):\\n  a,b=map(int,input().split())\\n  gra[a-1].append(b-1)\\n  gra[b-1].append(a-1)\\n  \\nfor _ in range(q):\\n  p,x=map(int,input().split())\\n  ans[p-1]+=x\\n  \\ndef dfs(st):\\n  temp=gra[st]\\n  for i in temp:\\n    if visited[i]==0:\\n      visited[i]+=1\\n      ans[i]+=ans[st]\\n      dfs(i)\\n      \\nvisited[0]+=1\\ndfs(0)\\n\\nfor h in range(n):\\n  ans[h]=str(ans[h])\\n  \\nprint(' '.join(ans))\", \"import collections\\n\\nn,q = map(int, input().split())\\n\\nvalue = ['-'] * (n+1)\\n\\nconnected = [set() for i in range(n+1)]\\n\\nfor i in range(n-1):\\n  a,b = map(int, input().split())\\n  connected[a].add(b)\\n  connected[b].add(a)\\n\\nfor i in range(q):\\n  p,x = map(int, input().split())\\n  if value[p] == '-':\\n    value[p] = 0\\n  value[p] -= x\\n\\nqueue = collections.deque(connected[1])\\n\\nif value[1] == '-':\\n  value[1] = 0\\nelse:\\n  value[1] = -value[1]\\n\\nfor i in queue:\\n  if value[i] == '-':\\n    value[i] = 0\\n  else:\\n    value[i] = -value[i]\\n  value[i] += value[1]\\n\\n\\nwhile queue:\\n  s = queue.popleft()\\n  v = value[s]\\n  for i in connected[s]:\\n    vi = value[i]\\n    if vi == '-' or vi < 0:\\n      if vi == '-':\\n        vi = 0\\n      value[i] = -(vi-v)\\n      queue.append(i)\\n    \\nprint(' '.join(str(i) for i in value[1:]))\", \"import sys\\nsys.setrecursionlimit(10**9)\\ndef f(p,q=-1):\\n  for i in c[p]:\\n    if i!=q:\\n      A[i]+=A[p]\\n      f(i,p)\\nn,q=map(int,input().split())\\nc=[[]for _ in range(n)]\\nfor i in range(n-1):\\n  a,b=map(int,input().split())\\n  c[a-1].append(b-1)\\n  c[b-1].append(a-1)\\nA=[0]*n\\nfor _ in range(q):\\n  p,x=map(int,input().split())\\n  A[p-1]+=x\\nf(0)\\nprint(*A)\", \"from collections import deque\\nn, q = list(map(int, input().split()))\\ngraphs = [set() for _ in range(n)]\\nfor _ in range(n - 1):\\n    a, b = [int(x) - 1 for x in input().split()]\\n    graphs[a].add(b)\\n    graphs[b].add(a)\\n\\nnodes = [0 for _ in range(n)]\\nfor _ in range(q):\\n    p, x = list(map(int, input().split()))\\n    nodes[p - 1] += x\\n\\ndone = [True for _ in range(n)]\\nD = deque()\\nD.append([0, 0])\\nans = [0 for _ in range(n)]\\nwhile D:\\n    node, value = D.popleft()\\n    value += nodes[node]\\n    ans[node] = value\\n    done[node] = False\\n    for g in graphs[node]:\\n        if done[g]:\\n            D.append([g, value])\\n\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN, Q = map(int, input().split())\\nT = [[] for _ in range(N)]\\n\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    T[a-1].append(b-1)\\n    T[b-1].append(a-1)\\n\\nS = [0] * N\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    S[p-1] += x\\n\\ndef search(p, q=-1):\\n    for a in T[p]:\\n        if a != q:\\n            S[a] += S[p]\\n            search(a, p)\\n\\nsearch(0)\\nprint(*S)\", \"def solve():\\n    import sys\\n    sys.setrecursionlimit(10**6)\\n    def dfs(tree, n, ans):\\n        for i in tree[n]:\\n            ans[i] += ans[n]\\n            tree[i].remove(n)\\n            dfs(tree, i, ans) #\\u518d\\u5e30\\u95a2\\u6570\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u3002\\u6b21\\u306e\\u6728\\u306e\\u4e2d\\u3092\\u63a2\\u7d22\\n     \\n    n, q = map(int,input().split())\\n    ab = [list(map(int,input().split())) for _ in range(n-1)]\\n    px = [list(map(int,input().split())) for _ in range(q)]\\n     \\n    g = [[] for _ in range(n)]\\n    for i in range(n-1):\\n        g[ab[i][0]-1].append(ab[i][1]-1)\\n        g[ab[i][1]-1].append(ab[i][0]-1)\\n    \\n    ans = [0]*n\\n    for i in range(q):\\n        ans[px[i-1][0]-1] += px[i-1][1]\\n    dfs(g, 0, ans)\\n     \\n    print(*ans)\\n    \\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"import sys\\nfrom functools import lru_cache\\nsys.setrecursionlimit(10**7)\\n \\nN,Q=map(int,input().split())\\nT=[[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n  a,b=map(int,input().split())\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nS=[0]*(N+1)\\nfor _ in range(Q):\\n  p,x=map(int,input().split())\\n  S[p]+=x\\n\\n@lru_cache(maxsize=None)\\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next==prev:\\n      continue\\n    S[next]+=S[now]\\n    dfs(next,now)\\n\\ndfs(1)\\nprint(*S[1:])\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10 ** 7)\\n \\n# \\u9045\\u5ef6\\u8a55\\u4fa1\\u3067\\u52a0\\u3048\\u3066\\u3042\\u3052\\u308b\\u3060\\u3051\\n \\nN,Q = map(int,input().split())\\nAB = [[int(x) for x in input().split()] for _ in range(N-1)]\\nPX = [[int(x) for x in input().split()] for _ in range(Q)]\\n \\ngraph = [[] for _ in range(N+1)]\\nfor a,b in AB:\\n    graph[a].append(b)\\n    graph[b].append(a)\\n \\nvalue = [0] * (N+1)\\nfor p,x in PX:\\n    value[p] += x\\n \\ndef dfs(v,parent,add):\\n    value[v] += add\\n    for x in graph[v]:\\n        if x == parent:\\n            continue\\n        dfs(x,v,value[v])\\n \\ndfs(1,0,0)\\n \\nanswer = ' '.join(map(str,value[1:]))\\nprint(answer)\", \"import collections\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n-1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nc = [[] for _ in range(n)]\\nfor a, b in ab:\\n    a, b = a-1, b-1\\n    c[a].append(b)\\n    c[b].append(a)\\n\\npoint = [0] * n\\nfor p, x in px:\\n    point[p-1] += x\\n\\nparents = [0] * n\\nans = [0] * n\\nq = collections.deque()\\nq.append(0)\\nwhile q:\\n    v = q.pop()\\n    ans[v] = ans[parents[v]] + point[v]\\n    for i in c[v]:\\n        if i == parents[v]:\\n            continue\\n        parents[i] = v\\n        q.append(i)\\nprint(' '.join(list(map(str, ans))))\", \"from collections import deque\\nn,q=map(int,input().split())\\ntree=[[] for _ in range(n)]\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    tree[a].append(b)\\n    tree[b].append(a)\\ncount=[0]*n\\nfor _ in range(q):\\n    p,cnt=map(int,input().split())\\n    count[p-1]+=cnt\\nstack=deque([[0,0,-1]])\\nwhile stack:\\n    num,cnt,pr=stack.pop()\\n    count[num]+=cnt\\n    for k in tree[num]:\\n        if k==pr:\\n            continue\\n        stack.append([k,count[num],num])\\nprint(*count)\", \"import sys\\nfrom collections import defaultdict\\nsys.setrecursionlimit(1000000)\\nN, Q = map(int,input().split())\\nedges = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a,b = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    edges[a].append(b)\\n    edges[b].append(a)\\n\\nd = defaultdict(int)\\nfor _ in range(Q):\\n    p,x = map(int,input().split())\\n    d[p-1] += x\\nvisited = [False]*N\\ncnt = [0]*N\\ndef dfs(n, acc):\\n    cnt[n] = acc\\n    for m in edges[n]:\\n        if not visited[m]:\\n            visited[m] = True\\n            dfs(m, acc+d[m])\\nvisited[0] = True\\ndfs(0, d[0])\\nprint(*cnt)\", \"from collections import deque\\nimport numpy as np\\n\\nN, Q = map(int, input().split())\\nA = [list(map(int, input().split())) for _ in range(N-1)]\\nP = [list(map(int, input().split())) for _ in range(Q)]\\n\\nlink = [[] for _ in range(N)]\\nfor i in range(N-1):\\n    link[A[i][0]-1].append(A[i][1]-1)\\n    link[A[i][1]-1].append(A[i][0]-1)\\n\\n\\nans = [0 for _ in range(N)]\\nfor i in range(Q):\\n  ans[P[i][0] - 1] += P[i][1]\\n\\ndist = [-1 for _ in range(N)]\\ndist[0] = 0\\nd = deque([0])\\nwhile d:\\n    now = d.pop()\\n    for i in range(len(link[now])):\\n        if dist[link[now][i]] == -1:\\n            ans[link[now][i]] += ans[now]\\n            dist[link[now][i]] = 0\\n            d.append(link[now][i])\\n\\nfor i in range(N):\\n    if i != N - 1:\\n        print(ans[i], end=\\\" \\\")\\n    else:\\n        print(ans[i])\\n\", \"import sys\\nread = sys.stdin.read\\nsys.setrecursionlimit(10**7)\\n#readlines = sys.stdin.readlines\\ndef main():\\n    def dfs(v, s):\\n        if notseen[v]:\\n            scores[v] += s\\n            notseen[v] = 0\\n            for nextv in gg[v]:\\n                dfs(nextv, scores[v])\\n\\n    data = tuple(map(int, read().split()))\\n    n, q = data[0], data[1]\\n    gg = {i: set() for i in range(1, n + 1)}\\n    # \\u6728\\u306a\\u306e\\u3067\\u201d\\u6b63\\u3057\\u304f\\u201d\\u51e6\\u7406\\u3059\\u308c\\u3070gg[x]\\u3078\\u306eadd\\u306f\\uff12\\u56de\\u3067\\u306f\\u306a\\u304f\\uff11\\u56de\\u3067\\u3044\\u3044\\u306f\\u305a\\u3002\\n    # \\u3060\\u304c\\u3046\\u307e\\u304f\\u884c\\u304b\\u306a\\u3044\\u305f\\u3081\\u3001\\u7121\\u5411\\u30b0\\u30e9\\u30d5\\u306e\\u3088\\u3046\\u306b\\uff12\\u56de\\u52a0\\u3048\\u3066\\u3001dfs\\u95a2\\u6570\\u3067\\u5404\\u9802\\u70b9\\u3092\\uff11\\u56de\\u3057\\u304b\\u307f\\u306a\\u3044\\u3088\\u3046\\u306b\\u5bfe\\u5fdc\\u3002\\n    for i1, v1 in zip(data[2:n*2:2], data[3:n*2:2]):\\n        gg[v1].add(i1)\\n        gg[i1].add(v1)\\n    scores = [0] * (n + 1)\\n    for j1, v1 in zip(data[n*2::2], data[n*2+1::2]):\\n        scores[j1] += v1\\n\\n    notseen = [1] * (n + 1)\\n    dfs(1, 0)\\n    print(*scores[1:], sep=' ')\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from collections import deque\\nN,Q = map(int,input().split())\\nab = [list(map(int,input().split())) for _ in range(N-1)]\\npx = [list(map(int,input().split())) for _ in range(Q)]\\n\\ntree = [[] for _ in range(N)]\\nfor a, b in ab:\\n  tree[a-1].append(b-1)\\n  tree[b-1].append(a-1)\\n\\ncounter = [0] * N\\nfor p, x in px:\\n  counter[p-1] += x\\n  \\nflag = [1] * N\\nt = deque()\\nt.append(0)\\n\\n#1(0)\\u304b\\u3089\\u9806\\u306b\\u63a2\\u7d22\\u3057\\u3066\\u3044\\u3051\\u3070\\u3001\\u81ea\\u7136\\u3068\\u89aa\\u2192\\u5b50\\u306e\\u9806\\u756a\\u306b\\u306a\\u308b\\nwhile t:\\n  v = t.popleft()\\n  flag[v] = 0\\n  for i in tree[v]:\\n    if flag[i]:\\n      counter[i] += counter[v]\\n      t.append(i)\\n      \\nprint(*counter)\", \"from collections import defaultdict\\nfrom sys import setrecursionlimit\\nsetrecursionlimit(10 ** 7)\\n\\nn,q = map(int,input().split())\\n\\nnode = [[] for _ in range(n+1)]\\ncounter = [0 for _ in range(n+1)]\\nvisited = [0 for _ in range(n+1)]\\npoint_dic = defaultdict(int)\\n\\nfor _ in range(n-1):\\n    a,b = map(int,input().split())\\n    node[a].append(b)\\n    node[b].append(a)\\n\\nfor _ in range(q):\\n    p,x = map(int,input().split())\\n    point_dic[p] += x\\n\\ndef dfs(now,before):\\n    nonlocal counter\\n    nonlocal visited\\n\\n    if visited[now]:\\n        return\\n    \\n    if point_dic[now]:\\n        point = point_dic[now]+before\\n    else:\\n        point = before\\n    \\n    visited[now] = 1\\n    counter[now] += point\\n\\n    for i in node[now]:\\n        if visited[i]:\\n            continue\\n\\n        dfs(i,point)\\n\\ndfs(1,0)\\n\\nprint(' '.join(map(str, counter[1:])))\", \"from collections import deque\\n\\n\\nN, Q = list(map(int, input().split()))\\nt = [[] for i in range(N + 1)]\\n\\nfor i in range(N - 1):\\n    a, b = list(map(int, input().split()))\\n    t[a].append(b)\\n    t[b].append(a)\\n\\nscore = [0] * (N + 1)\\nfor i in range(Q):\\n    p, x = list(map(int, input().split()))\\n    score[p] += x\\n\\nis_read = [False] * (N + 1)\\nd = deque()\\nd.append(1)\\n\\nwhile len(d) != 0:\\n    now = d.popleft()\\n    is_read[now] = True\\n    for c in t[now]:\\n        if is_read[c]:\\n            continue\\n        score[c] += score[now]\\n        d.append(c)\\nprint((\\\" \\\".join(map(str, score[1:]))))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef dfs(tree, n, ans):\\n    for i in tree[n]:\\n        ans[i] += ans[n]\\n        tree[i].remove(n)\\n        dfs(tree, i, ans)\\n \\nn, q = map(int,input().split())\\nab = [list(map(int,input().split())) for _ in range(n-1)]\\npx = [list(map(int,input().split())) for _ in range(q)]\\n \\ng = [[] for _ in range(n)]\\nfor i in range(n-1):\\n    g[ab[i][0]-1].append(ab[i][1]-1)\\n    g[ab[i][1]-1].append(ab[i][0]-1)\\n \\nans = [0]*n\\nfor i in range(q):\\n    ans[px[i-1][0]-1] += px[i-1][1]\\ndfs(g, 0, ans)\\n \\nprint(*ans)\", \"# import itertools\\n# import math\\nimport sys\\nsys.setrecursionlimit(500*500)\\n# import numpy as np\\n# from collections import deque\\n# import heapq\\n\\n# N = int(input())\\n# S = input()\\n# n, *a = map(int, open(0))\\nN, Q = map(int, input().split())\\n# A = list(map(int, input().split()))\\n# A = list(map(lambda x: int(x)*(-1), input().split()))\\n# B = list(map(int, input().split()))\\n# A_B = [list(map(int,input().split())) for _ in range(M)]\\n# S = input()\\n\\n# B_C = sorted(B_C, reverse=True, key=lambda x:x[1])\\n# all_cases = list(itertools.permutations(P))\\n# a = list(itertools.combinations_with_replacement(range(1, M + 1), N))\\n# itertools.product((0,1), repeat=n)\\n\\n# A = np.array(A)\\n# cum_A = np.cumsum(A)\\n# cum_A = np.insert(cum_A, 0, 0)\\n\\nedges = [list(map(int,input().split())) for _ in range(N - 1)]\\ntree = [[] for _ in range(N + 1)]\\n\\nfor edge in edges:\\n    tree[edge[0]].append(edge[1])\\n    tree[edge[1]].append(edge[0])\\n\\ndepth = [-1] * (N + 1)\\ndepth[1] = 0\\ncount = [0] * (N + 1)\\n\\nfor i in range(Q):\\n    p, x = map(int, input().split())\\n    count[p] += x\\n\\n# def dfs(tree, s):\\n#     for l in tree[s]:\\n#         if depth[l[0]] == -1:\\n#             depth[l[0]] = depth[s] + 1\\n#             dfs(tree, l[0])\\n# dfs(tree, 1)\\ndef dfs(tree, s):\\n    for l in tree[s]:\\n        if depth[l] == -1:\\n            depth[l] = 0\\n            count[l] += count[s]\\n            dfs(tree, l)\\ndfs(tree, 1)\\n\\n\\nfor i in count[1:]:\\n    print(i, end = ' ')\\n\\n# def factorization(n):\\n#     arr = []\\n#     temp = n\\n#     for i in range(2, int(-(-n**0.5//1))+1):\\n#         if temp%i==0:\\n#             cnt=0\\n#             while temp%i==0:\\n#                 cnt+=1\\n#                 temp //= i\\n#             arr.append([i, cnt])\\n#     if temp!=1:\\n#         arr.append([temp, 1])\\n#     if arr==[]:\\n#         arr.append([n, 1])\\n#     return arr\\n\\n\\n\\n# bfs\\n# tree = [[] for _ in range(N + 1)]\\n# edges = [list(map(int,input().split())) for _ in range(M)]\\n\\n# for edge in edges:\\n#     tree[edge[0]].append(edge[1])\\n#     tree[edge[1]].append(edge[0])\\n\\n# depth = [-1] * (N + 1)\\n# depth[1] = 0\\n\\n# d = deque()\\n# d.append(1)\\n\\n# ans = [0] * (N + 1)\\n# while d:\\n#  v = d.popleft()\\n#  for i in tree[v]:\\n#    if depth[i] != -1:\\n#      continue\\n#    depth[i] = depth[v] + 1\\n#    ans[i] = v\\n#    d.append(i)\\n\\n# # ans = depth[2:]\\n# print('Yes')\\n# print(*ans[2:], sep=\\\"\\\\n\\\")\", \"import sys\\nfrom functools import lru_cache\\nsys.setrecursionlimit(10**7)\\n\\nN, Q = map(int, input().split())\\nT = [[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n  a,b=map(int, input().split())\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nV = [0] * (N+1)\\nfor q in range(Q):\\n  p,x=map(int, input().split())\\n  V[p]+=x\\n\\n@lru_cache(maxsize=None)\\ndef dfs(i, parent, acc):\\n  V[i]+=acc\\n  for j in T[i]:\\n    if j != parent:\\n      dfs(j,i,V[i])\\n\\ncur,parent,acc=1,0,0\\ndfs(cur,parent,acc)\\nprint(*V[1:])\", \"import collections,sys\\ndef LI(): return list(map(int,sys.stdin.readline().rstrip().split()))\\nN,Q = LI()\\nab = [LI() for _ in range(N-1)]\\npx = [LI() for _ in range(Q)]\\nans = [0]*(N+1) #1_indexed\\ngraph = {i:collections.deque() for i in range(1,N+1)} #1_indexed\\nfor a,b in ab:\\n        graph[a].append(b)\\n        graph[b].append(a)\\nfor p,x in px:\\n    ans[p] += x\\nseen = [0]*(N+1) #1_indexed\\nstack = []\\ndef dfs():\\n    seen[1] = 1\\n    stack.append(1)\\n    while stack:\\n        s = stack.pop()\\n        if not graph[s]:\\n            continue\\n        for j in range(len(graph[s])):\\n            g_NO = graph[s].popleft()\\n            if seen[g_NO]:\\n                continue\\n            seen[g_NO] = 1\\n            stack.append(g_NO)\\n            ans[g_NO] += ans[s]\\ndfs()\\nprint(*ans[1:])\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN, Q = map(int,input().split())\\nG = [[] for _ in range(N)]\\ncounter = [0] * N\\n\\nfor _ in range(N-1):\\n    a, b = map(int,input().split())\\n    G[a-1].append(b-1)\\n    G[b-1].append(a-1)\\n\\nfor _ in range(Q):\\n    p, x = map(int,input().split())\\n    counter[p-1] += x\\n\\nseen = [False]*N\\n\\ndef dfs(v, seen, G, counter):\\n    seen[v] = True\\n    for u in G[v]:\\n        if not seen[u]:\\n            counter[u] += counter[v]\\n            dfs(u, seen, G, counter)\\n\\ndfs(0, seen, G, counter)\\n\\nfor c in counter:\\n    print(c, end = \\\" \\\")\\nprint(\\\"\\\")\\n\\n\", \"import sys\\nfrom collections import deque\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(s):\\n    stack = [s]\\n    visited = [False]*N\\n    while stack:\\n        v = stack.pop()\\n        if visited[v]:\\n            continue\\n        visited[v] = True\\n        for u in graph[v]:\\n            if not visited[u]:\\n                point[u] += point[v]\\n                stack.append(u)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [deque([]) for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"#!/usr/bin/env python3\\nfrom sys import setrecursionlimit\\n\\nsetrecursionlimit(10 ** 8)\\n\\n\\ndef dfs(now):\\n    seen[now] = True\\n    score[now] += operation[now]\\n    for next in branch[now]:\\n        if seen[next] is False:\\n            score[next] += score[now]\\n            dfs(next)\\n\\n\\nN, Q = map(int, input().split())\\nbranch = [set() for _ in range(N)]\\nfor _ in range(N - 1):\\n    a, b = map(int, input().split())\\n    branch[a - 1].add(b - 1)\\n    branch[b - 1].add(a - 1)\\noperation = [0] * N\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    operation[p - 1] += x\\n\\nseen = [False] * N\\nscore = [0] * N\\nfor i in range(N):\\n    if seen[i] is False:\\n        dfs(i)\\n\\nfor ans in score:\\n    print(ans, end=' ')\\n\", \"import sys\\nreadline = sys.stdin.readline\\n\\nN,Q = map(int,readline().split())\\n\\nG = [[] for i in range(N)]\\n\\nfor i in range(N - 1):\\n  a,b = map(int,readline().split())\\n  G[a - 1].append(b - 1)\\n  G[b - 1].append(a - 1)\\n  \\npoint = [0] * N\\nfor i in range(Q):\\n  p,x = map(int,readline().split())\\n  point[p - 1] += x\\n\\nstack = [(0, -1, 0)]\\nwhile stack:\\n  v,parent,p = stack.pop()\\n  point[v] += p\\n  p = point[v]\\n  for child in G[v]:\\n    if child == parent:\\n      continue\\n    stack.append([child, v, p])\\n  \\nprint(*point)\", \"def main():\\n\\tN, Q = [int(n) for n in input().split(\\\" \\\")]\\n\\tedges = [[] for i in range(N)]\\n\\tcounter = [0] * N\\n\\n\\tfor i in range(N - 1):\\n\\t\\ta, b = [int(x) for x in input().split(\\\" \\\")]\\n\\t\\tedges[a - 1].append(b - 1)\\n\\t\\tedges[b - 1].append(a - 1)\\n\\n\\tfor j in range(Q):\\n\\t\\tp, x = [int(q) for q in input().split(\\\" \\\")]\\n\\t\\tcounter[p - 1] += x\\n\\n\\tto_visit = [0]\\n\\tchecked = [1] + [0] * (N - 1)\\n\\ttree = [0] * N\\n\\n\\twhile len(to_visit) > 0:\\n\\t\\tvisiting = to_visit.pop()\\n\\t\\ttree[visiting] += counter[visiting]\\n\\t\\tfor e in edges[visiting]:\\n\\t\\t\\tif checked[e] == 0:\\n\\t\\t\\t\\tchecked[e] = 1\\n\\t\\t\\t\\tto_visit.append(e)\\n\\t\\t\\t\\ttree[e] = tree[visiting]\\n\\n\\tprint((\\\" \\\".join([str(s) for s in tree])))\\n\\nmain()\\n\", \"import sys\\nsys.setrecursionlimit(10**8)\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n-1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nt = dict()\\nfor i in range(1, n+1):\\n    t[i] = set()\\nfor a, b in ab:\\n    t[a].add(b)\\n    t[b].add(a)\\n\\ncnt = [0] * n\\ns = set()\\nfor p, x in px:\\n    cnt[p-1] += x\\n\\ndef solve(p):\\n    s.add(p)\\n    for c in t[p]:\\n        if c not in s:\\n            cnt[c-1] += cnt[p-1]\\n            solve(c)\\nsolve(1)\\nprint(*cnt, sep=' ')\\n\", \"from collections import deque\\n\\nN, Q = list(map(int, input().split()))\\ntree = [set() for _ in range(N+1)]\\n\\nfor i in range(N - 1):\\n    n1, n2 = list(map(int, input().split()))\\n    tree[n1].add(n2)\\n    tree[n2].add(n1)\\n\\nscores = [0]*(N+1)\\nfor _ in range(Q):\\n    p, x = list(map(int, input().split()))\\n    scores[p] += x\\n\\nq = deque()\\nq.append(1)\\nchecked = [False]*(N+1)\\nwhile q:\\n    v = q.pop()\\n    checked[v] = True\\n    s = scores[v]\\n    for k in tree[v]:\\n        if checked[k]:\\n            continue\\n        scores[k] += s\\n        q.append(k)\\n\\nprint((*scores[1:]))\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN,Q=map(int,input().split())\\nT=[[] for _ in range(N)]\\nfor _ in range(N-1):\\n  a,b=map(int,input().split())\\n  a-=1;b-=1\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nV=[0]*N\\nfor _ in range(Q):\\n  p,x=map(int,input().split())\\n  p-=1\\n  V[p]+=x\\n  \\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next==prev:\\n      continue\\n    V[next]+=V[now]\\n    dfs(next,now)\\n\\ndfs(0)\\nprint(*V)\", \"import sys\\nsys.setrecursionlimit(1<<30)\\nN,Q = map(int,input().split())\\ndef dfs(x, score):\\n    for y in Tree[x]:\\n        if Parent[x] != y:\\n            Parent[y] = x\\n            score[y] += score[x]\\n            dfs(y, score)\\nTree = [[] for i in range(N+1)]\\nParent = [0]*(N+1)\\nfor i in range(N-1):\\n    a,b = map(int,input().split())\\n    Tree[a].append(b)\\n    Tree[b].append(a)\\nscore = [0]*(N+1)\\nfor j in range(Q):\\n    p,x = map(int,input().split())\\n    score[p] += x\\ndfs(1,score)\\nprint(*score[1:],end=\\\"\\\\t\\\")\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN, Q = map(int, input().split())\\nL = [[] for _ in range(N)]\\n\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    L[a-1].append(b-1)\\n    L[b-1].append(a-1)\\n\\nS = [0] * N\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    S[p-1] += x\\n\\ndef search(p,q=-1):\\n    for a in L[p]:\\n        if a != q:\\n            S[a] += S[p]\\n            search(a,p)\\n\\nsearch(0)\\nprint(*S)\", \"import sys\\nsys.setrecursionlimit(10**7)\\nN, Q = [int(n) for n in input().split()]\\ntree_list = [[] for _ in range(N)]\\n\\nfor i in range(N-1):\\n    a, b = [int(n)-1 for n in input().split()]\\n    tree_list[a].append(b)\\n    tree_list[b].append(a)\\n\\nscore_list = [0] * N\\nfor j in range(Q):\\n    p, x = [int(n) for n in input().split()]\\n    score_list[p-1] += x\\n\\nreached = [False] * N\\ndef dfs(v):\\n    reached[v] = True\\n    for next_v in tree_list[v]:\\n        if reached[next_v] == True:\\n            continue\\n        score_list[next_v] += score_list[v]\\n        dfs(next_v)\\ndfs(0)\\nprint(*score_list)\", \"from collections import deque\\n\\nN, Q = list(map(int, input().split())) # N\\u306f\\u9802\\u70b9\\u306e\\u6570\\u3001Q\\u306f\\u64cd\\u4f5c\\u306e\\u56de\\u6570\\ngraph = [[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n    a, b = list(map(int, input().split()))\\n    graph[a].append(b)\\n    graph[b].append(a)\\n\\ncounts = [0] * (N+1)\\nfor _ in range(Q):\\n    p, x  = list(map(int, input().split()))\\n    counts[p] += x\\nvisited = [-1] * (N+1)\\n\\nq = deque()\\nq.append(1)\\nvisited[1] = 1\\nwhile q:\\n    node = q.pop()\\n\\n    next_nodes = graph[node]\\n    for next in next_nodes:\\n        if visited[next] != -1:\\n            continue\\n        q.append(next)\\n        visited[next] = 1\\n        counts[next] += counts[node]\\n\\nprint((*counts[1:]))\\n\\n\\n\", \"import sys\\nsys.setrecursionlimit(10000000)\\ndef main():\\n    n,q = list(map(int,input().split()))\\n    Ki = [[] for i in range(n)]\\n    for _ in range(n-1):\\n        a,b = list(map(int,input().split()))\\n        Ki[a-1]+=[b-1]\\n        Ki[b-1]+=[a-1]\\n    ans = [0]*n\\n    for _ in range(q):\\n        p,x = list(map(int,input().split()))\\n        ans[p-1]+=x\\n    def add(N=0,L=-1):\\n        for k in Ki[N]:\\n            if k!=L:\\n                ans[k]+=ans[N]\\n                add(k,N)\\n    add()\\n    print((' '.join([str(a) for a in ans])))\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\nn,q=map(int,input().split())\\nG=[[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    G[a].append(b)\\n    G[b].append(a)\\n\\nsubtree_size=[0]*n\\ncnt=[0]*n\\n\\nfor _ in range(q):\\n    p,x=map(int,input().split())\\n    p-=1\\n    cnt[p]+=x\\nans=[0]*n\\n\\ndef dfs(G,v,p):\\n    ans[v]+=cnt[v]\\n    for nv in G[v]:\\n        if nv==p:\\n            continue\\n        cnt[nv]+=cnt[v]\\n        dfs(G,nv,v)\\n    \\n    subtree_size[v]=1\\n    for c in G[v]:\\n        if c==p:\\n            continue\\n        subtree_size[v]+=subtree_size[c]\\n\\nroot=0\\ndfs(G,root,-1)\\n\\nfor i in range(n):\\n    print(ans[i],end=' ')\\n\\n\\n\\n\", \"import sys\\nsys.setrecursionlimit(10**8)\\nN, Q = map(int, input().split())\\ng = [[] for i in range(N)]\\nfor i in range(N-1):\\n    a, b = list(map(int, input().split()))\\n    g[a-1].append(b-1)\\n    g[b-1].append(a-1)\\ncnt = [0] * N\\nfor i in range(Q):\\n    p, x = list(map(int, input().split()))\\n    cnt[p-1] += x\\n\\ndef dfs(g, v):\\n    # nonlocal ans_cnt\\n    for nv in g[v]:\\n        if ans_cnt[nv] != -1: continue\\n        ans_cnt[nv] = cnt[nv] + ans_cnt[v]\\n        dfs(g, nv)\\n\\nans_cnt = [-1] * N\\nans_cnt[0] = cnt[0]\\ndfs(g, 0)\\n\\n# print(cnt)\\n# print(ans_cnt)\\nfor i, value in enumerate(ans_cnt):\\n    print(value, end=\\\"\\\")\\n    if i != len(ans_cnt)-1:\\n        print(\\\" \\\", end=\\\"\\\")\\nprint()\", \"import sys\\nsys.setrecursionlimit(10**7)\\nn,q=map(int,input().split())\\nedge=[[] for _ in range(n)]\\nfor i in range(n-1):\\n    x,y=map(int,input().split())\\n    edge[x-1].append(y-1)\\n    edge[y-1].append(x-1)\\nans=[0]*n\\nfor i in range(q):\\n    p,x=map(int,input().split())\\n    ans[p-1]+=x\\ndef dfs(c,p):\\n    for i in edge[c]:\\n        if i==p:\\n            continue\\n        ans[i]+=ans[c]\\n        dfs(i,c)\\ndfs(0,-1)\\nprint(*ans)\", \"N,Q = map(int,input().split())\\nG = [[] for n in range(N)]\\nans = N*[0]\\n \\nfor n in range(N-1):\\n  a,b = map(int,input().split())\\n  G[a-1].append(b-1)\\n  G[b-1].append(a-1)\\n \\nfor q in range(Q):\\n  p,x = map(int,input().split())\\n  ans[p-1]+=x\\n \\nf = N*[1]\\nt = [0]\\nwhile t:\\n  v = t.pop()\\n  f[v] = 0\\n  for k in G[v]:\\n    if f[k]:\\n      ans[k]+=ans[v]\\n      t.append(k)\\n \\nprint(*ans)\", \"import sys\\nsys.setrecursionlimit(10**8)\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n-1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nt = dict()\\nfor i in range(1, n+1):\\n    t[i] = []\\nfor a, b in ab:\\n    t[a].append(b)\\n    t[b].append(a)\\n\\ncnt = [0] * n\\ns = set()\\nfor p, x in px:\\n    cnt[p-1] += x\\n\\ndef solve(p):\\n    s.add(p)\\n    for c in t[p]:\\n        if c not in s:\\n            cnt[c-1] += cnt[p-1]\\n            solve(c)\\nsolve(1)\\nprint(*cnt, sep=' ')\\n\", \"import sys\\n# import math\\n# import bisect\\n# import numpy as np\\n# from decimal import Decimal\\n# from numba import njit, i8, u1, b1 #JIT compiler\\n# from itertools import combinations, product\\nfrom collections import Counter, deque, defaultdict\\n\\n# sys.setrecursionlimit(10 ** 6)\\nMOD = 10 ** 9 + 7\\nINF = 10 ** 9\\nPI = 3.14159265358979323846\\n\\ndef read_str():      return sys.stdin.readline().strip()\\ndef read_int():      return int(sys.stdin.readline().strip())\\ndef read_ints():     return map(int, sys.stdin.readline().strip().split())\\ndef read_ints2(x):   return map(lambda num: int(num) - x, sys.stdin.readline().strip().split())\\ndef read_str_list(): return list(sys.stdin.readline().strip().split())\\ndef read_int_list(): return list(map(int, sys.stdin.readline().strip().split()))\\ndef GCD(a: int, b: int) -> int: return b if a%b==0 else GCD(b, a%b)\\ndef LCM(a: int, b: int) -> int: return (a * b) // GCD(a, b)\\n\\ndef Main():\\n    n, q = read_ints()\\n    tree = [[] for _ in range(n)]\\n    c = [0] * n\\n    for _ in range(n-1):\\n        a, b = read_ints()\\n        tree[~-a].append(~-b)\\n        tree[~-b].append(~-a)\\n    for _ in range(q):\\n        p, x = read_ints()\\n        c[~-p] += x\\n    que = deque([0])\\n    seen = [False] * n\\n    while que:\\n        p = que.pop()\\n        seen[p] = True\\n        for x in tree[p]:\\n            if seen[x]: continue\\n            que.append(x)\\n            c[x] += c[p]\\n    \\n    print(*c)\\n\\ndef __starting_point():\\n    Main()\\n__starting_point()\", \"from collections import defaultdict\\nN,Q = list(map(int,input().split()))\\nd = defaultdict(list)\\n\\nfor _ in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    d[a].append(b)\\n    d[b].append(a)\\n\\nc = defaultdict(int)\\n\\nfor _ in range(Q):\\n    p,x = list(map(int,input().split()))\\n    c[p] += x\\n\\nans = [0 for i in range(N)]\\nque = [(1,0,-1)]\\nwhile len(que)>0:\\n    tmp = []\\n    for v,x,par in que:\\n        x += c[v]\\n        ans[v-1] = str(x)\\n        for w in d[v]:\\n            if w != par:\\n                tmp.append((w,x,v))\\n    que = tmp\\n\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\nn,q=map(int,input().split())\\nG=[[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    G[a].append(b)\\n    G[b].append(a)\\n\\nsubtree_size=[0]*n\\ncnt=[0]*n\\n\\nfor _ in range(q):\\n    p,x=map(int,input().split())\\n    p-=1\\n    cnt[p]+=x\\nans=[0]*n\\n\\ndef dfs(G,v,p):\\n    ans[v]+=cnt[v]\\n    for nv in G[v]:\\n        if nv==p:\\n            continue\\n        cnt[nv]+=cnt[v]\\n        dfs(G,nv,v)\\n    \\n    subtree_size[v]=1\\n    for c in G[v]:\\n        if c==p:\\n            continue\\n        subtree_size[v]+=subtree_size[c]\\n\\nroot=0\\ndfs(G,root,-1)\\n\\nfor i in range(n):\\n    print(ans[i],end=' ')\\n\\n\\n\\n\", \"import sys\\nimport math\\n\\n#https://atcoder.jp/contests/agc008/submissions/15248942\\nsys.setrecursionlimit(10 ** 8)\\nini = lambda: int(sys.stdin.readline())\\ninm = lambda: map(int, sys.stdin.readline().split())\\ninl = lambda: list(inm())\\nins = lambda: sys.stdin.readline().rstrip()\\ndebug = lambda *a, **kw: print(\\\"\\\\033[33m\\\", *a, \\\"\\\\033[0m\\\", **dict(file=sys.stderr, **kw))\\n\\nclass Node:\\n    def __init__(self):\\n        self.edge = []\\n        self.count = 0\\n\\ndef dfs(tree,node,p_node,p_count,counts):\\n    count = tree[node].count + p_count\\n    counts[node] = count\\n    for x in tree[node].edge:\\n        if x == p_node:\\n            continue\\n        dfs(tree,x,node,count,counts)\\n\\nN,Q = inm()\\n\\ntree = []\\nfor _ in range(N):\\n    tree.append(Node())\\n\\n#node = Node()\\n#tree = [node]*N\\n\\nfor _ in range(N-1):\\n    a,b = inm()\\n    tree[a-1].edge.append(b-1)\\n    tree[b-1].edge.append(a-1)\\n\\nfor _ in range(Q):\\n    p,x = inm()\\n    tree[p-1].count += x\\n\\nval = []\\nfor _ in range(N):\\n    val.append(0)\\n\\n#for i in range(N):\\n#    print(tree[i].edge)\\n#    print(tree[i].count)\\n\\ndfs(tree,0,-1,0,val)\\n\\ncounts_str = [str(n) for n in val]\\ns = \\\" \\\".join(counts_str)\\nprint(s)\\n\", \"from collections import defaultdict\\nfrom sys import setrecursionlimit # \\u518d\\u5e30\\u30a8\\u30e9\\u30fc\\u5bfe\\u7b56\\nsetrecursionlimit(10 ** 7)\\n\\nn,q = map(int,input().split())\\n\\nnode = [[] for _ in range(n+1)]\\ncounter = [0 for _ in range(n+1)]\\nvisited = [0 for _ in range(n+1)]\\npoint_dic = defaultdict(int)\\n\\nfor _ in range(n-1):\\n    a,b = map(int,input().split())\\n    node[a].append(b)\\n    node[b].append(a)\\n\\n# \\u5404\\u30ce\\u30fc\\u30c9\\u306e\\u52a0\\u7b97\\u5024\\u3092\\u8a18\\u9332\\nfor _ in range(q):\\n    p,x = map(int,input().split())\\n    point_dic[p] += x\\n\\n# \\u73fe\\u5728\\u306e\\u30ce\\u30fc\\u30c9(now)\\u3068\\u305d\\u308c\\u307e\\u3067\\u306b\\u52a0\\u7b97\\u3055\\u308c\\u305f\\u5024(before_point)\\ndef dfs(now,before_point):\\n    if visited[now]:\\n        return\\n    \\n    # \\u73fe\\u5728\\u306e\\u30ce\\u30fc\\u30c9\\u306e\\u52a0\\u7b97\\u5024\\u3092\\u8db3\\u3059\\n    if point_dic[now]:\\n        point = point_dic[now]+before_point\\n    else:\\n        point = before_point\\n    \\n    # \\u8a2a\\u554f\\u6e08\\u307f\\u306b\\u3059\\u308b\\n    visited[now] = 1\\n    counter[now] += point\\n\\n    for i in node[now]:\\n        if visited[i]:\\n            continue\\n\\n        dfs(i,point) # \\u518d\\u5e30\\u3057\\u3066\\u5b50\\u30ce\\u30fc\\u30c9\\u306b\\u5f15\\u304d\\u7d99\\u3050\\n\\ndfs(1,0)\\n\\nprint(' '.join(map(str, counter[1:])))\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\n\\ndef input():\\n    return sys.stdin.readline()[:-1]\\nN , Q = map(int,input().split())\\ngraph = [[] for _ in range(N)]\\npoint = [0] * N\\nfor _ in range(N - 1):\\n    a , b = map(int,input().split())\\n    graph[a - 1].append(b - 1)\\n    graph[b - 1].append(a - 1)\\n#print(graph)\\nfor _ in range(Q):\\n    a , b = map(int,input().split())\\n    a = a - 1\\n    point[a] += b\\n# dfs\\u3092\\u7528\\u3044\\u3066\\u7d2f\\u7a4d\\u548c\\u3092\\u8a08\\u7b97\\u3059\\u308b\\n# \\u521d\\u671f\\u72b6\\u614b\\u3060\\u3068\\u524d\\u306e\\u5024\\u304c\\u306a\\u3044\\u305f\\u3081\\u30c7\\u30d5\\u30a9\\u30eb\\u30c8\\u5f15\\u6570\\u306b-1\\u3092\\u4ee3\\u5165\\ndef dfs(now , prev = -1):\\n    for next in graph[now]:\\n        # \\u6b21\\u306e\\u30ce\\u30fc\\u30c9\\u304c\\u524d\\u306b\\u53c2\\u7167\\u3057\\u305f\\u5024\\u306e\\u6642\\u306fcontinue\\n        if next == prev:\\n            continue\\n        # \\u73fe\\u5728\\u306e\\u5024\\u3092\\u6b21\\u306e\\u30dd\\u30a4\\u30f3\\u30c8\\u306b\\u52a0\\u7b97\\u3059\\u308b\\u3053\\u3068\\u3067\\u7d2f\\u7a4d\\u548c\\u3092\\u3068\\u308b\\n        point[next] += point[now]\\n        # \\u6b21\\u306e\\u30ce\\u30fc\\u30c9\\u3068\\u73fe\\u5728\\u306e\\u30ce\\u30fc\\u30c9\\u3092\\u5f15\\u6570\\u306bdfs\\u3092\\u7d99\\u7d9a\\u3059\\u308b\\n        dfs(next , now)\\n\\ndfs(0)\\nprint(*point)\"]",
        "difficulty": "interview",
        "input": "4 3\n1 2\n2 3\n2 4\n2 10\n1 100\n3 1\n",
        "output": "100 110 111 110\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc138/tasks/abc138_d"
    },
    {
        "id": 1070,
        "task_id": 2653,
        "test_case_id": 2,
        "question": "Given is a rooted tree with N vertices numbered 1 to N.\nThe root is Vertex 1, and the i-th edge (1 \\leq i \\leq N - 1) connects Vertex a_i and b_i.\nEach of the vertices has a counter installed. Initially, the counters on all the vertices have the value 0.\nNow, the following Q operations will be performed:\n - Operation j (1 \\leq j \\leq Q): Increment by x_j the counter on every vertex contained in the subtree rooted at Vertex p_j.\nFind the value of the counter on each vertex after all operations.\n\n-----Constraints-----\n - 2 \\leq N \\leq 2 \\times 10^5\n - 1 \\leq Q \\leq 2 \\times 10^5\n - 1 \\leq a_i < b_i \\leq N\n - 1 \\leq p_j \\leq N\n - 1 \\leq x_j \\leq 10^4\n - The given graph is a tree.\n - All values in input are integers.\n\n-----Input-----\nInput is given from Standard Input in the following format:\nN Q\na_1 b_1\n:\na_{N-1} b_{N-1}\np_1 x_1\n:\np_Q x_Q\n\n-----Output-----\nPrint the values of the counters on Vertex 1, 2, \\ldots, N after all operations, in this order, with spaces in between.\n\n-----Sample Input-----\n4 3\n1 2\n2 3\n2 4\n2 10\n1 100\n3 1\n\n-----Sample Output-----\n100 110 111 110\n\nThe tree in this input is as follows:\n\nEach operation changes the values of the counters on the vertices as follows:\n - Operation 1: Increment by 10 the counter on every vertex contained in the subtree rooted at Vertex 2, that is, Vertex 2, 3, 4. The values of the counters on Vertex 1, 2, 3, 4 are now 0, 10, 10, 10, respectively.\n - Operation 2: Increment by 100 the counter on every vertex contained in the subtree rooted at Vertex 1, that is, Vertex 1, 2, 3, 4. The values of the counters on Vertex 1, 2, 3, 4 are now 100, 110, 110, 110, respectively.\n - Operation 3: Increment by 1 the counter on every vertex contained in the subtree rooted at Vertex 3, that is, Vertex 3. The values of the counters on Vertex 1, 2, 3, 4 are now 100, 110, 111, 110, respectively.",
        "solutions": "[\"import sys\\nsys.setrecursionlimit(10**7)\\n \\nN, Q = list(map(int, input().split()))\\nroot = [[] for _ in range(N)]\\nans = [0]*N\\n \\nfor _ in range(N-1):\\n  a,b=list(map(int, input().split()))\\n  a,b=a-1,b-1\\n  root[a].append(b)\\n  root[b].append(a)\\n\\nfor _ in range(Q):\\n  p,x=list(map(int, input().split()))\\n  ans[p-1]+=x\\n  \\nvisited=[False]*N\\ndef dfs(v):\\n  visited[v] = True\\n  for go in root[v]:\\n    if visited[go]:\\n      continue\\n    ans[go] += ans[v]\\n    dfs(go)\\n\\ndfs(0)\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\n\\ndef solve():\\n    N, Q = list(map(int, input().split()))\\n    graph = [[] for _ in range(N)]\\n    for i in range(N - 1):\\n        a, b = [int(x) - 1 for x in input().split()]\\n        graph[a].append(b)\\n        graph[b].append(a)\\n    ans = [0] * N\\n    for i in range(Q):\\n        p, x = list(map(int, input().split()))\\n        p -= 1\\n        ans[p] += x\\n    def dfs(cur, pre):\\n        for to in graph[cur]:\\n            if to == pre:\\n                continue\\n            ans[to] += ans[cur]\\n            dfs(to, cur)\\n    dfs(0, -1)\\n    print((*ans))\\n    return\\n\\nsolve()\\n\", \"def main():\\n    n,q=map(int,input().split())\\n    ab=[list(map(int,input().split())) for _ in range(n-1)]\\n    tree=[list() for _ in range(n)]\\n    score=[0]*n\\n    for a,b in ab:\\n        a,b=a-1,b-1\\n        tree[a].append(b)\\n        tree[b].append(a)\\n    e = [0]\\n    while len(e) > 0:\\n        i = e.pop()\\n        for j in tree[i]:\\n            tree[j].remove(i)\\n            e.append(j)\\n    px=[list(map(int,input().split())) for _ in range(q)]\\n    for p,x in px:\\n        p-=1\\n        score[p]+=x\\n    add(tree,score)\\n    print(*score)\\n\\ndef add(tree,score):\\n    s=tree[0][:]\\n    for i in s:\\n        score[i] += score[0]\\n    while len(s)>0:\\n        t = s.pop()\\n        for i in tree[t]:\\n            s.append(i)\\n            score[i] += score[t]\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"import collections,sys\\ndef LI(): return list(map(int,sys.stdin.readline().rstrip().split()))\\nN,Q = LI()\\nab = [LI() for _ in [None]*(N-1)]\\npx = [LI() for _ in [None]*(Q)]\\nans = [0]*(N+1) #1_indexed\\ngraph = {i:collections.deque() for i in range(1,N+1)} #1_indexed\\nfor a,b in ab:\\n        graph[a].append(b)\\n        graph[b].append(a)\\nfor p,x in px:\\n    ans[p] += x\\nseen = [0]*(N+1) #1_indexed\\nstack = []\\ndef dfs():\\n    seen[1] = 1\\n    stack.append(1)\\n    while stack:\\n        s = stack.pop()\\n        if not graph[s]:\\n            continue\\n        for j in range(len(graph[s])):\\n            g_NO = graph[s].popleft()\\n            if seen[g_NO]:\\n                continue\\n            seen[g_NO] = 1\\n            stack.append(g_NO)\\n            ans[g_NO] += ans[s]\\ndfs()\\nprint(*ans[1:])\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN,Q=map(int,input().split())\\nT=[[] for _ in range(N)]\\nfor _ in range(N-1):\\n  a,b=map(int,input().split())\\n  a-=1;b-=1\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nV=[0]*N\\nfor _ in range(Q):\\n  p,x=map(int,input().split())\\n  V[p-1]+=x\\n  \\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next!=prev:\\n      V[next]+=V[now]\\n      dfs(next,now)\\n\\ndfs(0)\\nprint(*V)\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10 ** 6)\\n\\nn, q = list(map(int, input().split()))\\nG = [[]*n for i in range(n)]\\nfor _ in range(n-1):\\n    a, b = list(map(int, input().split()))\\n    G[a-1].append(b-1)\\n    G[b-1].append(a-1)\\n\\n\\ndone = [False]*n\\npoints = [0]*n\\nd = defaultdict(lambda: 0)\\n\\nfor _ in range(q):\\n    a, b = list(map(int, input().split()))\\n    d[a-1] += b\\n\\n\\ndef dfs(i, cnt):\\n    cnt += d[i]\\n    points[i] += cnt\\n    done[i] = True\\n    for j in G[i]:\\n        if not done[j]:\\n            dfs(j, cnt)\\n\\n\\ndfs(0, 0)\\nprint((*points))\\n\", \"N,Q = map(int,input().split())\\nf = [[] for i in range(N)]\\nfor i in range(N-1):\\n    a,b = map(int,input().split())\\n    f[a-1].append(b-1)\\n    f[b-1].append(a-1)\\nans = [0] * N\\nfor i in range(Q):\\n    p,x = map(int,input().split())\\n    ans[p-1] += x\\nfrom collections import deque\\nd = deque()\\nd.append(0)\\nroot = [-1]*N\\nroot[0] = 0\\nwhile len(d) > 0:\\n    z = d.popleft()\\n    for i in f[z]:\\n        if root[i] == -1:\\n            root[i] = z\\n            ans[i] += ans[z]\\n            d.append(i)\\nprint(*ans)\", \"import sys\\nsys.setrecursionlimit(1<<30)\\ndef dfs(x):\\n    for y in edges[x]:\\n        if y != parent[x]:\\n            parent[y] = x\\n            count[y] += count[x]\\n            dfs(y)\\n\\nN, Q = map(int, input().split())\\nedges = [[] for _ in range(N+1)]\\ncount = [0]*(N+1)\\nfor i in range(N-1):\\n    a, b = map(int, input().split())\\n    edges[a].append(b)\\n    edges[b].append(a)\\nfor i in range(Q):\\n    p, x = map(int, input().split())\\n    count[p] += x\\n\\nparent = [-1]*(N+1)\\ndfs(1)\\nprint(*count[1:])\", \"def abc138d_ki():\\n    import heapq\\n    n, q = map(int, input().split())\\n    cnt = [0] * n\\n    graph = [[] for _ in range(n)]\\n    for _ in range(n - 1):\\n        a, b = map(int, input().split())\\n        graph[a - 1].append(b - 1)\\n        graph[b - 1].append(a - 1)\\n    for _ in range(q):\\n        p, x = map(int, input().split())\\n        cnt[p - 1] += x\\n    qu = [(0, 0)]\\n    heapq.heapify(qu)\\n    check = [False] * n\\n    while len(qu) != 0:\\n        h, no = heapq.heappop(qu)\\n        check[no] = True\\n        for nxt in graph[no]:\\n            if not check[nxt]:\\n                cnt[nxt] += cnt[no]\\n                heapq.heappush(qu, (h + 1, nxt))\\n    for c in cnt:\\n        print(c, end=' ')\\n\\n\\nabc138d_ki()\", \"import sys\\nsys.setrecursionlimit(10**9) #\\u518d\\u5e30\\u306e\\u4e0a\\u9650\\u6307\\u5b9a\\n# \\u6728\\u69cb\\u9020\\u306e\\u6a19\\u6e96\\u5165\\u529b\\nN,Q = list(map(int,input().split()))\\nT=[[] for _ in range(N)]\\nfor _ in range(N-1):\\n  a,b=list(map(int,input().split()))\\n  a-=1;b-=1\\n  T[a].append(b)\\n  T[b].append(a)\\n\\n# \\u5165\\u529b\\u4f8b1\\u306e\\u3068\\u304d\\u3001\\u7e4b\\u304c\\u308b\\u9802\\u70b9\\u306e\\u30ea\\u30b9\\u30c8 [[1], [0, 2, 3], [1], [1]]\\n# \\u8db3\\u7b97\\u90e8\\u5206\\u306e\\u6a19\\u6e96\\u5165\\u529b\\nV=[0]*N\\nfor _ in range(Q):\\n  p,x=list(map(int,input().split()))\\n  V[p-1]+=x\\n# \\u5165\\u529b\\u4f8b1\\u306e\\u3068\\u304d\\u3001\\u5404\\u30ce\\u30fc\\u30c9\\u4ee5\\u4e0b\\u306b\\u8db3\\u3059\\u5024 [100, 10, 1, 0]\\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next!=prev:# \\u89aa\\u3058\\u3083\\u306a\\u3051\\u308c\\u3070\\u3001(\\u5b50\\u3067\\u3042\\u308c\\u3070)\\u8db3\\u3057\\u7b97\\u3059\\u308b\\n      V[next]+=V[now]\\n      dfs(next,now)\\n\\ndfs(0)\\n\\\"\\\"\\\"\\n\\u5165\\u529b\\u4f8b1\\u306e\\u3068\\u304d\\ndfs(0,-1)\\u2192dfs(1,0)\\u2192dfs(2,1)\\n                  \\u2192dfs(3,1)\\n\\\"\\\"\\\"\\nprint((*V))\\n\", \"from collections import deque\\nN, Q = list(map(int, input().split()))\\ntree = [[] for _ in range(N+1)]\\ncounter = [0]*(N+1)\\nfor i in range(N-1):\\n    a, b = list(map(int, input().split()))\\n    tree[a].append(b)\\n    tree[b].append(a)\\nfor i in range(Q):\\n    p, x = list(map(int, input().split()))\\n    counter[p] += x\\nparent = [-1]*(N+1)\\nd = deque([1])\\n\\nwhile d:\\n    a = d.popleft()\\n    for b in tree[a]:\\n        if parent[a] != b:\\n            parent[b] = a\\n            counter[b] += counter[a]\\n            d.append(b)\\n\\nprint((*counter[1:]))\\n\", \"from collections import deque\\n\\n\\ndef main():\\n  n,q=map(int,input().split())\\n  node=[[]*n for i in range(n)]\\n  for i in range(n-1):\\n    a,b=map(lambda x:int(x)-1,input().split())\\n    node[b].append(a)\\n    node[a].append(b)\\n  \\n  ans=[0]*n\\n  for i in range(q):\\n    p,x=map(int,input().split())\\n    p-=1\\n    ans[p]+=x\\n\\n  root=[0]*n\\n  edge=[[]*n for i in range(n)]\\n  root[0]=0\\n  q=deque([0])\\n  visited=[False]*n\\n  visited[0]=True\\n  \\n  while q:\\n    r=q.popleft()\\n    for e in node[r]:\\n      if not visited[e]:\\n        visited[e]=True\\n        q.append(e)\\n        root[e]=r\\n        edge[r].append(e)\\n  \\n  \\n  p=[0]*n\\n  q=deque(edge[0])\\n  visited=[False]*n\\n  while q:\\n    e=q.popleft()\\n    p[e]=p[root[e]]+1\\n    for e2 in edge[e]:\\n      if not visited[e2]:\\n        visited[e2]=True\\n        q.append(e2)\\n  \\n  p=sorted(enumerate(p), key=lambda x:x[1])\\n  for i in range(1,n):\\n    e=p[i][0]\\n    r=root[e]\\n    ans[e]+=ans[r]\\n    \\n  print(*ans,sep=' ')\\n\\n\\ndef __starting_point():\\n  main()\\n__starting_point()\", \"import sys\\nimport math\\n\\n#https://atcoder.jp/contests/agc008/submissions/15248942\\nsys.setrecursionlimit(10 ** 8)\\nini = lambda: int(sys.stdin.readline())\\ninm = lambda: map(int, sys.stdin.readline().split())\\ninl = lambda: list(inm())\\nins = lambda: sys.stdin.readline().rstrip()\\ndebug = lambda *a, **kw: print(\\\"\\\\033[33m\\\", *a, \\\"\\\\033[0m\\\", **dict(file=sys.stderr, **kw))\\n\\ncounts = []\\n\\nclass Node:\\n    def __init__(self):\\n        self.edge = []\\n        self.count = 0\\n\\ndef dfs(tree,node,p_node,p_count):\\n    count = tree[node].count + p_count\\n    counts[node] = count\\n    for x in tree[node].edge:\\n        if x == p_node:\\n            continue\\n        dfs(tree,x,node,count)\\n\\nN,Q = inm()\\n\\ntree = []\\nfor _ in range(N):\\n    tree.append(Node())\\n\\n#node = Node()\\n#tree = [node]*N\\n\\nfor _ in range(N-1):\\n    a,b = inm()\\n    tree[a-1].edge.append(b-1)\\n    tree[b-1].edge.append(a-1)\\n\\nfor _ in range(Q):\\n    p,x = inm()\\n    tree[p-1].count += x\\n\\nfor _ in range(N):\\n    counts.append(0)\\n\\n#for i in range(N):\\n#    print(tree[i].edge)\\n#    print(tree[i].count)\\n\\n\\ndfs(tree,0,-1,0)\\n\\ncounts_str = [str(n) for n in counts]\\ns = \\\" \\\".join(counts_str)\\nprint(s)\\n\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\nN, Q = map(int, input().split())\\n\\nG = [[] for i in range(N)]\\n\\nfor _ in range(N - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    G[b].append(a)\\n    G[a].append(b)\\n\\nans = [0] * N\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    p -= 1\\n    ans[p] += x\\n\\ndef dfs(v, p):\\n    for to in G[v]:\\n        if to == p: continue\\n        ans[to] += ans[v]\\n        dfs(to, v)\\n\\ndfs(0, -1)\\nfor i in range(N):\\n    print(ans[i],end=' ')\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef dfs(tree, n, ans):\\n    for i in tree[n]:\\n        ans[i] += ans[n]\\n        tree[i].remove(n)\\n        dfs(tree, i, ans)\\n\\nn, q = map(int,input().split())\\nab = [list(map(int,input().split())) for _ in range(n-1)]\\npx = [list(map(int,input().split())) for _ in range(q)]\\n\\ng = [[] for _ in range(n)]\\nfor i in range(n-1):\\n    g[ab[i][0]-1].append(ab[i][1]-1)\\n    g[ab[i][1]-1].append(ab[i][0]-1)\\n\\nans = [0]*n\\nfor i in range(q):\\n    ans[px[i-1][0]-1] += px[i-1][1]\\ndfs(g, 0, ans)\\n\\nprint(*ans)\", \"import sys\\nread = sys.stdin.read\\nsys.setrecursionlimit(10**7)\\n#readlines = sys.stdin.readlines\\ndef main():\\n    def dfs(v, s):\\n        if added[v]:\\n            pass\\n        else:\\n            scores[v] += s\\n            added[v] = 1\\n            for nextv in gg[v]:\\n                dfs(nextv, scores[v])\\n\\n    data = tuple(map(int, read().split()))\\n    n, q = data[0], data[1]\\n    gg = {i: set() for i in range(1, n + 1)}\\n    for i1, v1 in zip(data[2:n*2:2], data[3:n*2:2]):\\n        gg[v1].add(i1)\\n        gg[i1].add(v1)\\n    scores = [0] * (n + 1)\\n    for j1, v1 in zip(data[n*2::2], data[n*2+1::2]):\\n        scores[j1] += v1\\n\\n    added = [0] * (n + 1)\\n    dfs(1, 0)\\n    print(*scores[1:], sep=' ')\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"n,q = map(int,input().split())\\ngraph = [[] for i in range(n)]\\nfor i in range(n-1):\\n  a,b = map(int,input().split())\\n  a -= 1\\n  b -= 1\\n  graph[a].append(b)\\n  graph[b].append(a)\\n\\nweight = [0 for i in range(n)]\\nfor i in range(q):\\n  p,x = map(int,input().split())\\n  weight[p-1] += x\\n\\nst = [0]\\nvisit = [False for i in range(n)]\\nwhile not len(st) == 0:\\n  now = st.pop()\\n  visit[now] = True\\n  for e in graph[now]:\\n    if visit[e]:\\n      continue\\n    weight[e] += weight[now]\\n    st.append(e)\\n    \\nprint(' '.join(map(str,weight)))\", \"\\nimport sys\\nsys.setrecursionlimit(10 ** 6)\\ndef main2():\\n    n, q = list(map(int, input().split()))\\n    tree = [[] for i in range(n)]\\n    point = [0] * n\\n\\n    for _ in range(n - 1):\\n        a, b = list(map(int, input().split()))\\n        tree[b - 1].append(a - 1)\\n        tree[a - 1].append(b - 1)\\n    for i in range(q):\\n        p, x = list(map(int, input().split()))\\n        point[p - 1] += x\\n\\n    def dfs(v, parent):\\n        nonlocal point, tree\\n        for next_val in tree[v]:\\n            if next_val == parent:\\n                continue\\n            point[next_val] += point[v]\\n            dfs(next_val, v)\\n    dfs(0, 0)\\n    print((*point))\\n\\n\\ndef __starting_point():\\n    main2()\\n\\n__starting_point()\", \"# abc138 D ki\\n\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\n# def input():\\n#     return sys.stdin.readline()[:-1]\\n\\nN,Q = map(int,input().split())\\ng=[[] for _ in range(N)]\\n\\npoint=[0]*N\\n\\nfor _ in range(N-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    g[a].append(b)\\n    g[b].append(a)\\n    \\nfor _ in range(Q):\\n    a,b=map(int,input().split())\\n    a-=1\\n    point[a]+=b\\n\\ndef dfs(n,pre=-1):\\n    for ne in g[n]:\\n        if ne == pre:\\n            continue\\n        \\n        point[ne]+=point[n]\\n        \\n        dfs(ne,n)\\n        \\ndfs(0)\\nprint(*point)\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(v,prev = -1):\\n    for u in graph[v]:\\n        if u == prev:\\n            continue\\n        point[u] += point[v]\\n        dfs(u,v)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [[] for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"from collections import defaultdict\\nN,Q = list(map(int,input().split()))\\nd = defaultdict(list)\\n\\nfor _ in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    d[a].append(b)\\n    d[b].append(a)\\n\\nc = defaultdict(int)\\n\\nfor _ in range(Q):\\n    p,x = list(map(int,input().split()))\\n    c[p] += x\\n\\nans = [0 for i in range(N)]\\nque = [(1,0,-1)]\\nwhile len(que)>0:\\n    tmp = []\\n    for v,x,par in que:\\n        x += c[v]\\n        ans[v-1] = str(x)\\n        for w in d[v]:\\n            if w != par:\\n                tmp.append((w,x,v))\\n    que = tmp\\n\\nprint((' '.join(ans)))\\n\", \"import sys\\nsys.setrecursionlimit(500000)\\n\\nn, q = list(map(int, input().split()))\\nab = [[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a, b = list(map(int, input().split()))\\n    ab[a-1].append(b-1)\\n    ab[b-1].append(a-1)\\nadd = [0]*n\\nfor _ in range(q):\\n    pp, xx = list(map(int, input().split()))\\n    add[pp-1] += xx\\n\\n\\ndef dfs(v, p, value):\\n    value += add[v]\\n    ans[v] = value\\n    for c in ab[v]:\\n        if c == p:\\n            continue\\n        dfs(c, v, value)\\n\\n\\nans = [0]*n\\ndfs(0, -1, 0)\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)# \\u5404\\u9802\\u70b9\\u3054\\u3068\\u306b\\u6240\\u5c5e\\u3059\\u308b\\u6728\\u306e\\u5927\\u304d\\u3055\\u3092\\u8a08\\u7b97\\u3059\\u308b\\nN,Q = list(map(int,input().split()))\\n\\nE = [[] for _ in range(N+1)]\\nvisited = [False for _ in range(N+1)]\\nNode_len = [1 for _ in range(N+1)]\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nAS = [0 for i in range(N+1)]\\ndef dfs(node,parent=-1):\\n    for child in E[node]:\\n        if child == parent:\\n            continue\\n        AS[child] += AS[node]\\n        dfs(child,node)\\n\\nfor q in range(Q):\\n    p,x = list(map(int,input().split()))\\n    AS[p] += x\\n\\ndfs(1)\\nprint((*AS[1:]))\\n\\n\\n\", \"import sys\\nfrom collections import deque\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(s):\\n    stack = [s]\\n    visited = [False]*N\\n    while stack:\\n        v = stack.pop()\\n        if visited[v]:\\n            continue\\n        visited[v] = True\\n        for u in graph[v]:\\n            if not visited[u]:\\n                point[u] += point[v]\\n                stack.append(u)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [deque([]) for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"from collections import defaultdict\\n\\nN, Q = list(map(int, input().split()))\\nd = defaultdict(list)\\n\\nfor _ in range(N - 1):\\n    a, b = list(map(int, input().split()))\\n    d[a].append(b)\\n    d[b].append(a)\\n\\nc = defaultdict(int)\\n\\nfor _ in range(Q):\\n    p, x = list(map(int, input().split()))\\n    c[p] += x\\n\\nans = [0 for i in range(N)]\\nque = [(1, 0, -1)]\\nwhile len(que) > 0:\\n    tmp = []\\n    for v, x, par in que:\\n        x += c[v]\\n        ans[v - 1] = str(x)\\n        for w in d[v]:\\n            if w != par:\\n                tmp.append((w, x, v))\\n    que = tmp\\n\\nprint((*ans))\\n\", \"n,q=map(int,input().split())\\ne=[[] for i in range(n)]\\nfor i in range(n-1):\\n  a,b=map(lambda x:int(x)-1,input().split())\\n  e[a].append(b)\\n  e[b].append(a)\\nps=[0 for i in range(n)]\\nfor i in range(q):\\n  p,x=map(int,input().split())\\n  ps[p-1]+=x\\n\\ns=[-1 for i in range(n)]\\nsc=[0]\\ns[0]=0\\nwhile sc:\\n  nsc=[]\\n  for i in sc:\\n    s[i]+=ps[i]\\n    for j in e[i]:\\n      if s[j]==-1:\\n        nsc.append(j)\\n        s[j]=s[i]\\n  sc=nsc\\n\\nr=\\\"\\\"\\nfor i in range(n):\\n  r+=str(s[i])+\\\" \\\"\\nprint(r[:-1])\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(v,prev = -1):\\n    for u in graph[v]:\\n        if u == prev:\\n            continue\\n        point[u] += point[v]\\n        dfs(u,v)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [[] for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"from collections import deque\\n\\n\\ndef nearlist(N, LIST):\\n    NEAR = [set() for _ in range(N)]\\n    for a, b in LIST:\\n        NEAR[a - 1].add(b - 1)\\n        NEAR[b - 1].add(a - 1)\\n    return NEAR\\n\\n\\ndef bfs(NEAR):\\n    que, frag = deque([0]), set([0])\\n    while que:\\n        q = que.popleft()\\n        for i in NEAR[q]:\\n            if i in frag:\\n                continue\\n            ans[i] += ans[q]\\n            que.append(i), frag.add(i)\\n    return\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n - 1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nans = [0] * n\\nfor p, x in px:\\n    ans[p - 1] += x\\n\\nnear = nearlist(n, ab)\\nbfs(near)\\nprint(*ans)\", \"#!/usr/bin/env python3\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 8)\\ninput = sys.stdin.readline\\n\\n\\ndef dfs(now):\\n    seen[now] = True\\n    for next in branch[now]:\\n        if seen[next] is False:\\n            score[next] += score[now]\\n            dfs(next)\\n\\n\\nN, Q = list(map(int, input().split()))\\nbranch = [set() for _ in range(N)]\\nfor _ in range(N - 1):\\n    a, b = list(map(int, input().split()))\\n    branch[a - 1].add(b - 1)\\n    branch[b - 1].add(a - 1)\\nscore = [0] * N\\nfor _ in range(Q):\\n    p, x = list(map(int, input().split()))\\n    score[p - 1] += x\\n\\nseen = [False] * N\\nfor i in range(N):\\n    if seen[i] is False:\\n        dfs(i)\\n\\nprint((*score))\\n\", \"from collections import deque\\nn,q = map(int,input().split())\\nz = [[] for i in range(n+1)]\\nfor i in range(n-1):\\n    a,b = map(int,input().split())\\n    z[a].append(b)\\n    z[b].append(a)\\nl = [list(map(int,input().split())) for i in range(q)]\\np,x = [list(i) for i in zip(*l)]\\nc = [0] * n\\nfor i in range(q):\\n    c[p[i]-1] += x[i]\\nqueue = deque([1])\\ncheck = [1] * (n+1)\\nwhile queue:\\n    p = queue.popleft()\\n    if check[p]:\\n        for i in z[p]:\\n            if check[i]:\\n                c[i-1] += c[p-1]\\n                queue.append(i)\\n    check[p] = 0\\nprint(' '.join(map(str,c)))\", \"#\\n# abc138 d\\n#\\nimport sys\\nfrom io import StringIO\\nimport unittest\\nfrom collections import deque\\n\\n\\nclass TestClass(unittest.TestCase):\\n    def assertIO(self, input, output):\\n        stdout, stdin = sys.stdout, sys.stdin\\n        sys.stdout, sys.stdin = StringIO(), StringIO(input)\\n        resolve()\\n        sys.stdout.seek(0)\\n        out = sys.stdout.read()[:-1]\\n        sys.stdout, sys.stdin = stdout, stdin\\n        self.assertEqual(out, output)\\n\\n    def test_\\u5165\\u529b\\u4f8b_1(self):\\n        input = \\\"\\\"\\\"4 3\\n1 2\\n2 3\\n2 4\\n2 10\\n1 100\\n3 1\\\"\\\"\\\"\\n        output = \\\"\\\"\\\"100 110 111 110\\\"\\\"\\\"\\n        self.assertIO(input, output)\\n\\n    def test_\\u5165\\u529b\\u4f8b_2(self):\\n        input = \\\"\\\"\\\"6 2\\n1 2\\n1 3\\n2 4\\n3 6\\n2 5\\n1 10\\n1 10\\\"\\\"\\\"\\n        output = \\\"\\\"\\\"20 20 20 20 20 20\\\"\\\"\\\"\\n        self.assertIO(input, output)\\n\\n\\ndef resolve():\\n    N, Q = list(map(int, input().split()))\\n    AB = [list(map(int, input().split())) for _ in range(N-1)]\\n    PX = [list(map(int, input().split())) for _ in range(Q)]\\n\\n    G = [[i+1, 0] for i in range(N)]\\n    for ab in AB:\\n        a, b = ab\\n        G[a-1][1] += 1\\n        G[b-1][1] += 1\\n        G[a-1].append(b)\\n        G[b-1].append(a)\\n\\n    ans = [0] * N\\n    for px in PX:\\n        p, x = px\\n        ans[p-1] += x\\n\\n    S = deque()\\n\\n    F = [False] * N\\n    S.append(1)\\n    F[0] = True\\n\\n    while S:\\n        p = S.pop()\\n        if G[p-1][1] == 0:\\n            continue\\n\\n        for np in G[p-1][2:]:\\n            if F[np-1]:\\n                continue\\n            S.append(np)\\n            F[np-1] = True\\n            ans[np-1] += ans[p-1]\\n\\n    print((*ans))\\n\\n\\ndef __starting_point():\\n    # unittest.main()\\n    resolve()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10 ** 9)\\nN, Q = map(int, input().split())\\n\\nAB_TREE = [[] for i in range(N)]\\n\\nfor _ in range(N - 1):\\n    a, b = map(int, input().split())\\n    a -= 1\\n    b -= 1\\n    AB_TREE[b].append(a)\\n    AB_TREE[a].append(b)\\n\\nans = [0] * (N + 1)\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    p -= 1\\n    ans[p] += x\\n\\n\\ndef dfs(v, p):\\n    for to in AB_TREE[v]:\\n        if to == p: continue\\n        ans[to] += ans[v]\\n        dfs(to, v)\\n\\n\\ndfs(0, -1)\\nfor i in range(N):\\n    print(ans[i],end=' ')\\n\", \"import collections\\nN,Q = map(int,input().split())\\nlski = [[] for i in range(N+1)]\\nfor i in range(N-1):\\n    a,b = map(int,input().split())\\n    lski[a].append(b)\\n    lski[b].append(a)\\ncountercost = collections.Counter()\\nfor i in range(Q):\\n    p,x = map(int,input().split())\\n    countercost[p] += x\\n\\ndist = [-1] * (N+1)\\nparent = [-1] * (N+1)\\ndist[0] = 0\\ndist[1] = countercost[1]\\n\\nd = collections.deque()\\nd.append(1)\\n\\nwhile d:\\n    v = d.popleft()\\n    for i in lski[v]:\\n        if parent[a] == b:\\n            continue\\n        if dist[i] != -1:\\n            continue\\n        parent[b] = a\\n        dist[i] = dist[v] + countercost[i]\\n        d.append(i)\\ndist.pop(0)\\nansls = [str(i) for i in dist]\\nprint(' '.join(ansls))\", \"N,Q = map(int,input().split())\\nG = [[] for n in range(N)]\\nans = N*[0]\\n\\nfor n in range(N-1):\\n  a,b = map(int,input().split())\\n  G[a-1].append(b-1)\\n  G[b-1].append(a-1)\\n\\nfor q in range(Q):\\n  p,x = map(int,input().split())\\n  ans[p-1]+=x\\n\\nf = N*[1]\\nt = [0]\\nwhile t:\\n  v = t.pop()\\n  f[v] = 0\\n  for k in G[v]:\\n    if f[k]:\\n      ans[k]+=ans[v]\\n      t.append(k)\\n\\nprint(*ans)\", \"import sys\\nfrom collections import deque\\nsys.setrecursionlimit(10**7)\\ndef I(): return int(sys.stdin.readline().rstrip())\\ndef MI(): return list(map(int,sys.stdin.readline().rstrip().split()))\\ndef LI(): return list(map(int,sys.stdin.readline().rstrip().split()))  #\\u7a7a\\u767d\\u3042\\u308a\\ndef LI2(): return list(map(int,sys.stdin.readline().rstrip()))  #\\u7a7a\\u767d\\u306a\\u3057\\ndef S(): return sys.stdin.readline().rstrip()\\ndef LS(): return list(sys.stdin.readline().rstrip().split())  #\\u7a7a\\u767d\\u3042\\u308a\\ndef LS2(): return list(sys.stdin.readline().rstrip())  #\\u7a7a\\u767d\\u306a\\u3057\\n\\n\\nN,Q = MI()\\nGraph = [[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n    a,b = MI()\\n    Graph[a].append(b)\\n    Graph[b].append(a)\\n\\ncount = [0]*(N+1)\\nadd_count = [0]*(N+1)\\nfor i in range(Q):\\n    p,x = MI()\\n    add_count[p] += x\\n\\nflag = [-1]*(N+1)\\ncount[1] = add_count[1]\\ndeq = deque([(1,add_count[1])])\\nflag[1] = 0\\n\\nwhile deq:\\n    n,r = deq.pop()\\n    for d in Graph[n]:\\n        if flag[d] == -1:\\n            flag[d] = 0\\n            count[d] = r+add_count[d]\\n            deq.appendleft((d,count[d]))\\n\\nprint((*count[1:]))\\n\", \"# D - Ki TLE\\nimport sys\\nsys.setrecursionlimit(10**7)\\nN,Q = map(int,input().split())\\n\\n# \\u6709\\u5411\\u30b0\\u30e9\\u30d5\\nG = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    a -= 1; b -= 1\\n    G[a].append(b)\\n    G[b].append(a)\\n    \\nlst = []\\nfor _ in range(Q):\\n    P,X = map(int,input().split())\\n    P-=1\\n    lst.append((P,X))\\n\\ncnt = [0]*N\\nfor p,x in lst:\\n    cnt[p] += x \\n    \\nseen = [False]*N\\ndef dfs(v):\\n    seen[v] = True# v \\u3092\\u8a2a\\u554f\\u6e08\\u307f\\u306b\\u3059\\u308b\\n    # v \\u304b\\u3089\\u884c\\u3051\\u308b\\u5404\\u9802\\u70b9 next_v \\u306b\\u3064\\u3044\\u3066\\n    for next_v in G[v]:\\n        # next_v \\u304c\\u63a2\\u7d22\\u6e08\\u307f\\u306a\\u3089\\u30b9\\u30eb\\u30fc\\n        if seen[next_v] == True:\\n             continue\\n        cnt[next_v] += cnt[v] \\n        dfs(next_v)\\ndfs(0)\\nprint(*cnt)\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\n\\ndef dfs(v,prev = -1):\\n    for u in graph[v]:\\n        if u == prev:\\n            continue\\n        point[u] += point[v]\\n        dfs(u,v)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [[] for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"import sys\\nfrom typing import List, Tuple\\nimport collections\\n\\nGraph = List[List[int]]\\n\\n\\ndef main():\\n    def input(): return sys.stdin.readline()[:-1]\\n    N, Q = list(map(int, input().split()))\\n\\n    graph = [[] for _ in range(N+1)]\\n    for _ in range(1, N):\\n        a, b = list(map(int, input().split()))\\n        graph[a].append(b)\\n        graph[b].append(a)\\n\\n    px = [0] * (N + 1)\\n    for _ in range(Q):\\n        p, x = list(map(int, input().split()))\\n        px[p] += x\\n\\n    queue = collections.deque()\\n    queue.append(1)\\n    checked = [0] * (N + 1)\\n    while queue:\\n        v = queue.pop()\\n        checked[v] = 1\\n        for next_v in graph[v]:\\n            if checked[next_v] == 1: continue\\n            px[next_v] += px[v]\\n            queue.append(next_v)\\n    print((*px[1:]))\\n\\ndef __starting_point():\\n    main()\\n\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**7)\\n\\nn,q=map(int,input().split())\\nans=[0 for _ in range(n)]\\ngra=[[] for _ in range(n)]\\nvisited=[0 for _ in range(n)]\\n\\nfor _ in range(n-1):\\n  a,b=map(int,input().split())\\n  gra[a-1].append(b-1)\\n  gra[b-1].append(a-1)\\n  \\nfor _ in range(q):\\n  p,x=map(int,input().split())\\n  ans[p-1]+=x\\n  \\ndef dfs(st):\\n  temp=gra[st]\\n  for i in temp:\\n    if visited[i]==0:\\n      visited[i]+=1\\n      ans[i]+=ans[st]\\n      dfs(i)\\n      \\nvisited[0]+=1\\ndfs(0)\\n\\nfor h in range(n):\\n  ans[h]=str(ans[h])\\n  \\nprint(' '.join(ans))\", \"import collections\\n\\nn,q = map(int, input().split())\\n\\nvalue = ['-'] * (n+1)\\n\\nconnected = [set() for i in range(n+1)]\\n\\nfor i in range(n-1):\\n  a,b = map(int, input().split())\\n  connected[a].add(b)\\n  connected[b].add(a)\\n\\nfor i in range(q):\\n  p,x = map(int, input().split())\\n  if value[p] == '-':\\n    value[p] = 0\\n  value[p] -= x\\n\\nqueue = collections.deque(connected[1])\\n\\nif value[1] == '-':\\n  value[1] = 0\\nelse:\\n  value[1] = -value[1]\\n\\nfor i in queue:\\n  if value[i] == '-':\\n    value[i] = 0\\n  else:\\n    value[i] = -value[i]\\n  value[i] += value[1]\\n\\n\\nwhile queue:\\n  s = queue.popleft()\\n  v = value[s]\\n  for i in connected[s]:\\n    vi = value[i]\\n    if vi == '-' or vi < 0:\\n      if vi == '-':\\n        vi = 0\\n      value[i] = -(vi-v)\\n      queue.append(i)\\n    \\nprint(' '.join(str(i) for i in value[1:]))\", \"import sys\\nsys.setrecursionlimit(10**9)\\ndef f(p,q=-1):\\n  for i in c[p]:\\n    if i!=q:\\n      A[i]+=A[p]\\n      f(i,p)\\nn,q=map(int,input().split())\\nc=[[]for _ in range(n)]\\nfor i in range(n-1):\\n  a,b=map(int,input().split())\\n  c[a-1].append(b-1)\\n  c[b-1].append(a-1)\\nA=[0]*n\\nfor _ in range(q):\\n  p,x=map(int,input().split())\\n  A[p-1]+=x\\nf(0)\\nprint(*A)\", \"from collections import deque\\nn, q = list(map(int, input().split()))\\ngraphs = [set() for _ in range(n)]\\nfor _ in range(n - 1):\\n    a, b = [int(x) - 1 for x in input().split()]\\n    graphs[a].add(b)\\n    graphs[b].add(a)\\n\\nnodes = [0 for _ in range(n)]\\nfor _ in range(q):\\n    p, x = list(map(int, input().split()))\\n    nodes[p - 1] += x\\n\\ndone = [True for _ in range(n)]\\nD = deque()\\nD.append([0, 0])\\nans = [0 for _ in range(n)]\\nwhile D:\\n    node, value = D.popleft()\\n    value += nodes[node]\\n    ans[node] = value\\n    done[node] = False\\n    for g in graphs[node]:\\n        if done[g]:\\n            D.append([g, value])\\n\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN, Q = map(int, input().split())\\nT = [[] for _ in range(N)]\\n\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    T[a-1].append(b-1)\\n    T[b-1].append(a-1)\\n\\nS = [0] * N\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    S[p-1] += x\\n\\ndef search(p, q=-1):\\n    for a in T[p]:\\n        if a != q:\\n            S[a] += S[p]\\n            search(a, p)\\n\\nsearch(0)\\nprint(*S)\", \"def solve():\\n    import sys\\n    sys.setrecursionlimit(10**6)\\n    def dfs(tree, n, ans):\\n        for i in tree[n]:\\n            ans[i] += ans[n]\\n            tree[i].remove(n)\\n            dfs(tree, i, ans) #\\u518d\\u5e30\\u95a2\\u6570\\u306b\\u306a\\u3063\\u3066\\u3044\\u308b\\u3002\\u6b21\\u306e\\u6728\\u306e\\u4e2d\\u3092\\u63a2\\u7d22\\n     \\n    n, q = map(int,input().split())\\n    ab = [list(map(int,input().split())) for _ in range(n-1)]\\n    px = [list(map(int,input().split())) for _ in range(q)]\\n     \\n    g = [[] for _ in range(n)]\\n    for i in range(n-1):\\n        g[ab[i][0]-1].append(ab[i][1]-1)\\n        g[ab[i][1]-1].append(ab[i][0]-1)\\n    \\n    ans = [0]*n\\n    for i in range(q):\\n        ans[px[i-1][0]-1] += px[i-1][1]\\n    dfs(g, 0, ans)\\n     \\n    print(*ans)\\n    \\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"import sys\\nfrom functools import lru_cache\\nsys.setrecursionlimit(10**7)\\n \\nN,Q=map(int,input().split())\\nT=[[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n  a,b=map(int,input().split())\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nS=[0]*(N+1)\\nfor _ in range(Q):\\n  p,x=map(int,input().split())\\n  S[p]+=x\\n\\n@lru_cache(maxsize=None)\\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next==prev:\\n      continue\\n    S[next]+=S[now]\\n    dfs(next,now)\\n\\ndfs(1)\\nprint(*S[1:])\", \"import sys\\ninput = sys.stdin.readline\\nsys.setrecursionlimit(10 ** 7)\\n \\n# \\u9045\\u5ef6\\u8a55\\u4fa1\\u3067\\u52a0\\u3048\\u3066\\u3042\\u3052\\u308b\\u3060\\u3051\\n \\nN,Q = map(int,input().split())\\nAB = [[int(x) for x in input().split()] for _ in range(N-1)]\\nPX = [[int(x) for x in input().split()] for _ in range(Q)]\\n \\ngraph = [[] for _ in range(N+1)]\\nfor a,b in AB:\\n    graph[a].append(b)\\n    graph[b].append(a)\\n \\nvalue = [0] * (N+1)\\nfor p,x in PX:\\n    value[p] += x\\n \\ndef dfs(v,parent,add):\\n    value[v] += add\\n    for x in graph[v]:\\n        if x == parent:\\n            continue\\n        dfs(x,v,value[v])\\n \\ndfs(1,0,0)\\n \\nanswer = ' '.join(map(str,value[1:]))\\nprint(answer)\", \"import collections\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n-1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nc = [[] for _ in range(n)]\\nfor a, b in ab:\\n    a, b = a-1, b-1\\n    c[a].append(b)\\n    c[b].append(a)\\n\\npoint = [0] * n\\nfor p, x in px:\\n    point[p-1] += x\\n\\nparents = [0] * n\\nans = [0] * n\\nq = collections.deque()\\nq.append(0)\\nwhile q:\\n    v = q.pop()\\n    ans[v] = ans[parents[v]] + point[v]\\n    for i in c[v]:\\n        if i == parents[v]:\\n            continue\\n        parents[i] = v\\n        q.append(i)\\nprint(' '.join(list(map(str, ans))))\", \"from collections import deque\\nn,q=map(int,input().split())\\ntree=[[] for _ in range(n)]\\nfor i in range(n-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    tree[a].append(b)\\n    tree[b].append(a)\\ncount=[0]*n\\nfor _ in range(q):\\n    p,cnt=map(int,input().split())\\n    count[p-1]+=cnt\\nstack=deque([[0,0,-1]])\\nwhile stack:\\n    num,cnt,pr=stack.pop()\\n    count[num]+=cnt\\n    for k in tree[num]:\\n        if k==pr:\\n            continue\\n        stack.append([k,count[num],num])\\nprint(*count)\", \"import sys\\nfrom collections import defaultdict\\nsys.setrecursionlimit(1000000)\\nN, Q = map(int,input().split())\\nedges = [[] for _ in range(N)]\\nfor _ in range(N-1):\\n    a,b = map(int,input().split())\\n    a -= 1\\n    b -= 1\\n    edges[a].append(b)\\n    edges[b].append(a)\\n\\nd = defaultdict(int)\\nfor _ in range(Q):\\n    p,x = map(int,input().split())\\n    d[p-1] += x\\nvisited = [False]*N\\ncnt = [0]*N\\ndef dfs(n, acc):\\n    cnt[n] = acc\\n    for m in edges[n]:\\n        if not visited[m]:\\n            visited[m] = True\\n            dfs(m, acc+d[m])\\nvisited[0] = True\\ndfs(0, d[0])\\nprint(*cnt)\", \"from collections import deque\\nimport numpy as np\\n\\nN, Q = map(int, input().split())\\nA = [list(map(int, input().split())) for _ in range(N-1)]\\nP = [list(map(int, input().split())) for _ in range(Q)]\\n\\nlink = [[] for _ in range(N)]\\nfor i in range(N-1):\\n    link[A[i][0]-1].append(A[i][1]-1)\\n    link[A[i][1]-1].append(A[i][0]-1)\\n\\n\\nans = [0 for _ in range(N)]\\nfor i in range(Q):\\n  ans[P[i][0] - 1] += P[i][1]\\n\\ndist = [-1 for _ in range(N)]\\ndist[0] = 0\\nd = deque([0])\\nwhile d:\\n    now = d.pop()\\n    for i in range(len(link[now])):\\n        if dist[link[now][i]] == -1:\\n            ans[link[now][i]] += ans[now]\\n            dist[link[now][i]] = 0\\n            d.append(link[now][i])\\n\\nfor i in range(N):\\n    if i != N - 1:\\n        print(ans[i], end=\\\" \\\")\\n    else:\\n        print(ans[i])\\n\", \"import sys\\nread = sys.stdin.read\\nsys.setrecursionlimit(10**7)\\n#readlines = sys.stdin.readlines\\ndef main():\\n    def dfs(v, s):\\n        if notseen[v]:\\n            scores[v] += s\\n            notseen[v] = 0\\n            for nextv in gg[v]:\\n                dfs(nextv, scores[v])\\n\\n    data = tuple(map(int, read().split()))\\n    n, q = data[0], data[1]\\n    gg = {i: set() for i in range(1, n + 1)}\\n    # \\u6728\\u306a\\u306e\\u3067\\u201d\\u6b63\\u3057\\u304f\\u201d\\u51e6\\u7406\\u3059\\u308c\\u3070gg[x]\\u3078\\u306eadd\\u306f\\uff12\\u56de\\u3067\\u306f\\u306a\\u304f\\uff11\\u56de\\u3067\\u3044\\u3044\\u306f\\u305a\\u3002\\n    # \\u3060\\u304c\\u3046\\u307e\\u304f\\u884c\\u304b\\u306a\\u3044\\u305f\\u3081\\u3001\\u7121\\u5411\\u30b0\\u30e9\\u30d5\\u306e\\u3088\\u3046\\u306b\\uff12\\u56de\\u52a0\\u3048\\u3066\\u3001dfs\\u95a2\\u6570\\u3067\\u5404\\u9802\\u70b9\\u3092\\uff11\\u56de\\u3057\\u304b\\u307f\\u306a\\u3044\\u3088\\u3046\\u306b\\u5bfe\\u5fdc\\u3002\\n    for i1, v1 in zip(data[2:n*2:2], data[3:n*2:2]):\\n        gg[v1].add(i1)\\n        gg[i1].add(v1)\\n    scores = [0] * (n + 1)\\n    for j1, v1 in zip(data[n*2::2], data[n*2+1::2]):\\n        scores[j1] += v1\\n\\n    notseen = [1] * (n + 1)\\n    dfs(1, 0)\\n    print(*scores[1:], sep=' ')\\n\\ndef __starting_point():\\n    main()\\n__starting_point()\", \"from collections import deque\\nN,Q = map(int,input().split())\\nab = [list(map(int,input().split())) for _ in range(N-1)]\\npx = [list(map(int,input().split())) for _ in range(Q)]\\n\\ntree = [[] for _ in range(N)]\\nfor a, b in ab:\\n  tree[a-1].append(b-1)\\n  tree[b-1].append(a-1)\\n\\ncounter = [0] * N\\nfor p, x in px:\\n  counter[p-1] += x\\n  \\nflag = [1] * N\\nt = deque()\\nt.append(0)\\n\\n#1(0)\\u304b\\u3089\\u9806\\u306b\\u63a2\\u7d22\\u3057\\u3066\\u3044\\u3051\\u3070\\u3001\\u81ea\\u7136\\u3068\\u89aa\\u2192\\u5b50\\u306e\\u9806\\u756a\\u306b\\u306a\\u308b\\nwhile t:\\n  v = t.popleft()\\n  flag[v] = 0\\n  for i in tree[v]:\\n    if flag[i]:\\n      counter[i] += counter[v]\\n      t.append(i)\\n      \\nprint(*counter)\", \"from collections import defaultdict\\nfrom sys import setrecursionlimit\\nsetrecursionlimit(10 ** 7)\\n\\nn,q = map(int,input().split())\\n\\nnode = [[] for _ in range(n+1)]\\ncounter = [0 for _ in range(n+1)]\\nvisited = [0 for _ in range(n+1)]\\npoint_dic = defaultdict(int)\\n\\nfor _ in range(n-1):\\n    a,b = map(int,input().split())\\n    node[a].append(b)\\n    node[b].append(a)\\n\\nfor _ in range(q):\\n    p,x = map(int,input().split())\\n    point_dic[p] += x\\n\\ndef dfs(now,before):\\n    nonlocal counter\\n    nonlocal visited\\n\\n    if visited[now]:\\n        return\\n    \\n    if point_dic[now]:\\n        point = point_dic[now]+before\\n    else:\\n        point = before\\n    \\n    visited[now] = 1\\n    counter[now] += point\\n\\n    for i in node[now]:\\n        if visited[i]:\\n            continue\\n\\n        dfs(i,point)\\n\\ndfs(1,0)\\n\\nprint(' '.join(map(str, counter[1:])))\", \"from collections import deque\\n\\n\\nN, Q = list(map(int, input().split()))\\nt = [[] for i in range(N + 1)]\\n\\nfor i in range(N - 1):\\n    a, b = list(map(int, input().split()))\\n    t[a].append(b)\\n    t[b].append(a)\\n\\nscore = [0] * (N + 1)\\nfor i in range(Q):\\n    p, x = list(map(int, input().split()))\\n    score[p] += x\\n\\nis_read = [False] * (N + 1)\\nd = deque()\\nd.append(1)\\n\\nwhile len(d) != 0:\\n    now = d.popleft()\\n    is_read[now] = True\\n    for c in t[now]:\\n        if is_read[c]:\\n            continue\\n        score[c] += score[now]\\n        d.append(c)\\nprint((\\\" \\\".join(map(str, score[1:]))))\\n\", \"import sys\\nsys.setrecursionlimit(10**6)\\ndef dfs(tree, n, ans):\\n    for i in tree[n]:\\n        ans[i] += ans[n]\\n        tree[i].remove(n)\\n        dfs(tree, i, ans)\\n \\nn, q = map(int,input().split())\\nab = [list(map(int,input().split())) for _ in range(n-1)]\\npx = [list(map(int,input().split())) for _ in range(q)]\\n \\ng = [[] for _ in range(n)]\\nfor i in range(n-1):\\n    g[ab[i][0]-1].append(ab[i][1]-1)\\n    g[ab[i][1]-1].append(ab[i][0]-1)\\n \\nans = [0]*n\\nfor i in range(q):\\n    ans[px[i-1][0]-1] += px[i-1][1]\\ndfs(g, 0, ans)\\n \\nprint(*ans)\", \"# import itertools\\n# import math\\nimport sys\\nsys.setrecursionlimit(500*500)\\n# import numpy as np\\n# from collections import deque\\n# import heapq\\n\\n# N = int(input())\\n# S = input()\\n# n, *a = map(int, open(0))\\nN, Q = map(int, input().split())\\n# A = list(map(int, input().split()))\\n# A = list(map(lambda x: int(x)*(-1), input().split()))\\n# B = list(map(int, input().split()))\\n# A_B = [list(map(int,input().split())) for _ in range(M)]\\n# S = input()\\n\\n# B_C = sorted(B_C, reverse=True, key=lambda x:x[1])\\n# all_cases = list(itertools.permutations(P))\\n# a = list(itertools.combinations_with_replacement(range(1, M + 1), N))\\n# itertools.product((0,1), repeat=n)\\n\\n# A = np.array(A)\\n# cum_A = np.cumsum(A)\\n# cum_A = np.insert(cum_A, 0, 0)\\n\\nedges = [list(map(int,input().split())) for _ in range(N - 1)]\\ntree = [[] for _ in range(N + 1)]\\n\\nfor edge in edges:\\n    tree[edge[0]].append(edge[1])\\n    tree[edge[1]].append(edge[0])\\n\\ndepth = [-1] * (N + 1)\\ndepth[1] = 0\\ncount = [0] * (N + 1)\\n\\nfor i in range(Q):\\n    p, x = map(int, input().split())\\n    count[p] += x\\n\\n# def dfs(tree, s):\\n#     for l in tree[s]:\\n#         if depth[l[0]] == -1:\\n#             depth[l[0]] = depth[s] + 1\\n#             dfs(tree, l[0])\\n# dfs(tree, 1)\\ndef dfs(tree, s):\\n    for l in tree[s]:\\n        if depth[l] == -1:\\n            depth[l] = 0\\n            count[l] += count[s]\\n            dfs(tree, l)\\ndfs(tree, 1)\\n\\n\\nfor i in count[1:]:\\n    print(i, end = ' ')\\n\\n# def factorization(n):\\n#     arr = []\\n#     temp = n\\n#     for i in range(2, int(-(-n**0.5//1))+1):\\n#         if temp%i==0:\\n#             cnt=0\\n#             while temp%i==0:\\n#                 cnt+=1\\n#                 temp //= i\\n#             arr.append([i, cnt])\\n#     if temp!=1:\\n#         arr.append([temp, 1])\\n#     if arr==[]:\\n#         arr.append([n, 1])\\n#     return arr\\n\\n\\n\\n# bfs\\n# tree = [[] for _ in range(N + 1)]\\n# edges = [list(map(int,input().split())) for _ in range(M)]\\n\\n# for edge in edges:\\n#     tree[edge[0]].append(edge[1])\\n#     tree[edge[1]].append(edge[0])\\n\\n# depth = [-1] * (N + 1)\\n# depth[1] = 0\\n\\n# d = deque()\\n# d.append(1)\\n\\n# ans = [0] * (N + 1)\\n# while d:\\n#  v = d.popleft()\\n#  for i in tree[v]:\\n#    if depth[i] != -1:\\n#      continue\\n#    depth[i] = depth[v] + 1\\n#    ans[i] = v\\n#    d.append(i)\\n\\n# # ans = depth[2:]\\n# print('Yes')\\n# print(*ans[2:], sep=\\\"\\\\n\\\")\", \"import sys\\nfrom functools import lru_cache\\nsys.setrecursionlimit(10**7)\\n\\nN, Q = map(int, input().split())\\nT = [[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n  a,b=map(int, input().split())\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nV = [0] * (N+1)\\nfor q in range(Q):\\n  p,x=map(int, input().split())\\n  V[p]+=x\\n\\n@lru_cache(maxsize=None)\\ndef dfs(i, parent, acc):\\n  V[i]+=acc\\n  for j in T[i]:\\n    if j != parent:\\n      dfs(j,i,V[i])\\n\\ncur,parent,acc=1,0,0\\ndfs(cur,parent,acc)\\nprint(*V[1:])\", \"import collections,sys\\ndef LI(): return list(map(int,sys.stdin.readline().rstrip().split()))\\nN,Q = LI()\\nab = [LI() for _ in range(N-1)]\\npx = [LI() for _ in range(Q)]\\nans = [0]*(N+1) #1_indexed\\ngraph = {i:collections.deque() for i in range(1,N+1)} #1_indexed\\nfor a,b in ab:\\n        graph[a].append(b)\\n        graph[b].append(a)\\nfor p,x in px:\\n    ans[p] += x\\nseen = [0]*(N+1) #1_indexed\\nstack = []\\ndef dfs():\\n    seen[1] = 1\\n    stack.append(1)\\n    while stack:\\n        s = stack.pop()\\n        if not graph[s]:\\n            continue\\n        for j in range(len(graph[s])):\\n            g_NO = graph[s].popleft()\\n            if seen[g_NO]:\\n                continue\\n            seen[g_NO] = 1\\n            stack.append(g_NO)\\n            ans[g_NO] += ans[s]\\ndfs()\\nprint(*ans[1:])\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN, Q = map(int,input().split())\\nG = [[] for _ in range(N)]\\ncounter = [0] * N\\n\\nfor _ in range(N-1):\\n    a, b = map(int,input().split())\\n    G[a-1].append(b-1)\\n    G[b-1].append(a-1)\\n\\nfor _ in range(Q):\\n    p, x = map(int,input().split())\\n    counter[p-1] += x\\n\\nseen = [False]*N\\n\\ndef dfs(v, seen, G, counter):\\n    seen[v] = True\\n    for u in G[v]:\\n        if not seen[u]:\\n            counter[u] += counter[v]\\n            dfs(u, seen, G, counter)\\n\\ndfs(0, seen, G, counter)\\n\\nfor c in counter:\\n    print(c, end = \\\" \\\")\\nprint(\\\"\\\")\\n\\n\", \"import sys\\nfrom collections import deque\\nsys.setrecursionlimit(10 ** 6)\\ndef input():\\n    return sys.stdin.readline()[:-1]\\n\\ndef dfs(s):\\n    stack = [s]\\n    visited = [False]*N\\n    while stack:\\n        v = stack.pop()\\n        if visited[v]:\\n            continue\\n        visited[v] = True\\n        for u in graph[v]:\\n            if not visited[u]:\\n                point[u] += point[v]\\n                stack.append(u)\\n\\nN, Q = list(map(int,input().split()))\\ngraph = [deque([]) for _ in range(N)]\\npoint = [0]*N\\n\\nfor i in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    graph[a-1].append(b-1)\\n    graph[b-1].append(a-1)\\n\\nfor i in range(Q):\\n    p,x = list(map(int,input().split()))\\n    p -= 1\\n    point[p] += x\\n\\ndfs(0)\\nprint((*point))\\n\", \"#!/usr/bin/env python3\\nfrom sys import setrecursionlimit\\n\\nsetrecursionlimit(10 ** 8)\\n\\n\\ndef dfs(now):\\n    seen[now] = True\\n    score[now] += operation[now]\\n    for next in branch[now]:\\n        if seen[next] is False:\\n            score[next] += score[now]\\n            dfs(next)\\n\\n\\nN, Q = map(int, input().split())\\nbranch = [set() for _ in range(N)]\\nfor _ in range(N - 1):\\n    a, b = map(int, input().split())\\n    branch[a - 1].add(b - 1)\\n    branch[b - 1].add(a - 1)\\noperation = [0] * N\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    operation[p - 1] += x\\n\\nseen = [False] * N\\nscore = [0] * N\\nfor i in range(N):\\n    if seen[i] is False:\\n        dfs(i)\\n\\nfor ans in score:\\n    print(ans, end=' ')\\n\", \"import sys\\nreadline = sys.stdin.readline\\n\\nN,Q = map(int,readline().split())\\n\\nG = [[] for i in range(N)]\\n\\nfor i in range(N - 1):\\n  a,b = map(int,readline().split())\\n  G[a - 1].append(b - 1)\\n  G[b - 1].append(a - 1)\\n  \\npoint = [0] * N\\nfor i in range(Q):\\n  p,x = map(int,readline().split())\\n  point[p - 1] += x\\n\\nstack = [(0, -1, 0)]\\nwhile stack:\\n  v,parent,p = stack.pop()\\n  point[v] += p\\n  p = point[v]\\n  for child in G[v]:\\n    if child == parent:\\n      continue\\n    stack.append([child, v, p])\\n  \\nprint(*point)\", \"def main():\\n\\tN, Q = [int(n) for n in input().split(\\\" \\\")]\\n\\tedges = [[] for i in range(N)]\\n\\tcounter = [0] * N\\n\\n\\tfor i in range(N - 1):\\n\\t\\ta, b = [int(x) for x in input().split(\\\" \\\")]\\n\\t\\tedges[a - 1].append(b - 1)\\n\\t\\tedges[b - 1].append(a - 1)\\n\\n\\tfor j in range(Q):\\n\\t\\tp, x = [int(q) for q in input().split(\\\" \\\")]\\n\\t\\tcounter[p - 1] += x\\n\\n\\tto_visit = [0]\\n\\tchecked = [1] + [0] * (N - 1)\\n\\ttree = [0] * N\\n\\n\\twhile len(to_visit) > 0:\\n\\t\\tvisiting = to_visit.pop()\\n\\t\\ttree[visiting] += counter[visiting]\\n\\t\\tfor e in edges[visiting]:\\n\\t\\t\\tif checked[e] == 0:\\n\\t\\t\\t\\tchecked[e] = 1\\n\\t\\t\\t\\tto_visit.append(e)\\n\\t\\t\\t\\ttree[e] = tree[visiting]\\n\\n\\tprint((\\\" \\\".join([str(s) for s in tree])))\\n\\nmain()\\n\", \"import sys\\nsys.setrecursionlimit(10**8)\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n-1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nt = dict()\\nfor i in range(1, n+1):\\n    t[i] = set()\\nfor a, b in ab:\\n    t[a].add(b)\\n    t[b].add(a)\\n\\ncnt = [0] * n\\ns = set()\\nfor p, x in px:\\n    cnt[p-1] += x\\n\\ndef solve(p):\\n    s.add(p)\\n    for c in t[p]:\\n        if c not in s:\\n            cnt[c-1] += cnt[p-1]\\n            solve(c)\\nsolve(1)\\nprint(*cnt, sep=' ')\\n\", \"from collections import deque\\n\\nN, Q = list(map(int, input().split()))\\ntree = [set() for _ in range(N+1)]\\n\\nfor i in range(N - 1):\\n    n1, n2 = list(map(int, input().split()))\\n    tree[n1].add(n2)\\n    tree[n2].add(n1)\\n\\nscores = [0]*(N+1)\\nfor _ in range(Q):\\n    p, x = list(map(int, input().split()))\\n    scores[p] += x\\n\\nq = deque()\\nq.append(1)\\nchecked = [False]*(N+1)\\nwhile q:\\n    v = q.pop()\\n    checked[v] = True\\n    s = scores[v]\\n    for k in tree[v]:\\n        if checked[k]:\\n            continue\\n        scores[k] += s\\n        q.append(k)\\n\\nprint((*scores[1:]))\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN,Q=map(int,input().split())\\nT=[[] for _ in range(N)]\\nfor _ in range(N-1):\\n  a,b=map(int,input().split())\\n  a-=1;b-=1\\n  T[a].append(b)\\n  T[b].append(a)\\n\\nV=[0]*N\\nfor _ in range(Q):\\n  p,x=map(int,input().split())\\n  p-=1\\n  V[p]+=x\\n  \\ndef dfs(now,prev=-1):\\n  for next in T[now]:\\n    if next==prev:\\n      continue\\n    V[next]+=V[now]\\n    dfs(next,now)\\n\\ndfs(0)\\nprint(*V)\", \"import sys\\nsys.setrecursionlimit(1<<30)\\nN,Q = map(int,input().split())\\ndef dfs(x, score):\\n    for y in Tree[x]:\\n        if Parent[x] != y:\\n            Parent[y] = x\\n            score[y] += score[x]\\n            dfs(y, score)\\nTree = [[] for i in range(N+1)]\\nParent = [0]*(N+1)\\nfor i in range(N-1):\\n    a,b = map(int,input().split())\\n    Tree[a].append(b)\\n    Tree[b].append(a)\\nscore = [0]*(N+1)\\nfor j in range(Q):\\n    p,x = map(int,input().split())\\n    score[p] += x\\ndfs(1,score)\\nprint(*score[1:],end=\\\"\\\\t\\\")\", \"import sys\\nsys.setrecursionlimit(10**9)\\n\\nN, Q = map(int, input().split())\\nL = [[] for _ in range(N)]\\n\\nfor _ in range(N-1):\\n    a, b = map(int, input().split())\\n    L[a-1].append(b-1)\\n    L[b-1].append(a-1)\\n\\nS = [0] * N\\n\\nfor _ in range(Q):\\n    p, x = map(int, input().split())\\n    S[p-1] += x\\n\\ndef search(p,q=-1):\\n    for a in L[p]:\\n        if a != q:\\n            S[a] += S[p]\\n            search(a,p)\\n\\nsearch(0)\\nprint(*S)\", \"import sys\\nsys.setrecursionlimit(10**7)\\nN, Q = [int(n) for n in input().split()]\\ntree_list = [[] for _ in range(N)]\\n\\nfor i in range(N-1):\\n    a, b = [int(n)-1 for n in input().split()]\\n    tree_list[a].append(b)\\n    tree_list[b].append(a)\\n\\nscore_list = [0] * N\\nfor j in range(Q):\\n    p, x = [int(n) for n in input().split()]\\n    score_list[p-1] += x\\n\\nreached = [False] * N\\ndef dfs(v):\\n    reached[v] = True\\n    for next_v in tree_list[v]:\\n        if reached[next_v] == True:\\n            continue\\n        score_list[next_v] += score_list[v]\\n        dfs(next_v)\\ndfs(0)\\nprint(*score_list)\", \"from collections import deque\\n\\nN, Q = list(map(int, input().split())) # N\\u306f\\u9802\\u70b9\\u306e\\u6570\\u3001Q\\u306f\\u64cd\\u4f5c\\u306e\\u56de\\u6570\\ngraph = [[] for _ in range(N+1)]\\nfor _ in range(N-1):\\n    a, b = list(map(int, input().split()))\\n    graph[a].append(b)\\n    graph[b].append(a)\\n\\ncounts = [0] * (N+1)\\nfor _ in range(Q):\\n    p, x  = list(map(int, input().split()))\\n    counts[p] += x\\nvisited = [-1] * (N+1)\\n\\nq = deque()\\nq.append(1)\\nvisited[1] = 1\\nwhile q:\\n    node = q.pop()\\n\\n    next_nodes = graph[node]\\n    for next in next_nodes:\\n        if visited[next] != -1:\\n            continue\\n        q.append(next)\\n        visited[next] = 1\\n        counts[next] += counts[node]\\n\\nprint((*counts[1:]))\\n\\n\\n\", \"import sys\\nsys.setrecursionlimit(10000000)\\ndef main():\\n    n,q = list(map(int,input().split()))\\n    Ki = [[] for i in range(n)]\\n    for _ in range(n-1):\\n        a,b = list(map(int,input().split()))\\n        Ki[a-1]+=[b-1]\\n        Ki[b-1]+=[a-1]\\n    ans = [0]*n\\n    for _ in range(q):\\n        p,x = list(map(int,input().split()))\\n        ans[p-1]+=x\\n    def add(N=0,L=-1):\\n        for k in Ki[N]:\\n            if k!=L:\\n                ans[k]+=ans[N]\\n                add(k,N)\\n    add()\\n    print((' '.join([str(a) for a in ans])))\\n\\ndef __starting_point():\\n    main()\\n\\n__starting_point()\", \"import sys\\nsys.setrecursionlimit(10**9)\\nn,q=map(int,input().split())\\nG=[[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    G[a].append(b)\\n    G[b].append(a)\\n\\nsubtree_size=[0]*n\\ncnt=[0]*n\\n\\nfor _ in range(q):\\n    p,x=map(int,input().split())\\n    p-=1\\n    cnt[p]+=x\\nans=[0]*n\\n\\ndef dfs(G,v,p):\\n    ans[v]+=cnt[v]\\n    for nv in G[v]:\\n        if nv==p:\\n            continue\\n        cnt[nv]+=cnt[v]\\n        dfs(G,nv,v)\\n    \\n    subtree_size[v]=1\\n    for c in G[v]:\\n        if c==p:\\n            continue\\n        subtree_size[v]+=subtree_size[c]\\n\\nroot=0\\ndfs(G,root,-1)\\n\\nfor i in range(n):\\n    print(ans[i],end=' ')\\n\\n\\n\\n\", \"import sys\\nsys.setrecursionlimit(10**8)\\nN, Q = map(int, input().split())\\ng = [[] for i in range(N)]\\nfor i in range(N-1):\\n    a, b = list(map(int, input().split()))\\n    g[a-1].append(b-1)\\n    g[b-1].append(a-1)\\ncnt = [0] * N\\nfor i in range(Q):\\n    p, x = list(map(int, input().split()))\\n    cnt[p-1] += x\\n\\ndef dfs(g, v):\\n    # nonlocal ans_cnt\\n    for nv in g[v]:\\n        if ans_cnt[nv] != -1: continue\\n        ans_cnt[nv] = cnt[nv] + ans_cnt[v]\\n        dfs(g, nv)\\n\\nans_cnt = [-1] * N\\nans_cnt[0] = cnt[0]\\ndfs(g, 0)\\n\\n# print(cnt)\\n# print(ans_cnt)\\nfor i, value in enumerate(ans_cnt):\\n    print(value, end=\\\"\\\")\\n    if i != len(ans_cnt)-1:\\n        print(\\\" \\\", end=\\\"\\\")\\nprint()\", \"import sys\\nsys.setrecursionlimit(10**7)\\nn,q=map(int,input().split())\\nedge=[[] for _ in range(n)]\\nfor i in range(n-1):\\n    x,y=map(int,input().split())\\n    edge[x-1].append(y-1)\\n    edge[y-1].append(x-1)\\nans=[0]*n\\nfor i in range(q):\\n    p,x=map(int,input().split())\\n    ans[p-1]+=x\\ndef dfs(c,p):\\n    for i in edge[c]:\\n        if i==p:\\n            continue\\n        ans[i]+=ans[c]\\n        dfs(i,c)\\ndfs(0,-1)\\nprint(*ans)\", \"N,Q = map(int,input().split())\\nG = [[] for n in range(N)]\\nans = N*[0]\\n \\nfor n in range(N-1):\\n  a,b = map(int,input().split())\\n  G[a-1].append(b-1)\\n  G[b-1].append(a-1)\\n \\nfor q in range(Q):\\n  p,x = map(int,input().split())\\n  ans[p-1]+=x\\n \\nf = N*[1]\\nt = [0]\\nwhile t:\\n  v = t.pop()\\n  f[v] = 0\\n  for k in G[v]:\\n    if f[k]:\\n      ans[k]+=ans[v]\\n      t.append(k)\\n \\nprint(*ans)\", \"import sys\\nsys.setrecursionlimit(10**8)\\n\\nn, q = map(int, input().split())\\nab = [list(map(int, input().split())) for _ in range(n-1)]\\npx = [list(map(int, input().split())) for _ in range(q)]\\n\\nt = dict()\\nfor i in range(1, n+1):\\n    t[i] = []\\nfor a, b in ab:\\n    t[a].append(b)\\n    t[b].append(a)\\n\\ncnt = [0] * n\\ns = set()\\nfor p, x in px:\\n    cnt[p-1] += x\\n\\ndef solve(p):\\n    s.add(p)\\n    for c in t[p]:\\n        if c not in s:\\n            cnt[c-1] += cnt[p-1]\\n            solve(c)\\nsolve(1)\\nprint(*cnt, sep=' ')\\n\", \"import sys\\n# import math\\n# import bisect\\n# import numpy as np\\n# from decimal import Decimal\\n# from numba import njit, i8, u1, b1 #JIT compiler\\n# from itertools import combinations, product\\nfrom collections import Counter, deque, defaultdict\\n\\n# sys.setrecursionlimit(10 ** 6)\\nMOD = 10 ** 9 + 7\\nINF = 10 ** 9\\nPI = 3.14159265358979323846\\n\\ndef read_str():      return sys.stdin.readline().strip()\\ndef read_int():      return int(sys.stdin.readline().strip())\\ndef read_ints():     return map(int, sys.stdin.readline().strip().split())\\ndef read_ints2(x):   return map(lambda num: int(num) - x, sys.stdin.readline().strip().split())\\ndef read_str_list(): return list(sys.stdin.readline().strip().split())\\ndef read_int_list(): return list(map(int, sys.stdin.readline().strip().split()))\\ndef GCD(a: int, b: int) -> int: return b if a%b==0 else GCD(b, a%b)\\ndef LCM(a: int, b: int) -> int: return (a * b) // GCD(a, b)\\n\\ndef Main():\\n    n, q = read_ints()\\n    tree = [[] for _ in range(n)]\\n    c = [0] * n\\n    for _ in range(n-1):\\n        a, b = read_ints()\\n        tree[~-a].append(~-b)\\n        tree[~-b].append(~-a)\\n    for _ in range(q):\\n        p, x = read_ints()\\n        c[~-p] += x\\n    que = deque([0])\\n    seen = [False] * n\\n    while que:\\n        p = que.pop()\\n        seen[p] = True\\n        for x in tree[p]:\\n            if seen[x]: continue\\n            que.append(x)\\n            c[x] += c[p]\\n    \\n    print(*c)\\n\\ndef __starting_point():\\n    Main()\\n__starting_point()\", \"from collections import defaultdict\\nN,Q = list(map(int,input().split()))\\nd = defaultdict(list)\\n\\nfor _ in range(N-1):\\n    a,b = list(map(int,input().split()))\\n    d[a].append(b)\\n    d[b].append(a)\\n\\nc = defaultdict(int)\\n\\nfor _ in range(Q):\\n    p,x = list(map(int,input().split()))\\n    c[p] += x\\n\\nans = [0 for i in range(N)]\\nque = [(1,0,-1)]\\nwhile len(que)>0:\\n    tmp = []\\n    for v,x,par in que:\\n        x += c[v]\\n        ans[v-1] = str(x)\\n        for w in d[v]:\\n            if w != par:\\n                tmp.append((w,x,v))\\n    que = tmp\\n\\nprint((*ans))\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\nn,q=map(int,input().split())\\nG=[[] for _ in range(n)]\\nfor _ in range(n-1):\\n    a,b=map(int,input().split())\\n    a-=1\\n    b-=1\\n    G[a].append(b)\\n    G[b].append(a)\\n\\nsubtree_size=[0]*n\\ncnt=[0]*n\\n\\nfor _ in range(q):\\n    p,x=map(int,input().split())\\n    p-=1\\n    cnt[p]+=x\\nans=[0]*n\\n\\ndef dfs(G,v,p):\\n    ans[v]+=cnt[v]\\n    for nv in G[v]:\\n        if nv==p:\\n            continue\\n        cnt[nv]+=cnt[v]\\n        dfs(G,nv,v)\\n    \\n    subtree_size[v]=1\\n    for c in G[v]:\\n        if c==p:\\n            continue\\n        subtree_size[v]+=subtree_size[c]\\n\\nroot=0\\ndfs(G,root,-1)\\n\\nfor i in range(n):\\n    print(ans[i],end=' ')\\n\\n\\n\\n\", \"import sys\\nimport math\\n\\n#https://atcoder.jp/contests/agc008/submissions/15248942\\nsys.setrecursionlimit(10 ** 8)\\nini = lambda: int(sys.stdin.readline())\\ninm = lambda: map(int, sys.stdin.readline().split())\\ninl = lambda: list(inm())\\nins = lambda: sys.stdin.readline().rstrip()\\ndebug = lambda *a, **kw: print(\\\"\\\\033[33m\\\", *a, \\\"\\\\033[0m\\\", **dict(file=sys.stderr, **kw))\\n\\nclass Node:\\n    def __init__(self):\\n        self.edge = []\\n        self.count = 0\\n\\ndef dfs(tree,node,p_node,p_count,counts):\\n    count = tree[node].count + p_count\\n    counts[node] = count\\n    for x in tree[node].edge:\\n        if x == p_node:\\n            continue\\n        dfs(tree,x,node,count,counts)\\n\\nN,Q = inm()\\n\\ntree = []\\nfor _ in range(N):\\n    tree.append(Node())\\n\\n#node = Node()\\n#tree = [node]*N\\n\\nfor _ in range(N-1):\\n    a,b = inm()\\n    tree[a-1].edge.append(b-1)\\n    tree[b-1].edge.append(a-1)\\n\\nfor _ in range(Q):\\n    p,x = inm()\\n    tree[p-1].count += x\\n\\nval = []\\nfor _ in range(N):\\n    val.append(0)\\n\\n#for i in range(N):\\n#    print(tree[i].edge)\\n#    print(tree[i].count)\\n\\ndfs(tree,0,-1,0,val)\\n\\ncounts_str = [str(n) for n in val]\\ns = \\\" \\\".join(counts_str)\\nprint(s)\\n\", \"from collections import defaultdict\\nfrom sys import setrecursionlimit # \\u518d\\u5e30\\u30a8\\u30e9\\u30fc\\u5bfe\\u7b56\\nsetrecursionlimit(10 ** 7)\\n\\nn,q = map(int,input().split())\\n\\nnode = [[] for _ in range(n+1)]\\ncounter = [0 for _ in range(n+1)]\\nvisited = [0 for _ in range(n+1)]\\npoint_dic = defaultdict(int)\\n\\nfor _ in range(n-1):\\n    a,b = map(int,input().split())\\n    node[a].append(b)\\n    node[b].append(a)\\n\\n# \\u5404\\u30ce\\u30fc\\u30c9\\u306e\\u52a0\\u7b97\\u5024\\u3092\\u8a18\\u9332\\nfor _ in range(q):\\n    p,x = map(int,input().split())\\n    point_dic[p] += x\\n\\n# \\u73fe\\u5728\\u306e\\u30ce\\u30fc\\u30c9(now)\\u3068\\u305d\\u308c\\u307e\\u3067\\u306b\\u52a0\\u7b97\\u3055\\u308c\\u305f\\u5024(before_point)\\ndef dfs(now,before_point):\\n    if visited[now]:\\n        return\\n    \\n    # \\u73fe\\u5728\\u306e\\u30ce\\u30fc\\u30c9\\u306e\\u52a0\\u7b97\\u5024\\u3092\\u8db3\\u3059\\n    if point_dic[now]:\\n        point = point_dic[now]+before_point\\n    else:\\n        point = before_point\\n    \\n    # \\u8a2a\\u554f\\u6e08\\u307f\\u306b\\u3059\\u308b\\n    visited[now] = 1\\n    counter[now] += point\\n\\n    for i in node[now]:\\n        if visited[i]:\\n            continue\\n\\n        dfs(i,point) # \\u518d\\u5e30\\u3057\\u3066\\u5b50\\u30ce\\u30fc\\u30c9\\u306b\\u5f15\\u304d\\u7d99\\u3050\\n\\ndfs(1,0)\\n\\nprint(' '.join(map(str, counter[1:])))\", \"import sys\\nsys.setrecursionlimit(10 ** 6)\\n\\ndef input():\\n    return sys.stdin.readline()[:-1]\\nN , Q = map(int,input().split())\\ngraph = [[] for _ in range(N)]\\npoint = [0] * N\\nfor _ in range(N - 1):\\n    a , b = map(int,input().split())\\n    graph[a - 1].append(b - 1)\\n    graph[b - 1].append(a - 1)\\n#print(graph)\\nfor _ in range(Q):\\n    a , b = map(int,input().split())\\n    a = a - 1\\n    point[a] += b\\n# dfs\\u3092\\u7528\\u3044\\u3066\\u7d2f\\u7a4d\\u548c\\u3092\\u8a08\\u7b97\\u3059\\u308b\\n# \\u521d\\u671f\\u72b6\\u614b\\u3060\\u3068\\u524d\\u306e\\u5024\\u304c\\u306a\\u3044\\u305f\\u3081\\u30c7\\u30d5\\u30a9\\u30eb\\u30c8\\u5f15\\u6570\\u306b-1\\u3092\\u4ee3\\u5165\\ndef dfs(now , prev = -1):\\n    for next in graph[now]:\\n        # \\u6b21\\u306e\\u30ce\\u30fc\\u30c9\\u304c\\u524d\\u306b\\u53c2\\u7167\\u3057\\u305f\\u5024\\u306e\\u6642\\u306fcontinue\\n        if next == prev:\\n            continue\\n        # \\u73fe\\u5728\\u306e\\u5024\\u3092\\u6b21\\u306e\\u30dd\\u30a4\\u30f3\\u30c8\\u306b\\u52a0\\u7b97\\u3059\\u308b\\u3053\\u3068\\u3067\\u7d2f\\u7a4d\\u548c\\u3092\\u3068\\u308b\\n        point[next] += point[now]\\n        # \\u6b21\\u306e\\u30ce\\u30fc\\u30c9\\u3068\\u73fe\\u5728\\u306e\\u30ce\\u30fc\\u30c9\\u3092\\u5f15\\u6570\\u306bdfs\\u3092\\u7d99\\u7d9a\\u3059\\u308b\\n        dfs(next , now)\\n\\ndfs(0)\\nprint(*point)\"]",
        "difficulty": "interview",
        "input": "6 2\n1 2\n1 3\n2 4\n3 6\n2 5\n1 10\n1 10\n",
        "output": "20 20 20 20 20 20\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://atcoder.jp/contests/abc138/tasks/abc138_d"
    },
    {
        "id": 1071,
        "task_id": 2767,
        "test_case_id": 1,
        "question": "Little Ivica got himself a summer job at a company that produces computer fonts. The branch of the company where Ivica works at specialises in testing computer fonts and Ivica’s team is responsible of testing only lowercase letters of the English alphabet.\n\nThe letters are tested so that various sentences using those letters are printed out and then manually (more accurately, visually) checked if everything is arranged properly. Only sentences which contain all 26 lowercase letter of the English alphabet (a–z) are used for testing purposes. These sentences are called test sentences.\n\nYou’ve probably already assumed that Ivica’s job is to find test sentences. He has a dictionary which consists of $N$ words and has to calculate how many different test sentences can be made out of those words. Every word from the dictionary can be used only once in the sentence and the order of the words in the sentence is irrelevant (i.e. “uvijek jedem sarmu” and “jedem sarmu uvijek” are equal sentences).\n\n-----Input-----\nThe first line of input contains the integer $N$ $(1 \\leq N \\leq 25)$, the number of words in the dictionary. Each of the following $N$ lines contains a single word from the dictionary, its length not exceeding 100. All the words from the dictionary are going to be unique.\n\n-----Output-----\nThe first and only line of output must contain the required number from the task.\n\n-----Examples-----\nSample Input 1:\n9\nthe\nquick\nbrown\nfox\njumps\nover\na\nsleazy\ndog\nSample Output 1:\n2\n\nSample Input 2:\n3\na\nb\nc\nSample Output 2:\n0",
        "solutions": "",
        "difficulty": "interview",
        "input": "9\nthe\nquick\nbrown\nfox\njumps\nover\na\nsleazy\ndog\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/font"
    },
    {
        "id": 1072,
        "task_id": 2767,
        "test_case_id": 2,
        "question": "Little Ivica got himself a summer job at a company that produces computer fonts. The branch of the company where Ivica works at specialises in testing computer fonts and Ivica’s team is responsible of testing only lowercase letters of the English alphabet.\n\nThe letters are tested so that various sentences using those letters are printed out and then manually (more accurately, visually) checked if everything is arranged properly. Only sentences which contain all 26 lowercase letter of the English alphabet (a–z) are used for testing purposes. These sentences are called test sentences.\n\nYou’ve probably already assumed that Ivica’s job is to find test sentences. He has a dictionary which consists of $N$ words and has to calculate how many different test sentences can be made out of those words. Every word from the dictionary can be used only once in the sentence and the order of the words in the sentence is irrelevant (i.e. “uvijek jedem sarmu” and “jedem sarmu uvijek” are equal sentences).\n\n-----Input-----\nThe first line of input contains the integer $N$ $(1 \\leq N \\leq 25)$, the number of words in the dictionary. Each of the following $N$ lines contains a single word from the dictionary, its length not exceeding 100. All the words from the dictionary are going to be unique.\n\n-----Output-----\nThe first and only line of output must contain the required number from the task.\n\n-----Examples-----\nSample Input 1:\n9\nthe\nquick\nbrown\nfox\njumps\nover\na\nsleazy\ndog\nSample Output 1:\n2\n\nSample Input 2:\n3\na\nb\nc\nSample Output 2:\n0",
        "solutions": "",
        "difficulty": "interview",
        "input": "3\na\nb\nc\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/font"
    },
    {
        "id": 1073,
        "task_id": 2767,
        "test_case_id": 3,
        "question": "Little Ivica got himself a summer job at a company that produces computer fonts. The branch of the company where Ivica works at specialises in testing computer fonts and Ivica’s team is responsible of testing only lowercase letters of the English alphabet.\n\nThe letters are tested so that various sentences using those letters are printed out and then manually (more accurately, visually) checked if everything is arranged properly. Only sentences which contain all 26 lowercase letter of the English alphabet (a–z) are used for testing purposes. These sentences are called test sentences.\n\nYou’ve probably already assumed that Ivica’s job is to find test sentences. He has a dictionary which consists of $N$ words and has to calculate how many different test sentences can be made out of those words. Every word from the dictionary can be used only once in the sentence and the order of the words in the sentence is irrelevant (i.e. “uvijek jedem sarmu” and “jedem sarmu uvijek” are equal sentences).\n\n-----Input-----\nThe first line of input contains the integer $N$ $(1 \\leq N \\leq 25)$, the number of words in the dictionary. Each of the following $N$ lines contains a single word from the dictionary, its length not exceeding 100. All the words from the dictionary are going to be unique.\n\n-----Output-----\nThe first and only line of output must contain the required number from the task.\n\n-----Examples-----\nSample Input 1:\n9\nthe\nquick\nbrown\nfox\njumps\nover\na\nsleazy\ndog\nSample Output 1:\n2\n\nSample Input 2:\n3\na\nb\nc\nSample Output 2:\n0",
        "solutions": "",
        "difficulty": "interview",
        "input": "15\nabcdefghijkl\nbcdefghijklm\ncdefghijklmn\ndefghijklmno\nefghijklmnop\nfghijklmnopq\nghijklmnopqr\nhijklmnopqrs\nijklmnopqrst\njklmnopqrstu\nklmnopqrstuv\nlmnopqrstuvw\nmnopqrstuvwx\nnopqrstuvwxy\nopqrstuvwxyz\n",
        "output": "8189\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/font"
    },
    {
        "id": 1074,
        "task_id": 2843,
        "test_case_id": 1,
        "question": "You are given $W$, a set of $N$ words that are anagrams of each other. There are no duplicate letters in any word. A set of words $S \\subseteq W$ is called “swap-free” if there is no way to turn a word $x \\in S$ into another word $y \\in S$ by swapping only a single pair of (not necessarily adjacent) letters in $x$. Find the size of the largest swap-free set $S$ chosen from the given set $W$.\n\n-----Input-----\nThe first line of input contains an integer $N$ ($1 \\le N \\le 500$). Following that are $N$ lines each with a single word. Every word contains only lowercase English letters and no duplicate letters. All $N$ words are unique, have at least one letter, and every word is an anagram of every other word.\n\n-----Output-----\nOutput the size of the largest swap-free set.\n\n-----Examples-----\nSample Input 1:\n6\nabc\nacb\ncab\ncba\nbac\nbca\nSample Output 1:\n3\n\nSample Input 2:\n11\nalerts\nalters\nartels\nestral\nlaster\nratels\nsalter\nslater\nstaler\nstelar\ntalers\nSample Output 2:\n8",
        "solutions": "",
        "difficulty": "interview",
        "input": "6\nabc\nacb\ncab\ncba\nbac\nbca\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/codenames"
    },
    {
        "id": 1075,
        "task_id": 2843,
        "test_case_id": 2,
        "question": "You are given $W$, a set of $N$ words that are anagrams of each other. There are no duplicate letters in any word. A set of words $S \\subseteq W$ is called “swap-free” if there is no way to turn a word $x \\in S$ into another word $y \\in S$ by swapping only a single pair of (not necessarily adjacent) letters in $x$. Find the size of the largest swap-free set $S$ chosen from the given set $W$.\n\n-----Input-----\nThe first line of input contains an integer $N$ ($1 \\le N \\le 500$). Following that are $N$ lines each with a single word. Every word contains only lowercase English letters and no duplicate letters. All $N$ words are unique, have at least one letter, and every word is an anagram of every other word.\n\n-----Output-----\nOutput the size of the largest swap-free set.\n\n-----Examples-----\nSample Input 1:\n6\nabc\nacb\ncab\ncba\nbac\nbca\nSample Output 1:\n3\n\nSample Input 2:\n11\nalerts\nalters\nartels\nestral\nlaster\nratels\nsalter\nslater\nstaler\nstelar\ntalers\nSample Output 2:\n8",
        "solutions": "",
        "difficulty": "interview",
        "input": "11\nalerts\nalters\nartels\nestral\nlaster\nratels\nsalter\nslater\nstaler\nstelar\ntalers\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/codenames"
    },
    {
        "id": 1076,
        "task_id": 2843,
        "test_case_id": 3,
        "question": "You are given $W$, a set of $N$ words that are anagrams of each other. There are no duplicate letters in any word. A set of words $S \\subseteq W$ is called “swap-free” if there is no way to turn a word $x \\in S$ into another word $y \\in S$ by swapping only a single pair of (not necessarily adjacent) letters in $x$. Find the size of the largest swap-free set $S$ chosen from the given set $W$.\n\n-----Input-----\nThe first line of input contains an integer $N$ ($1 \\le N \\le 500$). Following that are $N$ lines each with a single word. Every word contains only lowercase English letters and no duplicate letters. All $N$ words are unique, have at least one letter, and every word is an anagram of every other word.\n\n-----Output-----\nOutput the size of the largest swap-free set.\n\n-----Examples-----\nSample Input 1:\n6\nabc\nacb\ncab\ncba\nbac\nbca\nSample Output 1:\n3\n\nSample Input 2:\n11\nalerts\nalters\nartels\nestral\nlaster\nratels\nsalter\nslater\nstaler\nstelar\ntalers\nSample Output 2:\n8",
        "solutions": "",
        "difficulty": "interview",
        "input": "6\nates\neast\neats\netas\nsate\nteas\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/codenames"
    },
    {
        "id": 1077,
        "task_id": 3031,
        "test_case_id": 1,
        "question": "The Transit Authority of Greater Podunk is planning its holiday decorations. They want to create an illuminated display of their light rail map in which each stretch of track between stations can be illuminated in one of several colors. \n\nAt periodic intervals, the controlling software will choose two stations at random and illuminate all of the segments connecting those two stations. By design, for any two stations on the Greater Podunk Railway, there is a unique path connecting the two.\n\nFor maximum color and cheer, the display designers want to avoid having two adjacent segments of track lighting up in the same color. They fear, however, that they may have lost track of this guideline in the process of building the display. One of them has gone so far as to propose a means of measuring just how far from that ideal they may have fallen.\n\n-----Description-----\nYou are given a tree with $n$ nodes (stations), conveniently numbered from $1$ to $n$. Each edge in this tree has one of $n$ colors. A path in this tree is called a rainbow if all adjacent edges in the path have different colors. Also, a node is called good if every simple path with that node as one of its endpoints is a rainbow path. (A simple path is a path that does not repeat any vertex or edge.)\n\nFind all the good nodes in the given tree.\n\n-----Input-----\nThe first line of input contains a single integer $n$ ($1 \\le n \\le 50000$).\n\nEach of the next $n-1$ lines contains three space-separated integers $a_ i$, $b_ i$, and $c_ i$ ($1 \\le a_ i, b_ i, c_ i \\le n$; $a_ i \\ne b_ i$), describing an edge of color $c_ i$ that connects nodes $a_ i$ and $b_ i$.\n\nIt is guaranteed that the given edges form a tree.\n\n-----Output-----\nOn the first line of the output, print $k$, the number of good nodes.\n\nIn the next $k$ lines, print the indices of all good nodes in numerical order, one per line.\n\n-----Examples-----\n(For the first sample, node $3$ is good because all paths that have node $3$ as an endpoint are rainbow. In particular, even though the path $3 \\rightarrow 4 \\rightarrow 5 \\rightarrow 6$ has two edges of the same color (i.e. $3 \\rightarrow 4$, $5 \\rightarrow 6$), it is still rainbow because these edges are not adjacent.)\n\n-----Examples-----\nSample Input 1:\n8\n1 3 1\n2 3 1\n3 4 3\n4 5 4\n5 6 3\n6 7 2\n6 8 2\nSample Output 1:\n4\n3\n4\n5\n6\n\nSample Input 2:\n8\n1 2 2\n1 3 1\n2 4 3\n2 7 1\n3 5 2\n5 6 2\n7 8 1\nSample Output 2:\n0\n\nSample Input 3:\n9\n1 2 2\n1 3 1\n1 4 5\n1 5 5\n2 6 3\n3 7 3\n4 8 1\n5 9 2\nSample Output 3:\n5\n1\n2\n3\n6\n7",
        "solutions": "",
        "difficulty": "competition",
        "input": "8\n1 3 1\n2 3 1\n3 4 3\n4 5 4\n5 6 3\n6 7 2\n6 8 2\n",
        "output": "4\n3\n4\n5\n6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/rainbowroads"
    },
    {
        "id": 1078,
        "task_id": 3031,
        "test_case_id": 2,
        "question": "The Transit Authority of Greater Podunk is planning its holiday decorations. They want to create an illuminated display of their light rail map in which each stretch of track between stations can be illuminated in one of several colors. \n\nAt periodic intervals, the controlling software will choose two stations at random and illuminate all of the segments connecting those two stations. By design, for any two stations on the Greater Podunk Railway, there is a unique path connecting the two.\n\nFor maximum color and cheer, the display designers want to avoid having two adjacent segments of track lighting up in the same color. They fear, however, that they may have lost track of this guideline in the process of building the display. One of them has gone so far as to propose a means of measuring just how far from that ideal they may have fallen.\n\n-----Description-----\nYou are given a tree with $n$ nodes (stations), conveniently numbered from $1$ to $n$. Each edge in this tree has one of $n$ colors. A path in this tree is called a rainbow if all adjacent edges in the path have different colors. Also, a node is called good if every simple path with that node as one of its endpoints is a rainbow path. (A simple path is a path that does not repeat any vertex or edge.)\n\nFind all the good nodes in the given tree.\n\n-----Input-----\nThe first line of input contains a single integer $n$ ($1 \\le n \\le 50000$).\n\nEach of the next $n-1$ lines contains three space-separated integers $a_ i$, $b_ i$, and $c_ i$ ($1 \\le a_ i, b_ i, c_ i \\le n$; $a_ i \\ne b_ i$), describing an edge of color $c_ i$ that connects nodes $a_ i$ and $b_ i$.\n\nIt is guaranteed that the given edges form a tree.\n\n-----Output-----\nOn the first line of the output, print $k$, the number of good nodes.\n\nIn the next $k$ lines, print the indices of all good nodes in numerical order, one per line.\n\n-----Examples-----\n(For the first sample, node $3$ is good because all paths that have node $3$ as an endpoint are rainbow. In particular, even though the path $3 \\rightarrow 4 \\rightarrow 5 \\rightarrow 6$ has two edges of the same color (i.e. $3 \\rightarrow 4$, $5 \\rightarrow 6$), it is still rainbow because these edges are not adjacent.)\n\n-----Examples-----\nSample Input 1:\n8\n1 3 1\n2 3 1\n3 4 3\n4 5 4\n5 6 3\n6 7 2\n6 8 2\nSample Output 1:\n4\n3\n4\n5\n6\n\nSample Input 2:\n8\n1 2 2\n1 3 1\n2 4 3\n2 7 1\n3 5 2\n5 6 2\n7 8 1\nSample Output 2:\n0\n\nSample Input 3:\n9\n1 2 2\n1 3 1\n1 4 5\n1 5 5\n2 6 3\n3 7 3\n4 8 1\n5 9 2\nSample Output 3:\n5\n1\n2\n3\n6\n7",
        "solutions": "",
        "difficulty": "competition",
        "input": "8\n1 2 2\n1 3 1\n2 4 3\n2 7 1\n3 5 2\n5 6 2\n7 8 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/rainbowroads"
    },
    {
        "id": 1079,
        "task_id": 3031,
        "test_case_id": 3,
        "question": "The Transit Authority of Greater Podunk is planning its holiday decorations. They want to create an illuminated display of their light rail map in which each stretch of track between stations can be illuminated in one of several colors. \n\nAt periodic intervals, the controlling software will choose two stations at random and illuminate all of the segments connecting those two stations. By design, for any two stations on the Greater Podunk Railway, there is a unique path connecting the two.\n\nFor maximum color and cheer, the display designers want to avoid having two adjacent segments of track lighting up in the same color. They fear, however, that they may have lost track of this guideline in the process of building the display. One of them has gone so far as to propose a means of measuring just how far from that ideal they may have fallen.\n\n-----Description-----\nYou are given a tree with $n$ nodes (stations), conveniently numbered from $1$ to $n$. Each edge in this tree has one of $n$ colors. A path in this tree is called a rainbow if all adjacent edges in the path have different colors. Also, a node is called good if every simple path with that node as one of its endpoints is a rainbow path. (A simple path is a path that does not repeat any vertex or edge.)\n\nFind all the good nodes in the given tree.\n\n-----Input-----\nThe first line of input contains a single integer $n$ ($1 \\le n \\le 50000$).\n\nEach of the next $n-1$ lines contains three space-separated integers $a_ i$, $b_ i$, and $c_ i$ ($1 \\le a_ i, b_ i, c_ i \\le n$; $a_ i \\ne b_ i$), describing an edge of color $c_ i$ that connects nodes $a_ i$ and $b_ i$.\n\nIt is guaranteed that the given edges form a tree.\n\n-----Output-----\nOn the first line of the output, print $k$, the number of good nodes.\n\nIn the next $k$ lines, print the indices of all good nodes in numerical order, one per line.\n\n-----Examples-----\n(For the first sample, node $3$ is good because all paths that have node $3$ as an endpoint are rainbow. In particular, even though the path $3 \\rightarrow 4 \\rightarrow 5 \\rightarrow 6$ has two edges of the same color (i.e. $3 \\rightarrow 4$, $5 \\rightarrow 6$), it is still rainbow because these edges are not adjacent.)\n\n-----Examples-----\nSample Input 1:\n8\n1 3 1\n2 3 1\n3 4 3\n4 5 4\n5 6 3\n6 7 2\n6 8 2\nSample Output 1:\n4\n3\n4\n5\n6\n\nSample Input 2:\n8\n1 2 2\n1 3 1\n2 4 3\n2 7 1\n3 5 2\n5 6 2\n7 8 1\nSample Output 2:\n0\n\nSample Input 3:\n9\n1 2 2\n1 3 1\n1 4 5\n1 5 5\n2 6 3\n3 7 3\n4 8 1\n5 9 2\nSample Output 3:\n5\n1\n2\n3\n6\n7",
        "solutions": "",
        "difficulty": "competition",
        "input": "9\n1 2 2\n1 3 1\n1 4 5\n1 5 5\n2 6 3\n3 7 3\n4 8 1\n5 9 2\n",
        "output": "5\n1\n2\n3\n6\n7\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/rainbowroads"
    },
    {
        "id": 1080,
        "task_id": 3031,
        "test_case_id": 4,
        "question": "The Transit Authority of Greater Podunk is planning its holiday decorations. They want to create an illuminated display of their light rail map in which each stretch of track between stations can be illuminated in one of several colors. \n\nAt periodic intervals, the controlling software will choose two stations at random and illuminate all of the segments connecting those two stations. By design, for any two stations on the Greater Podunk Railway, there is a unique path connecting the two.\n\nFor maximum color and cheer, the display designers want to avoid having two adjacent segments of track lighting up in the same color. They fear, however, that they may have lost track of this guideline in the process of building the display. One of them has gone so far as to propose a means of measuring just how far from that ideal they may have fallen.\n\n-----Description-----\nYou are given a tree with $n$ nodes (stations), conveniently numbered from $1$ to $n$. Each edge in this tree has one of $n$ colors. A path in this tree is called a rainbow if all adjacent edges in the path have different colors. Also, a node is called good if every simple path with that node as one of its endpoints is a rainbow path. (A simple path is a path that does not repeat any vertex or edge.)\n\nFind all the good nodes in the given tree.\n\n-----Input-----\nThe first line of input contains a single integer $n$ ($1 \\le n \\le 50000$).\n\nEach of the next $n-1$ lines contains three space-separated integers $a_ i$, $b_ i$, and $c_ i$ ($1 \\le a_ i, b_ i, c_ i \\le n$; $a_ i \\ne b_ i$), describing an edge of color $c_ i$ that connects nodes $a_ i$ and $b_ i$.\n\nIt is guaranteed that the given edges form a tree.\n\n-----Output-----\nOn the first line of the output, print $k$, the number of good nodes.\n\nIn the next $k$ lines, print the indices of all good nodes in numerical order, one per line.\n\n-----Examples-----\n(For the first sample, node $3$ is good because all paths that have node $3$ as an endpoint are rainbow. In particular, even though the path $3 \\rightarrow 4 \\rightarrow 5 \\rightarrow 6$ has two edges of the same color (i.e. $3 \\rightarrow 4$, $5 \\rightarrow 6$), it is still rainbow because these edges are not adjacent.)\n\n-----Examples-----\nSample Input 1:\n8\n1 3 1\n2 3 1\n3 4 3\n4 5 4\n5 6 3\n6 7 2\n6 8 2\nSample Output 1:\n4\n3\n4\n5\n6\n\nSample Input 2:\n8\n1 2 2\n1 3 1\n2 4 3\n2 7 1\n3 5 2\n5 6 2\n7 8 1\nSample Output 2:\n0\n\nSample Input 3:\n9\n1 2 2\n1 3 1\n1 4 5\n1 5 5\n2 6 3\n3 7 3\n4 8 1\n5 9 2\nSample Output 3:\n5\n1\n2\n3\n6\n7",
        "solutions": "",
        "difficulty": "competition",
        "input": "10\n9 2 1\n9 3 1\n9 4 2\n9 5 2\n9 1 3\n9 6 4\n1 8 5\n1 10 5\n6 7 9\n",
        "output": "4\n1\n6\n7\n9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/rainbowroads"
    },
    {
        "id": 1081,
        "task_id": 3080,
        "test_case_id": 1,
        "question": "As an enterprising owner of a world-renowned ski resort in the US, you would like to increase your sales by stocking snack stands at key locations throughout your estate.\n\nThe ski resort is built on a mountain. A ski lift can take a guest to the top of the mountain. From there they can ski to a number of locations throughout the mountain.\n\nThere are $n$ areas on the mountain. The areas are labeled $1$ to $n$. The top of the mountain is area $1$. Areas are connected with ski runs that are strictly downhill. In other words, it is not possible to return to an area after leaving it without taking the ski lift. Every area (including area $1$) has exactly one snack stand.\n\nAs the owner of the resort, you want to know how effectively you can distribute snacks among snack stands to better serve your guests (and make more money). To this end you would like to run a survey, and analyze the result with a number of independent queries. Each guest in the survey has a favorite snack, and a list of favorite areas that they like to visit. You would like to know how to best stock your snack stands with their favorite snack.\n\nEach query is a set of a guest’s favorite areas, and a number $k$. You would like to know how many ways you can distribute this guest’s favorite snack to exactly $k$ snack stands on the mountain such that the snack stands meet a few conditions:\n - For each of this guest’s favorite areas, over all sequences of ski runs to reach that area from the top of the mountain, there must be exactly one snack stand with the guest’s favorite snack (In other words, they must not have a choice of more than one snack stand where their snack is available.)\n - Each of the $k$ snack stands stocked with this guest’s favorite snack must be on some sequence of ski runs from the top of the mountain to some area in the query set.\n\n-----Input-----\nEach input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will contain three space-separated integers $n$, $m$, and $q$, where $n$ ($1 \\le n \\le 10^5$) is the number of areas on the mountain, $m$ ($1 \\le m \\le n+50$) is the number of runs, and $q$ ($1 \\le q \\le 10^5$) is the number of queries.\n\nThe next $m$ lines each contain two integers $x$ and $y$ ($1 \\le x,y \\le n, x \\ne y$). This represents a ski run from area $x$ to area $y$. There will be at most one run between any two areas. It will be possible to reach each area from area 1 by some chain of ski runs.\n\nThe next $q$ lines are each a sequence of space-separated integers, starting with $k$ and $a$, which are followed by $a$ integers $i$. Here, $k$ ($1 \\le k \\le 4$) represents the number of snack stands to stock with this guest’s favorite snack, $a$ ($1 \\le a \\le n$) represents the number of areas in the query set, and the $a$ integers $i$ ($1 \\le i \\le n$) are the labels of the areas in the query set. In any given query, no integer $i$ will be repeated.\n\nThe sum of all $a$’s for all queries will not exceed $100000$.\n\n-----Output-----\nOutput $q$ integers, each on its own line with no blank lines in between. These represent the number of ways to select snack stands to stock for each query, in the order that they appear in the input. Two ways are considered different if an area is selected in one configuration but not the other.\n\n-----Examples-----\nSample Input 1:\n4 4 4\n1 2\n1 3\n2 4\n3 4\n1 1 4\n2 1 4\n1 1 3\n2 2 3 2\nSample Output 1:\n2\n0\n2\n1\n\nSample Input 2:\n8 10 4\n1 2\n2 3\n1 3\n3 6\n6 8\n2 4\n2 5\n4 7\n5 7\n7 8\n2 3 4 5 6\n2 2 6 8\n1 1 6\n1 1 8\nSample Output 2:\n0\n0\n3\n2",
        "solutions": "",
        "difficulty": "competition",
        "input": "4 4 4\n1 2\n1 3\n2 4\n3 4\n1 1 4\n2 1 4\n1 1 3\n2 2 3 2\n",
        "output": "2\n0\n2\n1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/skiresort"
    },
    {
        "id": 1082,
        "task_id": 3080,
        "test_case_id": 2,
        "question": "As an enterprising owner of a world-renowned ski resort in the US, you would like to increase your sales by stocking snack stands at key locations throughout your estate.\n\nThe ski resort is built on a mountain. A ski lift can take a guest to the top of the mountain. From there they can ski to a number of locations throughout the mountain.\n\nThere are $n$ areas on the mountain. The areas are labeled $1$ to $n$. The top of the mountain is area $1$. Areas are connected with ski runs that are strictly downhill. In other words, it is not possible to return to an area after leaving it without taking the ski lift. Every area (including area $1$) has exactly one snack stand.\n\nAs the owner of the resort, you want to know how effectively you can distribute snacks among snack stands to better serve your guests (and make more money). To this end you would like to run a survey, and analyze the result with a number of independent queries. Each guest in the survey has a favorite snack, and a list of favorite areas that they like to visit. You would like to know how to best stock your snack stands with their favorite snack.\n\nEach query is a set of a guest’s favorite areas, and a number $k$. You would like to know how many ways you can distribute this guest’s favorite snack to exactly $k$ snack stands on the mountain such that the snack stands meet a few conditions:\n - For each of this guest’s favorite areas, over all sequences of ski runs to reach that area from the top of the mountain, there must be exactly one snack stand with the guest’s favorite snack (In other words, they must not have a choice of more than one snack stand where their snack is available.)\n - Each of the $k$ snack stands stocked with this guest’s favorite snack must be on some sequence of ski runs from the top of the mountain to some area in the query set.\n\n-----Input-----\nEach input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will contain three space-separated integers $n$, $m$, and $q$, where $n$ ($1 \\le n \\le 10^5$) is the number of areas on the mountain, $m$ ($1 \\le m \\le n+50$) is the number of runs, and $q$ ($1 \\le q \\le 10^5$) is the number of queries.\n\nThe next $m$ lines each contain two integers $x$ and $y$ ($1 \\le x,y \\le n, x \\ne y$). This represents a ski run from area $x$ to area $y$. There will be at most one run between any two areas. It will be possible to reach each area from area 1 by some chain of ski runs.\n\nThe next $q$ lines are each a sequence of space-separated integers, starting with $k$ and $a$, which are followed by $a$ integers $i$. Here, $k$ ($1 \\le k \\le 4$) represents the number of snack stands to stock with this guest’s favorite snack, $a$ ($1 \\le a \\le n$) represents the number of areas in the query set, and the $a$ integers $i$ ($1 \\le i \\le n$) are the labels of the areas in the query set. In any given query, no integer $i$ will be repeated.\n\nThe sum of all $a$’s for all queries will not exceed $100000$.\n\n-----Output-----\nOutput $q$ integers, each on its own line with no blank lines in between. These represent the number of ways to select snack stands to stock for each query, in the order that they appear in the input. Two ways are considered different if an area is selected in one configuration but not the other.\n\n-----Examples-----\nSample Input 1:\n4 4 4\n1 2\n1 3\n2 4\n3 4\n1 1 4\n2 1 4\n1 1 3\n2 2 3 2\nSample Output 1:\n2\n0\n2\n1\n\nSample Input 2:\n8 10 4\n1 2\n2 3\n1 3\n3 6\n6 8\n2 4\n2 5\n4 7\n5 7\n7 8\n2 3 4 5 6\n2 2 6 8\n1 1 6\n1 1 8\nSample Output 2:\n0\n0\n3\n2",
        "solutions": "",
        "difficulty": "competition",
        "input": "8 10 4\n1 2\n2 3\n1 3\n3 6\n6 8\n2 4\n2 5\n4 7\n5 7\n7 8\n2 3 4 5 6\n2 2 6 8\n1 1 6\n1 1 8\n",
        "output": "0\n0\n3\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/skiresort"
    },
    {
        "id": 1083,
        "task_id": 3080,
        "test_case_id": 3,
        "question": "As an enterprising owner of a world-renowned ski resort in the US, you would like to increase your sales by stocking snack stands at key locations throughout your estate.\n\nThe ski resort is built on a mountain. A ski lift can take a guest to the top of the mountain. From there they can ski to a number of locations throughout the mountain.\n\nThere are $n$ areas on the mountain. The areas are labeled $1$ to $n$. The top of the mountain is area $1$. Areas are connected with ski runs that are strictly downhill. In other words, it is not possible to return to an area after leaving it without taking the ski lift. Every area (including area $1$) has exactly one snack stand.\n\nAs the owner of the resort, you want to know how effectively you can distribute snacks among snack stands to better serve your guests (and make more money). To this end you would like to run a survey, and analyze the result with a number of independent queries. Each guest in the survey has a favorite snack, and a list of favorite areas that they like to visit. You would like to know how to best stock your snack stands with their favorite snack.\n\nEach query is a set of a guest’s favorite areas, and a number $k$. You would like to know how many ways you can distribute this guest’s favorite snack to exactly $k$ snack stands on the mountain such that the snack stands meet a few conditions:\n - For each of this guest’s favorite areas, over all sequences of ski runs to reach that area from the top of the mountain, there must be exactly one snack stand with the guest’s favorite snack (In other words, they must not have a choice of more than one snack stand where their snack is available.)\n - Each of the $k$ snack stands stocked with this guest’s favorite snack must be on some sequence of ski runs from the top of the mountain to some area in the query set.\n\n-----Input-----\nEach input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will contain three space-separated integers $n$, $m$, and $q$, where $n$ ($1 \\le n \\le 10^5$) is the number of areas on the mountain, $m$ ($1 \\le m \\le n+50$) is the number of runs, and $q$ ($1 \\le q \\le 10^5$) is the number of queries.\n\nThe next $m$ lines each contain two integers $x$ and $y$ ($1 \\le x,y \\le n, x \\ne y$). This represents a ski run from area $x$ to area $y$. There will be at most one run between any two areas. It will be possible to reach each area from area 1 by some chain of ski runs.\n\nThe next $q$ lines are each a sequence of space-separated integers, starting with $k$ and $a$, which are followed by $a$ integers $i$. Here, $k$ ($1 \\le k \\le 4$) represents the number of snack stands to stock with this guest’s favorite snack, $a$ ($1 \\le a \\le n$) represents the number of areas in the query set, and the $a$ integers $i$ ($1 \\le i \\le n$) are the labels of the areas in the query set. In any given query, no integer $i$ will be repeated.\n\nThe sum of all $a$’s for all queries will not exceed $100000$.\n\n-----Output-----\nOutput $q$ integers, each on its own line with no blank lines in between. These represent the number of ways to select snack stands to stock for each query, in the order that they appear in the input. Two ways are considered different if an area is selected in one configuration but not the other.\n\n-----Examples-----\nSample Input 1:\n4 4 4\n1 2\n1 3\n2 4\n3 4\n1 1 4\n2 1 4\n1 1 3\n2 2 3 2\nSample Output 1:\n2\n0\n2\n1\n\nSample Input 2:\n8 10 4\n1 2\n2 3\n1 3\n3 6\n6 8\n2 4\n2 5\n4 7\n5 7\n7 8\n2 3 4 5 6\n2 2 6 8\n1 1 6\n1 1 8\nSample Output 2:\n0\n0\n3\n2",
        "solutions": "",
        "difficulty": "competition",
        "input": "8 9 4\n1 2\n1 3\n3 6\n6 8\n2 4\n2 5\n4 7\n5 7\n7 8\n2 3 4 5 6\n2 2 6 8\n1 1 6\n1 1 8\n",
        "output": "2\n0\n3\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/skiresort"
    },
    {
        "id": 1084,
        "task_id": 3112,
        "test_case_id": 1,
        "question": "Your factory has $N$ junctions (numbered from $1$ to $N$) connected by $M$ conveyor belts. Each conveyor belt transports any product automatically from one junction to another junction in exactly one minute. Note that each conveyor belt only works in one direction. There can be more than one conveyor belt connecting two junctions, and there can be a conveyor belt connecting a junction to itself.\n\nThere are $K$ producers (machines which produce the products) located at the first $K$ junctions, i.e. junctions $1, 2, \\ldots , K$. The producer at junction $j$ produces an product each minute $(x \\cdot K + j)$ for all integers $x \\ge 0$ and $j = 1, 2, \\ldots , K$. All products are transported immediately via the conveyor belts to the warehouse at junction $N$, except for those produced at junction $N$ (if any). Items produced at junction $N$ are directly delivered to the warehouse (there is no need to use the conveyor belts).\n\nAt each junction, there is a robot deciding which conveyor belts the incoming product should go to in a negligible time (instantly). The robots can be programmed such that all products produced by a producer are always delivered to the warehouse via the same route. Once the robots are programmed, the routing can no longer be changed. Items from different producers may have the same or different routes.\n\nA prudent potential investor comes and wants to inspect the factory before making any decision. You want to show to the potential investor that your factory employs a good risk management policy. Thus, you want to make sure that each conveyor belt only transports at most one product at any time; i.e. two products cannot be on the same conveyor belt at the same time. On the other hand, there is no limit on the number of products at the junctions (the robots have a lot of arms!). To achieve this, you may turn off zero or more producers, but you still want to maximize the production, hence, this problem.\n\nFind the maximum number of producers that can be left running such that all the produced products can be delivered to the warehouse and each conveyor belt transports at most $1$ product at any time.\n\n-----Input-----\nThe first line contains three integers $N$, $K$, and $M$ ($1 \\le K \\le N \\le 300$; $0 \\le M \\le 1000$) representing the number of junctions, the number of producers, and the number of conveyor belts, respectively.\n\nThe next $M$ lines, each contains two integers $a$ and $b$ ($1 \\le a, b \\le N$) representing a conveyor belt connecting junction $a$ and junction $b$ with the direction from $a$ to $b$.\n\n-----Output-----\nThe output contains an integer denoting the maximum number of producers which can be left running such that all the produced products can be delivered to the warehouse and each conveyor belt transports at most one product at any time.\n\n-----Explanation-----\nIn Sample Input $1$, $N = 4$, $K = 2$, $M = 3$, and the directed edges are $\\{ (1,3)$, $(2,3)$, $(3,4)\\} $. There is only one possible delivery route for each producer, i.e. $1 \\rightarrow 3 \\rightarrow 4$ for producer $1$, and $2 \\rightarrow 3 \\rightarrow 4$ for producer $2$. Both producers are using conveyor belt $(3,4)$, however, the products from producer $1$ are on the conveyor belt $(3,4)$ on minutes $2, 4, 6, \\dots $ (even minutes), while the products from producer $2$ are on the conveyor belt $(3,4)$ on minutes $3, 5, 7, \\dots $ (odd minutes). Therefore, both producers can be left running.\n\nIn Sample Input $2$, $N = 5$, $K = 2$, $M = 4$, and the directed edges are $\\{ (1,3)$, $(3,4)$, $(2,4)$, $(4,5)\\} $. Similar to the previous example, there is only one possible delivery route for each product produced by each producer. In this example, only one producer can be left running as products from both producers ($1$ and $2$) are on the conveyor belt $(4,5)$ at the same time if both are running.\n\n-----Examples-----\nSample Input 1:\n4 2 3\n1 3\n2 3\n3 4\nSample Output 1:\n2\n\nSample Input 2:\n5 2 4\n1 3\n3 4\n2 4\n4 5\nSample Output 2:\n1",
        "solutions": "",
        "difficulty": "competition",
        "input": "4 2 3\n1 3\n2 3\n3 4\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/conveyorbelts"
    },
    {
        "id": 1085,
        "task_id": 3112,
        "test_case_id": 2,
        "question": "Your factory has $N$ junctions (numbered from $1$ to $N$) connected by $M$ conveyor belts. Each conveyor belt transports any product automatically from one junction to another junction in exactly one minute. Note that each conveyor belt only works in one direction. There can be more than one conveyor belt connecting two junctions, and there can be a conveyor belt connecting a junction to itself.\n\nThere are $K$ producers (machines which produce the products) located at the first $K$ junctions, i.e. junctions $1, 2, \\ldots , K$. The producer at junction $j$ produces an product each minute $(x \\cdot K + j)$ for all integers $x \\ge 0$ and $j = 1, 2, \\ldots , K$. All products are transported immediately via the conveyor belts to the warehouse at junction $N$, except for those produced at junction $N$ (if any). Items produced at junction $N$ are directly delivered to the warehouse (there is no need to use the conveyor belts).\n\nAt each junction, there is a robot deciding which conveyor belts the incoming product should go to in a negligible time (instantly). The robots can be programmed such that all products produced by a producer are always delivered to the warehouse via the same route. Once the robots are programmed, the routing can no longer be changed. Items from different producers may have the same or different routes.\n\nA prudent potential investor comes and wants to inspect the factory before making any decision. You want to show to the potential investor that your factory employs a good risk management policy. Thus, you want to make sure that each conveyor belt only transports at most one product at any time; i.e. two products cannot be on the same conveyor belt at the same time. On the other hand, there is no limit on the number of products at the junctions (the robots have a lot of arms!). To achieve this, you may turn off zero or more producers, but you still want to maximize the production, hence, this problem.\n\nFind the maximum number of producers that can be left running such that all the produced products can be delivered to the warehouse and each conveyor belt transports at most $1$ product at any time.\n\n-----Input-----\nThe first line contains three integers $N$, $K$, and $M$ ($1 \\le K \\le N \\le 300$; $0 \\le M \\le 1000$) representing the number of junctions, the number of producers, and the number of conveyor belts, respectively.\n\nThe next $M$ lines, each contains two integers $a$ and $b$ ($1 \\le a, b \\le N$) representing a conveyor belt connecting junction $a$ and junction $b$ with the direction from $a$ to $b$.\n\n-----Output-----\nThe output contains an integer denoting the maximum number of producers which can be left running such that all the produced products can be delivered to the warehouse and each conveyor belt transports at most one product at any time.\n\n-----Explanation-----\nIn Sample Input $1$, $N = 4$, $K = 2$, $M = 3$, and the directed edges are $\\{ (1,3)$, $(2,3)$, $(3,4)\\} $. There is only one possible delivery route for each producer, i.e. $1 \\rightarrow 3 \\rightarrow 4$ for producer $1$, and $2 \\rightarrow 3 \\rightarrow 4$ for producer $2$. Both producers are using conveyor belt $(3,4)$, however, the products from producer $1$ are on the conveyor belt $(3,4)$ on minutes $2, 4, 6, \\dots $ (even minutes), while the products from producer $2$ are on the conveyor belt $(3,4)$ on minutes $3, 5, 7, \\dots $ (odd minutes). Therefore, both producers can be left running.\n\nIn Sample Input $2$, $N = 5$, $K = 2$, $M = 4$, and the directed edges are $\\{ (1,3)$, $(3,4)$, $(2,4)$, $(4,5)\\} $. Similar to the previous example, there is only one possible delivery route for each product produced by each producer. In this example, only one producer can be left running as products from both producers ($1$ and $2$) are on the conveyor belt $(4,5)$ at the same time if both are running.\n\n-----Examples-----\nSample Input 1:\n4 2 3\n1 3\n2 3\n3 4\nSample Output 1:\n2\n\nSample Input 2:\n5 2 4\n1 3\n3 4\n2 4\n4 5\nSample Output 2:\n1",
        "solutions": "",
        "difficulty": "competition",
        "input": "5 2 4\n1 3\n3 4\n2 4\n4 5\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/conveyorbelts"
    },
    {
        "id": 1086,
        "task_id": 3112,
        "test_case_id": 3,
        "question": "Your factory has $N$ junctions (numbered from $1$ to $N$) connected by $M$ conveyor belts. Each conveyor belt transports any product automatically from one junction to another junction in exactly one minute. Note that each conveyor belt only works in one direction. There can be more than one conveyor belt connecting two junctions, and there can be a conveyor belt connecting a junction to itself.\n\nThere are $K$ producers (machines which produce the products) located at the first $K$ junctions, i.e. junctions $1, 2, \\ldots , K$. The producer at junction $j$ produces an product each minute $(x \\cdot K + j)$ for all integers $x \\ge 0$ and $j = 1, 2, \\ldots , K$. All products are transported immediately via the conveyor belts to the warehouse at junction $N$, except for those produced at junction $N$ (if any). Items produced at junction $N$ are directly delivered to the warehouse (there is no need to use the conveyor belts).\n\nAt each junction, there is a robot deciding which conveyor belts the incoming product should go to in a negligible time (instantly). The robots can be programmed such that all products produced by a producer are always delivered to the warehouse via the same route. Once the robots are programmed, the routing can no longer be changed. Items from different producers may have the same or different routes.\n\nA prudent potential investor comes and wants to inspect the factory before making any decision. You want to show to the potential investor that your factory employs a good risk management policy. Thus, you want to make sure that each conveyor belt only transports at most one product at any time; i.e. two products cannot be on the same conveyor belt at the same time. On the other hand, there is no limit on the number of products at the junctions (the robots have a lot of arms!). To achieve this, you may turn off zero or more producers, but you still want to maximize the production, hence, this problem.\n\nFind the maximum number of producers that can be left running such that all the produced products can be delivered to the warehouse and each conveyor belt transports at most $1$ product at any time.\n\n-----Input-----\nThe first line contains three integers $N$, $K$, and $M$ ($1 \\le K \\le N \\le 300$; $0 \\le M \\le 1000$) representing the number of junctions, the number of producers, and the number of conveyor belts, respectively.\n\nThe next $M$ lines, each contains two integers $a$ and $b$ ($1 \\le a, b \\le N$) representing a conveyor belt connecting junction $a$ and junction $b$ with the direction from $a$ to $b$.\n\n-----Output-----\nThe output contains an integer denoting the maximum number of producers which can be left running such that all the produced products can be delivered to the warehouse and each conveyor belt transports at most one product at any time.\n\n-----Explanation-----\nIn Sample Input $1$, $N = 4$, $K = 2$, $M = 3$, and the directed edges are $\\{ (1,3)$, $(2,3)$, $(3,4)\\} $. There is only one possible delivery route for each producer, i.e. $1 \\rightarrow 3 \\rightarrow 4$ for producer $1$, and $2 \\rightarrow 3 \\rightarrow 4$ for producer $2$. Both producers are using conveyor belt $(3,4)$, however, the products from producer $1$ are on the conveyor belt $(3,4)$ on minutes $2, 4, 6, \\dots $ (even minutes), while the products from producer $2$ are on the conveyor belt $(3,4)$ on minutes $3, 5, 7, \\dots $ (odd minutes). Therefore, both producers can be left running.\n\nIn Sample Input $2$, $N = 5$, $K = 2$, $M = 4$, and the directed edges are $\\{ (1,3)$, $(3,4)$, $(2,4)$, $(4,5)\\} $. Similar to the previous example, there is only one possible delivery route for each product produced by each producer. In this example, only one producer can be left running as products from both producers ($1$ and $2$) are on the conveyor belt $(4,5)$ at the same time if both are running.\n\n-----Examples-----\nSample Input 1:\n4 2 3\n1 3\n2 3\n3 4\nSample Output 1:\n2\n\nSample Input 2:\n5 2 4\n1 3\n3 4\n2 4\n4 5\nSample Output 2:\n1",
        "solutions": "",
        "difficulty": "competition",
        "input": "5 2 6\n1 4\n2 3\n3 4\n4 5\n2 4\n3 3\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/conveyorbelts"
    },
    {
        "id": 1087,
        "task_id": 3268,
        "test_case_id": 3,
        "question": "One day, a flock of hungry birds find a huge tree of elderberries. They want to eat all berries of this tree. These birds quickly alight on some leaves of the tree and eat all berries at where they stand. Now they are thinking of who will eat other berries …\n\nSince the flock has some giant birds and some tiny birds, and they observe that the tree has some big branches and some small branches, fairly distributing these berries among all birds turns out to be a tough problem.\n\nFormally, the tree of elderberries can be presented as a tree with the following properties:\n - The tree has $n$ vertices numbered from $1$ to $n$. The root of the tree is vertex $1$.\n - Each non-leaf vertex (vertex having at least one child) is called a branch. A branch is classified as either big or small. The root is always a big branch.\n - On each leaf of the tree, there is either one giant bird, one tiny bird or one elderberry.\n\nThe process of determining which bird eats which berry is as below:\n - First, every bird and every berry has a label. A label is a non-empty string of at most five lowercase English characters.\n - Second, every bird has a controlled area. Consider the bird at vertex $v$:\n - If it is a tiny bird, its controlled area is the subtree rooted at vertex $p_ v$, where $p_ v$ is the parent of $v$.\n - If it is a giant bird, its controlled area is the subtree rooted at vertex $b_ v$, where $b_ v$ is the closest vertex to $v$ among all ancestors of $v$ which are big branches.\n - Finally, for each berry, the bird who eats it is determined using the following rules:\n - A bird can only eat berries having the same label with it.\n - A bird can only eat berries inside its controlled area.\n - If there is more than one bird satisfying the two above conditions, only one bird with smallest controlled area eats it.\n\nAll birds also set some rules for labeling birds and berries:\n - A bird or a berry can have same label with other birds and berries. However, no groups of eight or more birds have the same label.\n - Each berry is inside the controlled area of at least one bird with same label.\n - No two birds with same label have the same controlled area.\n - From all of the above, it can be proven that the bird eating every berry can be uniquely determined.\n\nSadly, tiny birds think that these rules give advantages to giant birds over them. They claim that, by the rules of setting controlled areas, giant birds usually control larger areas, which means having more berries. (Although their thought is not always true.) They want their controlled areas to be determined using the rules of giant birds. (In other words, they want all tiny birds become giant birds).\n\nGiant birds accept the tiny birds’ claim. However, they try to change the labels of some birds and berries, so that if we assume all tiny birds become giant, the new set of labels still satisfy all the above conditions. Moreover, after changing labels, no berries have changed owner (i.e, every berry is still eaten by the same bird).\n\nThey want to change as few labels as possible. Please help them!\n\n-----Input-----\nThe first line contains one integer $n$ $(3 \\leq n \\leq 150000)$ — the number of vertices in the elderberry tree. The $i$-th of the next $n$ lines describes the $i$-th vertex in either of the following formats:\n - ‘$p_ i$$t_ i$’, where $t_ i$ is either ‘B’ or ‘S’: meaning that $i$ is a branch. $t_ i = B$ and $t_ i = S$ mean that $i$ is a big or small branch, respectively.\n - ‘$p_ i$$T_ i$ $L$’, where $T_ i$ is either ‘G’, ‘T’ or ‘E’ and $L$ is a non-empty string of at most five English characters: meaning that $i$ is a leaf. $t_ i = G, t_ i = T$ and $t_ i = E$ mean that in vertex $i$ there is either a giant bird, a tiny bird or an elderberry. $L$ is the label of this bird or berry.\n\nIn both formats, $p_ i$ satisfies $1 \\leq p_ i < i$ and denotes the parent of vertex $i$. We assume that $p_1 = 0$.\n\nIt is guaranteed that the input describes a valid tree, there is at least one bird and one berry, and all labels satisfy the rules presented above. Remember that no eight birds have the same label.\n\n-----Output-----\n - The first line contains an integer $k$ — the minimum number of labels need changing.\n - Each of the next $k$ lines contains an integer $x$ $(1 \\leq x \\leq n)$ and a label $S$ — meaning that the label of the bird or berry at vertex $x$ is assigned to $S$. $x$ should be a leaf of the tree. $S$ should be a non-empty string of at most five lowercase English characters.\n\nIf there are multiple optimal solutions, you can output any one of them.\n\n-----Explanation for examples-----\nBelow are figures depicting these three examples. Red vertices are big branches, green vertices are small branches, violet vertices have giant birds, yellow vertices have tiny birds and white vertices have berries.\n - In the first example:\n - Initially:\n - There are $3$ birds with label ‘a’ at vertices $6$, $7$ and $13$. Their controlled areas are the subtrees rooted at vertices $2$, $5$ and $1$, respectively.\n - There is $1$ bird with label ‘b’ at vertex $12$. Its controlled area is the subtree rooted at vertex $1$.\n - There are $3$ elderberries with label ‘a’ at vertices $3$, $8$ and $11$. They are eaten by the birds at vertices $6$, $7$ and $13$, respectively.\n - There are $2$ elderberries with label ‘b’ at vertices $4$ and $9$. They are both eaten by the bird at vertex $12$.\n - If all tiny birds become giant birds, both the controlled area of the birds at vertices $6$ and $7$ become the subtree rooted at vertex $2$, violating the rules (No two birds with same label having same controlled area). Hence, at least one bird’s label needs changing. We can change the label of the bird at vertex $6$ to ‘c’. As the bird at vertex $6$ eats the berry at vertex $3$, we need to change the label of the bird at vertex $3$ as well. The solution in which the bird/berry at vertices $7$ and $8$ get changed is also accepted.\n - In the second example:\n - Initially:\n - The controlled areas of the birds at vertices $3$ and $6$ are subtrees rooted at vertices $1$ and $5$, respectively.\n - The bird at vertex $3$ eats the berry at vertex $4$.\n - If all tiny birds become giant birds, their controlled areas are subtrees rooted at vertices $1$ and $2$, respectively. As a result, the bird at vertex $6$ eats the berry at vertex $4$, which is invalid, (The owner of every berry must not change). Hence the label of the bird at vertex $6$ must be changed.\n - In the third example, no changes are necessary.\n\n-----Examples-----\nSample Input 1:\n13\n0 B\n1 B\n2 E a\n2 E b\n2 S\n5 G a\n5 T a\n5 E a\n5 E b\n1 S\n10 E a\n10 G b\n1 T a\nSample Output 1:\n2\n3 c\n6 c\n\nSample Input 2:\n6\n0 B\n1 B\n1 T a\n2 E a\n2 S\n5 T a\nSample Output 2:\n1\n6 b",
        "solutions": "",
        "difficulty": "competition",
        "input": "6\n0 B\n1 G y\n1 E y\n1 E z\n1 T z\n1 E z\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/enjoyingelderberries"
    },
    {
        "id": 1088,
        "task_id": 3497,
        "test_case_id": 1,
        "question": "You have all heard of the story of the three little pigs. However, this time they’re facing another predicament yet again!\n\nThe three pigs brought a bunch of their pig friends into a tree, which is a connected undirected graph with $V$ vertices and $V-1$ edges. Originally there were $V$ pigs, one on each vertex of the tree. However, $V-P$ big bad wolves suddenly teleported into $V-P$ distinct vertices of $T$ and ate the pigs originally on these vertices. Now there are only $P$ pigs left, and every vertex of $T$ is occupied by a single pig or a single wolf (but not both).\n\nOf course, the surviving pigs now wants to escape. A pig is considered to have escaped if he is on any leaf vertex (a vertex connected to only one edge). A pig currently on vertex $u$ can move to vertex $v$ only if there is an edge connecting $u$ to $v$ and $v$ isn’t occupied by a wolf. However, more than one pig may be on the same vertex at the same time. The pigs want you to help them compute the minimum number of wolves which must be removed such that every pig can escape.\n\n-----Input-----\nThe first line of the input contains two integers $V$ and $P$, $3\\leq P\\leq V\\leq 200000$. This is followed by $V-1$ lines, each containing two integers $u$ and $v$, where $0\\leq u,v\\leq V-1$, indicating that there is an edge between vertex $u$ and vertex $v$. It is guaranteed that the graph represented by the input is a tree.\n\nThe last line of the input contains $P$ integers, where $p_ i$ ($0\\leq p_ i\\leq V-1$) denotes the initial vertex occupied by the $i^\\mathrm {th}$ pig. It is guaranteed that no two pigs occupy the same vertex.\n\n-----Output-----\nOutput $W$, an integer denoting the minimum number of wolves to remove such that every pig can escape.\n\n-----Examples-----\nSample Input:\n6 3\n0 1\n1 2\n2 3\n2 4\n1 5\n1 2 5 \nSample Output:\n0",
        "solutions": "",
        "difficulty": "competition",
        "input": "6 3\n0 1\n1 2\n2 3\n2 4\n1 5\n1 2 5 \n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/theescape"
    },
    {
        "id": 1089,
        "task_id": 3497,
        "test_case_id": 2,
        "question": "You have all heard of the story of the three little pigs. However, this time they’re facing another predicament yet again!\n\nThe three pigs brought a bunch of their pig friends into a tree, which is a connected undirected graph with $V$ vertices and $V-1$ edges. Originally there were $V$ pigs, one on each vertex of the tree. However, $V-P$ big bad wolves suddenly teleported into $V-P$ distinct vertices of $T$ and ate the pigs originally on these vertices. Now there are only $P$ pigs left, and every vertex of $T$ is occupied by a single pig or a single wolf (but not both).\n\nOf course, the surviving pigs now wants to escape. A pig is considered to have escaped if he is on any leaf vertex (a vertex connected to only one edge). A pig currently on vertex $u$ can move to vertex $v$ only if there is an edge connecting $u$ to $v$ and $v$ isn’t occupied by a wolf. However, more than one pig may be on the same vertex at the same time. The pigs want you to help them compute the minimum number of wolves which must be removed such that every pig can escape.\n\n-----Input-----\nThe first line of the input contains two integers $V$ and $P$, $3\\leq P\\leq V\\leq 200000$. This is followed by $V-1$ lines, each containing two integers $u$ and $v$, where $0\\leq u,v\\leq V-1$, indicating that there is an edge between vertex $u$ and vertex $v$. It is guaranteed that the graph represented by the input is a tree.\n\nThe last line of the input contains $P$ integers, where $p_ i$ ($0\\leq p_ i\\leq V-1$) denotes the initial vertex occupied by the $i^\\mathrm {th}$ pig. It is guaranteed that no two pigs occupy the same vertex.\n\n-----Output-----\nOutput $W$, an integer denoting the minimum number of wolves to remove such that every pig can escape.\n\n-----Examples-----\nSample Input:\n6 3\n0 1\n1 2\n2 3\n2 4\n1 5\n1 2 5 \nSample Output:\n0",
        "solutions": "",
        "difficulty": "competition",
        "input": "11 3\n0 1\n1 2\n0 3\n3 4\n4 5\n5 6\n0 7\n7 8\n8 9\n9 10\n1 3 9 \n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/theescape"
    },
    {
        "id": 1090,
        "task_id": 3642,
        "test_case_id": 1,
        "question": "You are given an undirected tree1 with each of its node assigned a magic $X_ i$. The magic of a path2 is defined as the product of the magic of the nodes on that path divided by the number of the nodes on the path. For example, the magic of a path that consists of nodes with magic $3$ and $5$ is $7.5$ ($3\\cdot 5 / 2$). In the given tree, find the path with the minimal magic and output the magic of that path.\n\n-----Input-----\nThe first line of input contains the integer $N$ ($1 \\leq N \\leq 10^6$), the number of nodes in the tree. Each of the following $N - 1$ lines contains two integers, $A_ i$ and $B_ i$ ($1 \\leq A_ i, B_ i \\leq N$), the labels of nodes connected with an edge. The $i$-th of the following $N$ lines contains the integer $X_ i$ ($1 \\leq X_ i \\leq 10^9$), magic of the $i$-th node.\n\n-----Output-----\nOutput the magic of the path with minimal magic in the form of a completely reduced fraction $P/Q$ ($P$ and $Q$ are relatively prime integers).\n\nIn all test cases, it will hold that the required $P$ and $Q$ are smaller than $10^{18}$.\n\n-----Examples-----\nSample Input:\n2\n1 2\n3\n4\nSample Output:\n3/1",
        "solutions": "",
        "difficulty": "competition",
        "input": "2\n1 2\n3\n4\n",
        "output": "3/1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/mag"
    },
    {
        "id": 1091,
        "task_id": 3642,
        "test_case_id": 2,
        "question": "You are given an undirected tree1 with each of its node assigned a magic $X_ i$. The magic of a path2 is defined as the product of the magic of the nodes on that path divided by the number of the nodes on the path. For example, the magic of a path that consists of nodes with magic $3$ and $5$ is $7.5$ ($3\\cdot 5 / 2$). In the given tree, find the path with the minimal magic and output the magic of that path.\n\n-----Input-----\nThe first line of input contains the integer $N$ ($1 \\leq N \\leq 10^6$), the number of nodes in the tree. Each of the following $N - 1$ lines contains two integers, $A_ i$ and $B_ i$ ($1 \\leq A_ i, B_ i \\leq N$), the labels of nodes connected with an edge. The $i$-th of the following $N$ lines contains the integer $X_ i$ ($1 \\leq X_ i \\leq 10^9$), magic of the $i$-th node.\n\n-----Output-----\nOutput the magic of the path with minimal magic in the form of a completely reduced fraction $P/Q$ ($P$ and $Q$ are relatively prime integers).\n\nIn all test cases, it will hold that the required $P$ and $Q$ are smaller than $10^{18}$.\n\n-----Examples-----\nSample Input:\n2\n1 2\n3\n4\nSample Output:\n3/1",
        "solutions": "",
        "difficulty": "competition",
        "input": "5\n1 2\n2 4\n1 3\n5 2\n2\n1\n1\n1\n3\n",
        "output": "1/2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/mag"
    },
    {
        "id": 1092,
        "task_id": 3784,
        "test_case_id": 1,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "3 2\n",
        "output": "6\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1093,
        "task_id": 3784,
        "test_case_id": 2,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "4 4\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1094,
        "task_id": 3784,
        "test_case_id": 3,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 3\n",
        "output": "1196\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1095,
        "task_id": 3784,
        "test_case_id": 4,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "31 8\n",
        "output": "64921457\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1096,
        "task_id": 3784,
        "test_case_id": 6,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "10 2\n",
        "output": "141356\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1097,
        "task_id": 3784,
        "test_case_id": 7,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "33 22\n",
        "output": "804201731\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1098,
        "task_id": 3784,
        "test_case_id": 8,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "50 50\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1099,
        "task_id": 3784,
        "test_case_id": 14,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "3 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1100,
        "task_id": 3784,
        "test_case_id": 15,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "3 3\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1101,
        "task_id": 3784,
        "test_case_id": 16,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "3 4\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1102,
        "task_id": 3784,
        "test_case_id": 17,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "4 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1103,
        "task_id": 3784,
        "test_case_id": 18,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "4 2\n",
        "output": "20\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1104,
        "task_id": 3784,
        "test_case_id": 19,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "4 3\n",
        "output": "15\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1105,
        "task_id": 3784,
        "test_case_id": 20,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "4 5\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1106,
        "task_id": 3784,
        "test_case_id": 21,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "5 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1107,
        "task_id": 3784,
        "test_case_id": 22,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "5 2\n",
        "output": "78\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1108,
        "task_id": 3784,
        "test_case_id": 23,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "5 3\n",
        "output": "60\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1109,
        "task_id": 3784,
        "test_case_id": 24,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "5 4\n",
        "output": "18\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1110,
        "task_id": 3784,
        "test_case_id": 25,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "5 5\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1111,
        "task_id": 3784,
        "test_case_id": 26,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "5 6\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1112,
        "task_id": 3784,
        "test_case_id": 27,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "6 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1113,
        "task_id": 3784,
        "test_case_id": 28,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "6 2\n",
        "output": "320\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1114,
        "task_id": 3784,
        "test_case_id": 29,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "6 3\n",
        "output": "269\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1115,
        "task_id": 3784,
        "test_case_id": 30,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "6 4\n",
        "output": "90\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1116,
        "task_id": 3784,
        "test_case_id": 31,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "6 5\n",
        "output": "19\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1117,
        "task_id": 3784,
        "test_case_id": 32,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "6 6\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1118,
        "task_id": 3784,
        "test_case_id": 33,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "6 7\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1119,
        "task_id": 3784,
        "test_case_id": 34,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 1\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1120,
        "task_id": 3784,
        "test_case_id": 35,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 2\n",
        "output": "1404\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1121,
        "task_id": 3784,
        "test_case_id": 36,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 4\n",
        "output": "452\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1122,
        "task_id": 3784,
        "test_case_id": 37,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 5\n",
        "output": "102\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1123,
        "task_id": 3784,
        "test_case_id": 38,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 6\n",
        "output": "19\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1124,
        "task_id": 3784,
        "test_case_id": 39,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 7\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1125,
        "task_id": 3784,
        "test_case_id": 40,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "7 8\n",
        "output": "1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1126,
        "task_id": 3784,
        "test_case_id": 41,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "8 5\n",
        "output": "566\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1127,
        "task_id": 3784,
        "test_case_id": 42,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "9 2\n",
        "output": "29660\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1128,
        "task_id": 3784,
        "test_case_id": 43,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "10 4\n",
        "output": "55564\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1129,
        "task_id": 3784,
        "test_case_id": 44,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "15 12\n",
        "output": "625\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1130,
        "task_id": 3784,
        "test_case_id": 45,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "45 19\n",
        "output": "486112971\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1131,
        "task_id": 3784,
        "test_case_id": 46,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "48 20\n",
        "output": "804531912\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1132,
        "task_id": 3784,
        "test_case_id": 47,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "49 2\n",
        "output": "987390633\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1133,
        "task_id": 3784,
        "test_case_id": 48,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "50 2\n",
        "output": "637245807\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1134,
        "task_id": 3784,
        "test_case_id": 49,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "50 33\n",
        "output": "805999139\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1135,
        "task_id": 3784,
        "test_case_id": 50,
        "question": "A never-ending, fast-changing and dream-like world unfolds, as the secret door opens.\n\nA world is an unordered graph G, in whose vertex set V(G) there are two special vertices s(G) and t(G). An initial world has vertex set {s(G), t(G)} and an edge between them.\n\nA total of n changes took place in an initial world. In each change, a new vertex w is added into V(G), an existing edge (u, v) is chosen, and two edges (u, w) and (v, w) are added into E(G). Note that it's possible that some edges are chosen in more than one change.\n\nIt's known that the capacity of the minimum s-t cut of the resulting graph is m, that is, at least m edges need to be removed in order to make s(G) and t(G) disconnected.\n\nCount the number of non-similar worlds that can be built under the constraints, modulo 10^9 + 7. We define two worlds similar, if they are isomorphic and there is isomorphism in which the s and t vertices are not relabelled. Formally, two worlds G and H are considered similar, if there is a bijection between their vertex sets $f : V(G) \\rightarrow V(H)$, such that:   f(s(G)) = s(H);  f(t(G)) = t(H);  Two vertices u and v of G are adjacent in G if and only if f(u) and f(v) are adjacent in H. \n\n\n-----Input-----\n\nThe first and only line of input contains two space-separated integers n, m (1 ≤ n, m ≤ 50) — the number of operations performed and the minimum cut, respectively.\n\n\n-----Output-----\n\nOutput one integer — the number of non-similar worlds that can be built, modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n3 2\n\nOutput\n6\n\nInput\n4 4\n\nOutput\n3\n\nInput\n7 3\n\nOutput\n1196\n\nInput\n31 8\n\nOutput\n64921457\n\n\n\n-----Note-----\n\nIn the first example, the following 6 worlds are pairwise non-similar and satisfy the constraints, with s(G) marked in green, t(G) marked in blue, and one of their minimum cuts in light blue.\n\n [Image] \n\nIn the second example, the following 3 worlds satisfy the constraints.\n\n [Image]",
        "solutions": "[\"mod = int(1e9 + 7)\\nn, m = map(int, input().split())\\nf = [ [0 for i in range(60)] for j in range(60) ]\\ng = [ [0 for i in range(60)] for j in range(60) ]\\ns = [ [0 for i in range(60)] for j in range(60) ]\\ninv = [ 1 ]\\nf[0][0] = s[0][0] = 1\\n\\ndef pow(x, exp) :\\n    res = 1\\n    for i in range(0, 31) :\\n        if exp & 1 : res = res * x % mod\\n        exp >>= 1\\n        if exp == 0 : break\\n        x = x * x % mod\\n    return res\\n\\nfor i in range(1, n + 1) :\\n    inv.append( pow(i, mod - 2) )\\n\\nfor node in range(1, n + 1) :\\n    for cut in range(1, n + 1) :\\n        tmp = 0\\n        for ln in range(node) :\\n            for lc in range(cut - 1, n + 1) :\\n                if f[ln][lc] == 0 : continue\\n                if lc == cut - 1 :\\n                    tmp = ( tmp + f[ln][lc] * s[node - ln - 1][cut - 1] ) % mod\\n                else :\\n                    tmp = ( tmp + f[ln][lc] * f[node - ln - 1][cut - 1] ) % mod\\n        cnt = 1\\n        if tmp != 0 :\\n            cn, cc = 0, 0\\n            for i in range(1, n + 1) :\\n                cn += node\\n                cc += cut\\n                cnt = cnt * (tmp + i - 1) % mod * inv[i] % mod\\n                if cn > n or cc > n : break\\n                for j in range(n - cn, -1, -1) :\\n                    for k in range(n - cc, -1, -1) :\\n                        if f[j][k] == 0 : continue\\n                        g[j + cn][k + cc] += f[j][k] * cnt\\n                        g[j + cn][k + cc] %= mod\\n            for i in range(n + 1) :\\n                for j in range(n + 1) :\\n                    f[i][j] = (f[i][j] + g[i][j]) % mod\\n                    g[i][j] = 0\\n            \\n    for cut in range(n, -1, -1) :\\n        s[node][cut] = ( s[node][cut + 1] + f[node][cut] ) % mod\\nprint(f[n][m - 1])\"]",
        "difficulty": "competition",
        "input": "50 49\n",
        "output": "19\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/848/D"
    },
    {
        "id": 1136,
        "task_id": 3801,
        "test_case_id": 1,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "2 1\n0 1\n2 1\n",
        "output": "332748119\n332748119\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1137,
        "task_id": 3801,
        "test_case_id": 2,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "1 2\n1\n1\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1138,
        "task_id": 3801,
        "test_case_id": 3,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "3 3\n0 1 1\n4 3 5\n",
        "output": "160955686\n185138929\n974061117\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1139,
        "task_id": 3801,
        "test_case_id": 4,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "5 5\n0 1 0 0 1\n9 8 3 8 8\n",
        "output": "45170585\n105647559\n680553097\n483815788\n105647559\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1140,
        "task_id": 3801,
        "test_case_id": 5,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n0 1 0 0 1 1 1 1 1 1\n12 18 6 18 7 2 9 18 1 9\n",
        "output": "199115375\n823101465\n598679864\n797795239\n486469073\n424203836\n910672909\n823101465\n212101918\n910672909\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1141,
        "task_id": 3801,
        "test_case_id": 6,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "20 20\n1 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1\n1 13 7 11 17 15 19 18 14 11 15 1 12 4 5 16 14 11 18 9\n",
        "output": "688505688\n964619120\n826562404\n585852097\n851622699\n345141790\n104431483\n414170148\n349014804\n585852097\n516550769\n688505688\n13942874\n670143860\n447795381\n684086734\n654880455\n585852097\n20914311\n207085074\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1142,
        "task_id": 3801,
        "test_case_id": 7,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 50\n0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0\n1 2 1 1 2 1 1 1 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 2 2 1 1 2 2 2\n",
        "output": "346646202\n693292404\n346646202\n346646202\n693292404\n346646202\n346646202\n346646202\n346646202\n346646202\n346646202\n693292404\n346646202\n346646202\n693292404\n346646202\n346646202\n346646202\n693292404\n346646202\n346646202\n693292404\n346646202\n542025302\n693292404\n346646202\n346646202\n693292404\n693292404\n693292404\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1143,
        "task_id": 3801,
        "test_case_id": 8,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "40 40\n1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n28 13 22 35 22 13 23 35 14 36 30 10 10 15 3 9 35 35 9 29 14 28 8 29 22 30 4 31 39 24 4 19 37 4 20 7 11 17 3 25\n",
        "output": "368107101\n848286965\n360530176\n210572788\n199380339\n848286965\n195418938\n210572788\n683175727\n45461550\n37884625\n544374860\n345376326\n518064489\n502910639\n510487564\n210572788\n210572788\n510487564\n202995863\n683175727\n526005255\n675598802\n202995863\n360530176\n37884625\n337799401\n871017740\n548372189\n30307700\n337799401\n855863890\n878594665\n337799401\n690752652\n840710040\n180265088\n187842013\n502910639\n863440815\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1144,
        "task_id": 3801,
        "test_case_id": 9,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n1 1 1 1 1 1 1 1 1 1\n1 2 2 1 2 2 2 1 1 1\n",
        "output": "665496237\n332748121\n332748121\n665496237\n332748121\n332748121\n332748121\n665496237\n665496237\n665496237\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1145,
        "task_id": 3801,
        "test_case_id": 10,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n1 1 1 1 1 1 1 0 1 1\n2 1 2 2 1 1 1 1 1 1\n",
        "output": "771370640\n385685320\n771370640\n771370640\n385685320\n385685320\n385685320\n635246407\n385685320\n385685320\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1146,
        "task_id": 3801,
        "test_case_id": 11,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n0 0 0 1 0 0 0 0 0 0\n2 2 2 2 2 2 2 1 2 2\n",
        "output": "973938381\n973938381\n973938381\n791643586\n973938381\n973938381\n973938381\n986091367\n973938381\n973938381\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1147,
        "task_id": 3801,
        "test_case_id": 12,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n0 0 1 0 0 0 1 0 0 0\n2 1 2 1 1 2 1 1 1 1\n",
        "output": "44896189\n521570271\n482402083\n521570271\n521570271\n44896189\n740323218\n521570271\n521570271\n521570271\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1148,
        "task_id": 3801,
        "test_case_id": 13,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n1 0 0 0 1 1 1 0 1 0\n1 2 1 2 1 1 2 2 2 1\n",
        "output": "910950063\n595918255\n797081304\n595918255\n910950063\n910950063\n823655773\n595918255\n823655773\n797081304\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1149,
        "task_id": 3801,
        "test_case_id": 14,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n1 1 1 1 1 1 1 1 1 1\n17 10 8 34 5 4 3 44 20 14\n",
        "output": "709444118\n6278277\n803618104\n420643883\n502261315\n401809052\n301356789\n426922160\n12556554\n408087329\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1150,
        "task_id": 3801,
        "test_case_id": 15,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n1 1 1 1 1 1 1 1 0 1\n40 36 29 4 36 35 9 38 40 18\n",
        "output": "59109317\n951618303\n17898146\n105735367\n951618303\n675623373\n487465664\n505363810\n736385984\n974931328\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1151,
        "task_id": 3801,
        "test_case_id": 16,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n0 0 0 0 0 0 0 1 0 0\n8 33 37 18 30 48 45 34 25 48\n",
        "output": "211347083\n497465085\n104016450\n725092025\n542990473\n269838145\n315363533\n227335634\n286118002\n269838145\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1152,
        "task_id": 3801,
        "test_case_id": 17,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n0 0 1 0 0 0 0 0 1 0\n47 34 36 9 3 16 17 46 47 1\n",
        "output": "167709201\n57603825\n597597985\n690531016\n562925123\n673030499\n527924089\n312815611\n253346183\n853137943\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1153,
        "task_id": 3801,
        "test_case_id": 18,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n1 0 0 1 1 0 1 0 0 1\n24 7 10 9 6 13 27 17 6 39\n",
        "output": "976715988\n573793375\n391885813\n865390672\n244178997\n209978251\n599683310\n965679188\n634429229\n89796951\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1154,
        "task_id": 3801,
        "test_case_id": 19,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n0 0 0 0 0 1 0 0 0 0\n34 34 34 34 34 34 34 34 34 34\n",
        "output": "971203339\n971203339\n971203339\n971203339\n971203339\n754874965\n971203339\n971203339\n971203339\n971203339\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1155,
        "task_id": 3801,
        "test_case_id": 20,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 10\n1 1 1 1 1 1 1 1 1 1\n43 43 43 43 43 43 43 43 43 43\n",
        "output": "44\n44\n44\n44\n44\n44\n44\n44\n44\n44\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1156,
        "task_id": 3801,
        "test_case_id": 21,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n1 2 2 2 1 1 2 1 1 1 2 1 1 2 1 2 1 2 1 2 1 2 2 2 1 2 2 2 2 1\n",
        "output": "260411572\n520823144\n520823144\n520823144\n260411572\n260411572\n520823144\n260411572\n260411572\n260411572\n520823144\n260411572\n260411572\n520823144\n260411572\n520823144\n260411572\n520823144\n260411572\n520823144\n260411572\n520823144\n520823144\n520823144\n260411572\n520823144\n520823144\n520823144\n520823144\n260411572\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1157,
        "task_id": 3801,
        "test_case_id": 22,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1\n2 1 1 1 2 1 1 2 1 1 1 2 1 1 1 1 2 2 2 2 1 2 1 2 1 1 1 1 2 1\n",
        "output": "720162001\n859203177\n859203177\n859203177\n720162001\n859203177\n859203177\n720162001\n859203177\n859203177\n859203177\n720162001\n859203177\n859203177\n859203177\n859203177\n720162001\n720162001\n720162001\n720162001\n859203177\n720162001\n859203177\n720162001\n427819009\n859203177\n859203177\n859203177\n720162001\n859203177\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1158,
        "task_id": 3801,
        "test_case_id": 23,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n2 1 1 2 1 2 2 2 2 2 1 1 2 2 2 1 2 2 2 1 2 1 1 1 1 1 2 1 1 1\n",
        "output": "188114875\n593179614\n593179614\n550614566\n593179614\n188114875\n188114875\n188114875\n188114875\n188114875\n593179614\n593179614\n188114875\n188114875\n188114875\n593179614\n188114875\n188114875\n188114875\n593179614\n188114875\n593179614\n593179614\n593179614\n593179614\n593179614\n188114875\n593179614\n593179614\n593179614\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1159,
        "task_id": 3801,
        "test_case_id": 24,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0\n1 1 1 1 2 1 2 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 2 2 1 1 2 1 2 2\n",
        "output": "593179614\n593179614\n593179614\n593179614\n188114875\n593179614\n188114875\n188114875\n593179614\n188114875\n593179614\n188114875\n593179614\n275307283\n188114875\n188114875\n593179614\n188114875\n275307283\n188114875\n593179614\n188114875\n188114875\n188114875\n593179614\n593179614\n188114875\n593179614\n188114875\n188114875\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1160,
        "task_id": 3801,
        "test_case_id": 25,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0\n1 1 1 2 2 1 2 1 2 1 1 2 2 2 1 2 1 1 1 1 1 2 1 2 1 2 1 2 1 1\n",
        "output": "297674502\n297674502\n297674502\n101192689\n595349004\n549718521\n101192689\n297674502\n595349004\n297674502\n549718521\n101192689\n101192689\n101192689\n549718521\n595349004\n297674502\n549718521\n297674502\n549718521\n297674502\n101192689\n549718521\n595349004\n297674502\n101192689\n297674502\n101192689\n297674502\n549718521\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1161,
        "task_id": 3801,
        "test_case_id": 26,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n23 45 44 49 17 36 32 26 40 8 36 11 5 19 41 16 7 38 23 40 13 16 24 44 22 13 1 2 32 31\n",
        "output": "42365832\n603712812\n124449607\n524276926\n161519661\n283321379\n362757265\n481911094\n203885493\n839372581\n283321379\n280673490\n399827319\n121801718\n683148698\n680500809\n360109376\n243603436\n42365832\n203885493\n240955547\n680500809\n521629037\n124449607\n561346980\n240955547\n479263205\n958526410\n362757265\n881738413\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1162,
        "task_id": 3801,
        "test_case_id": 27,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n41 39 15 34 45 27 18 7 48 33 46 11 24 16 35 43 7 31 26 17 30 15 5 9 29 20 21 37 3 7\n",
        "output": "61128841\n655563720\n98563838\n955457225\n295691514\n377063779\n916872088\n578393446\n115755411\n17191573\n235712813\n338478642\n556999882\n38585137\n895478524\n415648916\n578393446\n137148975\n437042480\n976850789\n197127676\n98563838\n698350848\n458436044\n257106377\n796914686\n736935985\n775521122\n818308250\n578393446\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1163,
        "task_id": 3801,
        "test_case_id": 28,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n29 38 18 19 46 28 12 5 46 17 31 20 24 33 9 6 47 2 2 41 34 2 50 5 47 10 40 21 49 28\n",
        "output": "528451192\n658031067\n259159750\n828137710\n218632982\n957717585\n838269402\n848401094\n218632982\n688426143\n942792071\n398871317\n678294451\n807874326\n129579875\n419134701\n787610942\n139711567\n139711567\n368476241\n378607933\n139711567\n498056116\n848401094\n787610942\n698557835\n797742634\n967849277\n927322509\n957717585\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1164,
        "task_id": 3801,
        "test_case_id": 29,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0\n1 12 9 1 5 32 38 25 34 31 27 43 13 38 48 40 5 42 20 45 1 4 35 38 1 44 31 42 8 37\n",
        "output": "399967190\n806628868\n604971651\n399967190\n3347244\n800038448\n225087925\n16736220\n621707871\n420050654\n816670600\n228435169\n208351705\n225087925\n231782413\n26777952\n3347244\n51806110\n13388976\n30125196\n399967190\n601624407\n23430708\n225087925\n399967190\n628402359\n420050654\n826712332\n205004461\n823365088\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1165,
        "task_id": 3801,
        "test_case_id": 30,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0\n5 20 47 27 17 5 18 30 43 23 44 6 47 8 23 41 2 46 49 33 45 27 33 16 36 2 42 36 8 23\n",
        "output": "114252107\n760713694\n489959522\n18014766\n787754905\n689300600\n484993454\n142826188\n936763395\n126261951\n805769671\n827160720\n475023194\n781749983\n176049701\n138271795\n444998584\n252523902\n765679762\n354766165\n214239282\n727490181\n354766165\n565255613\n24019688\n275720240\n798903275\n969986908\n104636607\n126261951\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1166,
        "task_id": 3801,
        "test_case_id": 31,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1\n39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39\n",
        "output": "417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n417992317\n142843895\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1167,
        "task_id": 3801,
        "test_case_id": 32,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "30 30\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22\n",
        "output": "23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n23\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1168,
        "task_id": 3801,
        "test_case_id": 33,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "50 50\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8\n",
        "output": "9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1169,
        "task_id": 3801,
        "test_case_id": 34,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "5 50\n1 1 1 1 1\n1 1 4 2 3\n",
        "output": "635246412\n635246412\n544496942\n272248471\n907494883\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1170,
        "task_id": 3801,
        "test_case_id": 35,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "10 50\n0 0 0 0 0 0 0 0 1 0\n3 1 3 3 1 3 1 2 2 1\n",
        "output": "187134581\n727874429\n187134581\n187134581\n727874429\n187134581\n727874429\n457504505\n124563167\n727874429\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1171,
        "task_id": 3801,
        "test_case_id": 36,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "20 50\n1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1\n1 2 2 1 2 2 2 2 2 2 1 1 2 2 2 1 1 1 2 2\n",
        "output": "853605709\n708967065\n708967065\n853605709\n708967065\n708967065\n708967065\n922030188\n708967065\n922030188\n853605709\n853605709\n708967065\n922030188\n708967065\n461015094\n853605709\n853605709\n708967065\n708967065\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1172,
        "task_id": 3801,
        "test_case_id": 37,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "20 50\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n1 2 2 2 2 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1\n",
        "output": "436731907\n873463814\n873463814\n873463814\n873463814\n873463814\n873463814\n436731907\n873463814\n436731907\n873463814\n436731907\n873463814\n436731907\n436731907\n873463814\n436731907\n873463814\n873463814\n436731907\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1173,
        "task_id": 3801,
        "test_case_id": 38,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "40 50\n0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0\n33 26 4 19 42 43 19 32 13 23 19 1 18 43 43 43 19 31 4 25 28 23 33 37 36 23 5 12 18 32 34 1 21 22 34 35 37 16 41 39\n",
        "output": "729284231\n60340485\n239647233\n389641092\n20685064\n829280137\n389641092\n918933511\n529292419\n629288325\n366487398\n808595073\n579290372\n829280137\n829280137\n41331201\n389641092\n110338438\n239647233\n249989765\n679286278\n629288325\n426374038\n968931464\n160336391\n629288325\n49997953\n718941699\n579290372\n918933511\n539634951\n808595073\n89829960\n818937605\n539634951\n349985671\n968931464\n958588932\n210334344\n589632904\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1174,
        "task_id": 3801,
        "test_case_id": 39,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "41 50\n0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n1 2 4 2 3 2 4 3 1 1 3 1 3 2 5 3 3 5 4 4 1 1 2 3 2 1 5 5 5 4 2 2 2 1 2 4 4 5 2 1 4\n",
        "output": "394710173\n789420346\n580596339\n789420346\n185886166\n789420346\n580596339\n185886166\n394710173\n394710173\n185886166\n394710173\n581788048\n789420346\n636898629\n185886166\n185886166\n975306512\n580596339\n580596339\n394710173\n394710173\n55110581\n185886166\n55110581\n394710173\n975306512\n975306512\n975306512\n580596339\n789420346\n789420346\n789420346\n394710173\n789420346\n580596339\n580596339\n975306512\n789420346\n394710173\n580596339\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1175,
        "task_id": 3801,
        "test_case_id": 40,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "42 50\n0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0\n2 4 6 8 1 3 6 1 4 1 3 4 3 7 6 6 8 7 4 1 7 4 6 9 3 1 9 7 1 2 9 3 1 6 1 5 1 8 2 6 8 8\n",
        "output": "11284873\n329090227\n33854619\n45139492\n504764613\n995500935\n33854619\n504764613\n22569746\n504764613\n516049486\n22569746\n516049486\n538619232\n33854619\n33854619\n45139492\n538619232\n22569746\n504764613\n538619232\n22569746\n33854619\n549904105\n516049486\n504764613\n549904105\n538619232\n504764613\n11284873\n990014099\n516049486\n504764613\n33854619\n504764613\n527334359\n504764613\n45139492\n663667290\n33854619\n45139492\n45139492\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1176,
        "task_id": 3801,
        "test_case_id": 41,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "43 50\n1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n13 7 13 15 8 9 11 1 15 9 3 7 3 15 4 7 7 16 9 13 12 16 16 1 5 5 14 5 17 2 1 13 4 13 10 17 17 6 11 15 14 3 6\n",
        "output": "175780254\n94650906\n163530008\n802992688\n561362014\n881093354\n522311681\n319731340\n802992688\n881093354\n959194020\n241630674\n959194020\n802992688\n280681007\n241630674\n241630674\n124479675\n881093354\n163530008\n842043021\n124479675\n124479675\n13521558\n600412347\n600412347\n483261348\n67607790\n444211015\n639462680\n319731340\n163530008\n280681007\n163530008\n202580341\n444211015\n444211015\n920143687\n522311681\n802992688\n483261348\n959194020\n920143687\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1177,
        "task_id": 3801,
        "test_case_id": 42,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "44 50\n0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0\n2 6 6 11 2 4 11 10 5 15 15 20 20 7 9 8 17 4 16 19 12 16 12 13 2 11 20 2 6 10 2 18 7 5 18 10 15 6 11 9 7 5 17 11\n",
        "output": "327775237\n983325711\n983325711\n305397274\n327775237\n853173373\n305397274\n640631832\n320315916\n960947748\n960947748\n272889453\n283019311\n648091153\n975866390\n312856595\n290478632\n655550474\n625713190\n618253869\n968407069\n625713190\n968407069\n633172511\n327775237\n305397274\n283019311\n327775237\n983325711\n640631832\n327775237\n953488427\n648091153\n816905628\n953488427\n640631832\n960947748\n983325711\n305397274\n975866390\n648091153\n320315916\n290478632\n305397274\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1178,
        "task_id": 3801,
        "test_case_id": 43,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "45 50\n0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0\n4 4 23 23 13 23 9 16 4 18 20 15 21 24 22 20 22 1 15 7 10 17 20 6 15 7 4 10 16 7 14 9 13 17 10 14 22 23 3 5 20 11 4 24 24\n",
        "output": "630266647\n555616275\n379739073\n948743787\n301438985\n948743787\n669416691\n225976394\n555616275\n340589029\n156600176\n835755590\n563727926\n786866823\n560278630\n781592669\n970855676\n388465157\n835755590\n853405544\n889918511\n614441551\n156600176\n446277794\n117450132\n853405544\n630266647\n78300088\n225976394\n722767393\n708566735\n669416691\n58825276\n931705632\n78300088\n708566735\n970855676\n948743787\n223138897\n39150044\n781592669\n280139315\n555616275\n338964591\n786866823\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1179,
        "task_id": 3801,
        "test_case_id": 44,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "46 50\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n29 22 30 33 13 31 19 11 12 21 5 4 24 21 20 6 28 16 27 18 21 11 3 24 21 8 8 33 24 7 34 12 13 32 26 33 33 22 18 2 3 7 24 17 9 30\n",
        "output": "265429165\n98093399\n859759619\n646262275\n738585431\n455845720\n311590743\n548168876\n144254977\n502007298\n975163564\n380833110\n288509954\n502007298\n905921197\n571249665\n669343064\n525088087\n75012610\n715504642\n502007298\n548168876\n784747009\n288509954\n502007298\n761666220\n761666220\n646262275\n288509954\n167335766\n242348376\n144254977\n738585431\n51931821\n478926509\n646262275\n646262275\n98093399\n715504642\n190416555\n784747009\n167335766\n288509954\n121174188\n357752321\n859759619\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1180,
        "task_id": 3801,
        "test_case_id": 45,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "50 50\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n",
        "output": "2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1181,
        "task_id": 3801,
        "test_case_id": 46,
        "question": "The only difference between easy and hard versions is constraints.\n\nNauuo is a girl who loves random picture websites.\n\nOne day she made a random picture website by herself which includes $n$ pictures.\n\nWhen Nauuo visits the website, she sees exactly one picture. The website does not display each picture with equal probability. The $i$-th picture has a non-negative weight $w_i$, and the probability of the $i$-th picture being displayed is $\\frac{w_i}{\\sum_{j=1}^nw_j}$. That is to say, the probability of a picture to be displayed is proportional to its weight.\n\nHowever, Nauuo discovered that some pictures she does not like were displayed too often. \n\nTo solve this problem, she came up with a great idea: when she saw a picture she likes, she would add $1$ to its weight; otherwise, she would subtract $1$ from its weight.\n\nNauuo will visit the website $m$ times. She wants to know the expected weight of each picture after all the $m$ visits modulo $998244353$. Can you help her?\n\nThe expected weight of the $i$-th picture can be denoted by $\\frac {q_i} {p_i}$ where $\\gcd(p_i,q_i)=1$, you need to print an integer $r_i$ satisfying $0\\le r_i<998244353$ and $r_i\\cdot p_i\\equiv q_i\\pmod{998244353}$. It can be proved that such $r_i$ exists and is unique.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1\\le n\\le 50$, $1\\le m\\le 50$) — the number of pictures and the number of visits to the website.\n\nThe second line contains $n$ integers $a_1,a_2,\\ldots,a_n$ ($a_i$ is either $0$ or $1$) — if $a_i=0$ , Nauuo does not like the $i$-th picture; otherwise Nauuo likes the $i$-th picture. It is guaranteed that there is at least one picture which Nauuo likes.\n\nThe third line contains $n$ integers $w_1,w_2,\\ldots,w_n$ ($1\\le w_i\\le50$) — the initial weights of the pictures.\n\n\n-----Output-----\n\nThe output contains $n$ integers $r_1,r_2,\\ldots,r_n$ — the expected weights modulo $998244353$.\n\n\n-----Examples-----\nInput\n2 1\n0 1\n2 1\n\nOutput\n332748119\n332748119\n\nInput\n1 2\n1\n1\n\nOutput\n3\n\nInput\n3 3\n0 1 1\n4 3 5\n\nOutput\n160955686\n185138929\n974061117\n\n\n\n-----Note-----\n\nIn the first example, if the only visit shows the first picture with a probability of $\\frac 2 3$, the final weights are $(1,1)$; if the only visit shows the second picture with a probability of $\\frac1 3$, the final weights are $(2,2)$.\n\nSo, both expected weights are $\\frac2 3\\cdot 1+\\frac 1 3\\cdot 2=\\frac4 3$ .\n\nBecause $332748119\\cdot 3\\equiv 4\\pmod{998244353}$, you need to print $332748119$ instead of $\\frac4 3$ or $1.3333333333$.\n\nIn the second example, there is only one picture which Nauuo likes, so every time Nauuo visits the website, $w_1$ will be increased by $1$.\n\nSo, the expected weight is $1+2=3$.\n\nNauuo is very naughty so she didn't give you any hint of the third example.",
        "solutions": "[\"P = 998244353\\nN, M = list(map(int, input().split()))\\nA = [int(a) for a in input().split()]\\nB = [int(a) for a in input().split()]\\nli = sum([A[i]*B[i] for i in range(N)])\\ndi = sum([(A[i]^1)*B[i] for i in range(N)])\\nX = [[] for _ in range(M+1)]\\n\\nX[0] = [1]\\ndef calc(L):\\n    su = sum(L)\\n    pl = 0\\n    pd = 0\\n    RE = []\\n    for i in range(len(L)):\\n        a = li + i\\n        b = di - (len(L) - 1 - i)\\n        pd = b * L[i] * pow(su*(a+b), P-2, P)\\n        RE.append((pl+pd)%P)\\n        pl = a * L[i] * pow(su*(a+b), P-2, P)\\n    RE.append(pl%P)\\n    return RE\\n\\nfor i in range(M):\\n    X[i+1] = calc(X[i])\\nne = 0\\npo = 0\\nfor i in range(M+1):\\n    po = (po + X[M][i] * (li + i)) % P\\n    ne = (ne + X[M][i] * (di - M + i)) % P\\nfor i in range(N):\\n    print(po * B[i] * pow(li, P-2, P) % P if A[i] else ne * B[i] * pow(di, P-2, P) % P)\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef inv(x):\\n    return pow(x, mod - 2, mod)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    def plus(a,b):\\n        c = a[0] * b[1] % mod\\n        d = b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def sub(a,b):\\n        c = a[0] * b[1] % mod\\n        d = -b[0] * a[1] % mod\\n        e = a[1] * b[1] % mod\\n        return [(c+d)%mod, e]\\n\\n    def mul(a,b):\\n        c = a[0] * b[0] % mod\\n        d = a[1] * b[1] % mod\\n        return [c,d]\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return [c, 1]\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = mul(x, [c, s])\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = plus(r, mul(y, [p, s]))\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = plus(r, mul(z, [q, s]))\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v[0]*inv(v[1]) % mod)\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\", \"import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,copy,functools\\n\\nsys.setrecursionlimit(10**7)\\ninf = 10**20\\neps = 1.0 / 10**10\\nmod = 998244353\\ndd = [(-1,0),(0,1),(1,0),(0,-1)]\\nddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]\\n\\ndef LI(): return list(map(int, sys.stdin.readline().split()))\\ndef LLI(): return [list(map(int, l.split())) for l in sys.stdin.readlines()]\\ndef LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]\\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\\ndef LS(): return sys.stdin.readline().split()\\ndef I(): return int(sys.stdin.readline())\\ndef F(): return float(sys.stdin.readline())\\ndef S(): return input()\\ndef pf(s): return print(s, flush=True)\\ndef pe(s): return print(str(s), file=sys.stderr)\\ndef JA(a, sep): return sep.join(map(str, a))\\n\\ndef MF(n,d): return ModFraction(n,d)\\n\\nclass ModFraction():\\n    def __init__(self, n, d):\\n        self.n = n\\n        self.d = d\\n\\n    def __add__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __sub__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = -xf.n * self.d % mod\\n        c = self.d * xf.d % mod\\n        return ModFraction((a+b) % mod, c)\\n\\n    def __mul__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.n % mod\\n        b = self.d * xf.d % mod\\n        return ModFraction(a, b)\\n\\n    def __truediv__(self, x):\\n        xf = ModFraction.xf(x)\\n        a = self.n * xf.d % mod\\n        b = self.d * xf.n % mod\\n        return ModFraction(a, b)\\n\\n    @classmethod\\n    def xf(cls, x):\\n        if isinstance(x, int):\\n            return ModFraction(x, 1)\\n        return x\\n\\n    @classmethod\\n    def inv(cls, x):\\n        return pow(x, mod - 2, mod)\\n\\n    def int(self):\\n        return self.n * ModFraction.inv(self.d) % mod\\n\\n    def __str__(self):\\n        return \\\"{} / {}\\\".format(self.n, self.d)\\n\\ndef main():\\n    n,m = LI()\\n    a = LI()\\n    w = LI()\\n\\n    fm = {}\\n    def f(c,a,m,p,q):\\n        if m == 0 or c == 0:\\n            return MF(c, 1)\\n\\n        key = (c,a,m,p,q)\\n        if key in fm:\\n            return fm[key]\\n\\n        s = c + p + q\\n        x = f(c+a,a,m-1,p,q)\\n        r = x * MF(c, s)\\n        if p > 0:\\n            y = f(c,a,m-1,p+1,q)\\n            r = r + y * MF(p, s)\\n        if q > 0:\\n            z = f(c,a,m-1,p,q-1)\\n            r = r + z * MF(q, s)\\n\\n        fm[key] = r\\n        return r\\n\\n    ps = 0\\n    qs = 0\\n    for i in range(n):\\n        if a[i] == 1:\\n            ps += w[i]\\n        else:\\n            qs += w[i]\\n\\n    r = []\\n    for i in range(n):\\n        if a[i] == 1:\\n            v = f(w[i],1,m,ps-w[i],qs)\\n        else:\\n            v = f(w[i],-1,m,ps,qs-w[i])\\n\\n        r.append(v.int())\\n\\n    return JA(r,'\\\\n')\\n\\n\\nprint(main())\\n\\n\"]",
        "difficulty": "competition",
        "input": "50 50\n1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1\n50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50\n",
        "output": "51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n51\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1172/C1"
    },
    {
        "id": 1182,
        "task_id": 3981,
        "test_case_id": 1,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1183,
        "task_id": 3981,
        "test_case_id": 2,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1184,
        "task_id": 3981,
        "test_case_id": 3,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "8 8\n53212775 61830464\n32047271 31958754\n46845363 47341547\n46097536 79640734\n25559819 90360169\n5442822 40309249\n57656501 31139219\n1191345 7784770\n61052639 71686018\n53542361 10165576\n61963163 25946318\n15498921 83261369\n97981085 40697107\n35269685 20583089\n50275077 3751318\n40877349 23290816\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1185,
        "task_id": 3981,
        "test_case_id": 4,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "10 8\n1446 2090916\n0 0\n1060 1123600\n2939 8946081\n1405 8946081\n2842 8946081\n2979 8946081\n2850 8946081\n0 100000000\n2991 8946081\n1446 2090916\n0 0\n1060 1123600\n2575 8946081\n1586 8946081\n460 8946081\n0 100000000\n2991 8946081\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1186,
        "task_id": 3981,
        "test_case_id": 5,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "8 9\n4965 24651225\n0 0\n0 36000000\n5004 25040016\n4354 30107169\n1309 30107169\n2301 30107169\n5487 30107169\n30107169 513\n30107169 2849\n30107169 998\n25040016 996\n30107169 2249\n30107169 1567\n36000000 6000\n0 6000\n24651225 1035\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1187,
        "task_id": 3981,
        "test_case_id": 6,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "3 3\n0 100000000\n0 0\n100000000 0\n100000000 100000000\n100000000 0\n0 100000000\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1188,
        "task_id": 3981,
        "test_case_id": 7,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "3 3\n1 1\n2 2\n3 3\n0 10\n1 9\n2 8\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1189,
        "task_id": 3981,
        "test_case_id": 8,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "3 3\n1 1\n2 2\n3 3\n2 2\n6 6\n7 7\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1190,
        "task_id": 3981,
        "test_case_id": 9,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "3 3\n1 1\n2 2\n3 3\n1 1\n1 2\n1 3\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1191,
        "task_id": 3981,
        "test_case_id": 10,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "4 4\n0 5\n5 10\n5 0\n10 5\n0 1\n7 2\n14 1\n7 0\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1192,
        "task_id": 3981,
        "test_case_id": 11,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "3 3\n0 0\n100000000 0\n100000000 1\n100000000 100000000\n0 100000000\n0 99999999\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1193,
        "task_id": 3981,
        "test_case_id": 12,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "4 3\n0 0\n100000000 0\n100000000 2\n50000000 1\n100000000 100000000\n0 100000000\n0 99999998\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1194,
        "task_id": 3981,
        "test_case_id": 13,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "4 3\n0 0\n100000000 0\n100000000 2\n50000001 1\n100000000 100000000\n0 100000000\n0 99999998\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1195,
        "task_id": 3981,
        "test_case_id": 14,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "4 3\n0 0\n100000000 0\n100000000 2\n49999999 1\n100000000 100000000\n0 100000000\n0 99999998\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1196,
        "task_id": 3981,
        "test_case_id": 15,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "6 3\n0 0\n1 1\n2 2\n3 3\n4 4\n6 6\n0 6\n3 3\n6 0\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1197,
        "task_id": 3981,
        "test_case_id": 16,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "6 3\n0 0\n1 1\n2 2\n3 3\n4 4\n5 5\n0 6\n3 3\n6 0\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1198,
        "task_id": 3981,
        "test_case_id": 17,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "5 5\n2 0\n0 4\n4 8\n8 4\n10 0\n0 2\n2 6\n6 10\n10 6\n8 2\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1199,
        "task_id": 3981,
        "test_case_id": 18,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "8 8\n0 1\n0 3\n1 4\n3 4\n4 3\n4 1\n3 0\n1 0\n2 5\n3 5\n5 3\n5 2\n3 0\n2 0\n0 2\n0 3\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1200,
        "task_id": 3981,
        "test_case_id": 19,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "6 6\n2 0\n4 0\n6 1\n4 2\n2 2\n0 1\n1 0\n3 0\n4 2\n3 4\n1 4\n0 2\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1201,
        "task_id": 3981,
        "test_case_id": 20,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "4 4\n0 0\n0 5\n5 0\n5 5\n0 3\n4 0\n7 4\n3 7\n",
        "output": "YES\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1202,
        "task_id": 3981,
        "test_case_id": 21,
        "question": "After the war, the supersonic rocket became the most common public transportation.\n\nEach supersonic rocket consists of two \"engines\". Each engine is a set of \"power sources\". The first engine has $n$ power sources, and the second one has $m$ power sources. A power source can be described as a point $(x_i, y_i)$ on a 2-D plane. All points in each engine are different.\n\nYou can manipulate each engine separately. There are two operations that you can do with each engine. You can do each operation as many times as you want. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i+a, y_i+b)$, $a$ and $b$ can be any real numbers. In other words, all power sources will be shifted. For every power source as a whole in that engine: $(x_i, y_i)$ becomes $(x_i \\cos \\theta - y_i \\sin \\theta, x_i \\sin \\theta + y_i \\cos \\theta)$, $\\theta$ can be any real number. In other words, all power sources will be rotated.\n\nThe engines work as follows: after the two engines are powered, their power sources are being combined (here power sources of different engines may coincide). If two power sources $A(x_a, y_a)$ and $B(x_b, y_b)$ exist, then for all real number $k$ that $0 \\lt k \\lt 1$, a new power source will be created $C_k(kx_a+(1-k)x_b,ky_a+(1-k)y_b)$. Then, this procedure will be repeated again with all new and old power sources. After that, the \"power field\" from all power sources will be generated (can be considered as an infinite set of all power sources occurred).\n\nA supersonic rocket is \"safe\" if and only if after you manipulate the engines, destroying any power source and then power the engine, the power field generated won't be changed (comparing to the situation where no power source erased). Two power fields are considered the same if and only if any power source in one field belongs to the other one as well.\n\nGiven a supersonic rocket, check whether it is safe or not.\n\n\n-----Input-----\n\nThe first line contains two integers $n$, $m$ ($3 \\le n, m \\le 10^5$) — the number of power sources in each engine.\n\nEach of the next $n$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the first engine.\n\nEach of the next $m$ lines contains two integers $x_i$ and $y_i$ ($0\\leq x_i, y_i\\leq 10^8$) — the coordinates of the $i$-th power source in the second engine.\n\nIt is guaranteed that there are no two or more power sources that are located in the same point in each engine.\n\n\n-----Output-----\n\nPrint \"YES\" if the supersonic rocket is safe, otherwise \"NO\".\n\nYou can print each letter in an arbitrary case (upper or lower).\n\n\n-----Examples-----\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n1 1\n\nOutput\nYES\n\nInput\n3 4\n0 0\n0 2\n2 0\n0 2\n2 2\n2 0\n0 0\n\nOutput\nNO\n\n\n\n-----Note-----\n\nThe first sample: [Image] Those near pairs of blue and orange points actually coincide. \n\nFirst, manipulate the first engine: use the second operation with $\\theta = \\pi$ (to rotate all power sources $180$ degrees).\n\nThe power sources in the first engine become $(0, 0)$, $(0, -2)$, and $(-2, 0)$. [Image] \n\nSecond, manipulate the second engine: use the first operation with $a = b = -2$.\n\nThe power sources in the second engine become $(-2, 0)$, $(0, 0)$, $(0, -2)$, and $(-1, -1)$. [Image] \n\nYou can examine that destroying any point, the power field formed by the two engines are always the solid triangle $(0, 0)$, $(-2, 0)$, $(0, -2)$.\n\nIn the second sample, no matter how you manipulate the engines, there always exists a power source in the second engine that power field will shrink if you destroy it.",
        "solutions": "[\"import sys\\n\\n\\n# > 0 anti-clock, < 0 clockwise, == 0 same line\\ndef orientation(p1, p2, p3):\\n    return (p2[0] - p1[0])*(p3[1] - p1[1]) - (p2[1] - p1[1])*(p3[0] - p1[0])\\n\\n\\ndef dot(p1, p2, p3, p4):\\n    return (p2[0]-p1[0])*(p4[0]-p3[0]) + (p2[1]-p1[1])*(p4[1]-p3[1])\\n\\n\\ndef theta(p1, p2):\\n    dx = p2[0] - p1[0]\\n    dy = p2[1] - p1[1]\\n    if abs(dx) < 0.1 and abs(dy) < 0.1:\\n        t = 0\\n    else:\\n        t = dy/(abs(dx) + abs(dy))\\n        if abs(t) < 0.1 ** 10:\\n            t = 0\\n    if dx < 0:\\n        t = 2 - t\\n    elif dy < 0:\\n        t = 4 + t\\n\\n    return t*90\\n\\n\\ndef dist_sq(p1, p2):\\n    return (p1[0] - p2[0])*(p1[0] - p2[0]) + (p1[1] - p2[1])*(p1[1] - p2[1])\\n\\n\\ndef chull(points):\\n    # let 0 element to be smallest, reorder elements\\n    pi = [x for x in range(len(points))]\\n    min_y = points[0][1]\\n    min_x = points[0][0]\\n    min_ind = 0\\n    for i in range(len(points)):\\n        if points[i][1] < min_y or points[i][1] == min_y and points[i][0] < min_x:\\n            min_y = points[i][1]\\n            min_x = points[i][0]\\n            min_ind = i\\n    pi[0] = min_ind\\n    pi[min_ind] = 0\\n    th = [theta(points[pi[0]], points[x]) for x in range(len(points))]\\n    pi.sort(key=lambda x: th[x])\\n    # process equals\\n    unique = [pi[0], pi[1]]\\n    for i in range(2, len(pi)):\\n        if th[pi[i]] != th[unique[-1]]:\\n            unique.append(pi[i])\\n        else:\\n            if dist_sq(points[pi[0]], points[unique[-1]]) < dist_sq(points[pi[0]], points[pi[i]]):\\n                unique[-1] = pi[i] # put max\\n    pi = unique\\n    stack = []\\n    for i in range(min(len(pi), 3)):\\n        stack.append(points[pi[i]])\\n    if len(stack) < 3:\\n        return stack\\n    for i in range(3, len(pi)):\\n        while len(stack) >= 2:\\n            o = orientation(stack[-2], stack[-1], points[pi[i]])\\n            if o > 0:\\n                stack.append(points[pi[i]])\\n                break\\n            elif o < 0:\\n                stack.pop()\\n            else:  # ==\\n                if dist_sq(stack[-2], stack[-1]) < dist_sq(stack[-2], points[pi[i]]):\\n                    stack.pop()\\n                else:\\n                    break  # skip i-th point\\n    return stack\\n\\n\\ndef z_func(s):\\n    slen, l, r = len(s), 0, 0\\n    z = [0]*slen\\n    z[0] = slen\\n    for i in range(1, slen):\\n        if i <= r: z[i] = min(r-i+1, z[i-l])\\n        while i+z[i] < slen and s[z[i]] == s[i+z[i]]: z[i] += 1\\n        if i+z[i]-1 > r: l, r = i, i+z[i]-1\\n    return z\\n\\nn,m = map(int, sys.stdin.readline().strip().split())\\na = []\\nfor _ in range(n):\\n    x,y = map(int, sys.stdin.readline().strip().split())\\n    a.append((x, y))\\nb = []\\nfor _ in range(m):\\n    x, y = map(int, sys.stdin.readline().strip().split())\\n    b.append((x, y))\\n\\nah = chull(a)\\nbh = chull(b)\\nif len(ah) == len(bh):\\n    if len(ah) == 2:\\n        if dist_sq(ah[0], ah[1]) == dist_sq(bh[0], bh[1]):\\n            print('YES')\\n        else:\\n            print('NO')\\n    else:\\n        da = []\\n        for i in range(len(ah)):\\n            dot_a = dot(ah[i-2], ah[i-1], ah[i-1], ah[i])\\n            da.append((dist_sq(ah[i], ah[i-1]), dot_a))\\n        db = []\\n        for i in range(len(bh)):\\n            dot_b = dot(bh[i - 2], bh[i - 1], bh[i - 1], bh[i])\\n            db.append((dist_sq(bh[i], bh[i-1]), dot_b))\\n        l = r = 0\\n        daab = []\\n        daab.extend(db)\\n        daab.append(-1)\\n        daab.extend(da)\\n        daab.extend(da)\\n        zab = z_func(daab)\\n        found = False\\n        for i in range(len(db)+1, len(daab)-len(db)+1):\\n            if zab[i] == len(db):\\n                found = True\\n                break\\n        if found:\\n            print('YES')\\n        else:\\n            print('NO')\\nelse:\\n    print('NO')\"]",
        "difficulty": "competition",
        "input": "4 4\n0 0\n1 0\n1 1\n2 1\n0 1\n1 1\n1 0\n2 0\n",
        "output": "NO\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1017/E"
    },
    {
        "id": 1203,
        "task_id": 3991,
        "test_case_id": 1,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "2\n4 7\n",
        "output": "3\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1204,
        "task_id": 3991,
        "test_case_id": 2,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "3\n4 3 1\n",
        "output": "9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1205,
        "task_id": 3991,
        "test_case_id": 3,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "20\n8 11 13 19 21 34 36 44 57 58 61 63 76 78 79 81 85 86 90 95\n",
        "output": "83396599\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1206,
        "task_id": 3991,
        "test_case_id": 4,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "20\n1 8 9 12 15 17 18 24 30 33 36 41 53 54 59 62 64 66 72 73\n",
        "output": "68059140\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1207,
        "task_id": 3991,
        "test_case_id": 5,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "20\n2 6 8 9 20 23 27 36 43 49 63 65 70 71 85 87 89 91 94 97\n",
        "output": "92743989\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1208,
        "task_id": 3991,
        "test_case_id": 6,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "1\n78091781\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1209,
        "task_id": 3991,
        "test_case_id": 7,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "2\n1000000000 1\n",
        "output": "999999999\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1210,
        "task_id": 3991,
        "test_case_id": 8,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "3\n999999998 999999999 999999992\n",
        "output": "21\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1211,
        "task_id": 3991,
        "test_case_id": 9,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "3\n465343471 465343474 465343473\n",
        "output": "9\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1212,
        "task_id": 3991,
        "test_case_id": 10,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "10\n10 3 6 2 1 9 8 4 5 7\n",
        "output": "7181\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1213,
        "task_id": 3991,
        "test_case_id": 11,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "10\n756734546 756734524 756734550 756734529 756734553 756734538 756734541 756734536 756734579 756734537\n",
        "output": "36489\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1214,
        "task_id": 3991,
        "test_case_id": 12,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "10\n877105545 939360757 849826701 845946140 803128820 926787996 967305000 904694971 921301848 971203310\n",
        "output": "861364152\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1215,
        "task_id": 3991,
        "test_case_id": 13,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "5\n4 7 13 17 18\n",
        "output": "270\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1216,
        "task_id": 3991,
        "test_case_id": 14,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "5\n20 17 13 7 2\n",
        "output": "330\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1217,
        "task_id": 3991,
        "test_case_id": 15,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "5\n3 17 2 5 4\n",
        "output": "237\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1218,
        "task_id": 3991,
        "test_case_id": 16,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "5\n999999980 999999985 999999986 999999990 999999992\n",
        "output": "210\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1219,
        "task_id": 3991,
        "test_case_id": 17,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "5\n1000000000 999999988 999999982 999999981 999999980\n",
        "output": "342\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1220,
        "task_id": 3991,
        "test_case_id": 18,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "5\n999999984 999999997 999999994 999999991 999999982\n",
        "output": "285\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1221,
        "task_id": 3991,
        "test_case_id": 19,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "1\n2\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1222,
        "task_id": 3991,
        "test_case_id": 20,
        "question": "Leha decided to move to a quiet town Vičkopolis, because he was tired by living in Bankopolis. Upon arrival he immediately began to expand his network of hacked computers. During the week Leha managed to get access to n computers throughout the town. Incidentally all the computers, which were hacked by Leha, lie on the same straight line, due to the reason that there is the only one straight street in Vičkopolis.\n\nLet's denote the coordinate system on this street. Besides let's number all the hacked computers with integers from 1 to n. So the i-th hacked computer is located at the point x_{i}. Moreover the coordinates of all computers are distinct. \n\nLeha is determined to have a little rest after a hard week. Therefore he is going to invite his friend Noora to a restaurant. However the girl agrees to go on a date with the only one condition: Leha have to solve a simple task.\n\nLeha should calculate a sum of F(a) for all a, where a is a non-empty subset of the set, that consists of all hacked computers. Formally, let's denote A the set of all integers from 1 to n. Noora asks the hacker to find value of the expression $\\sum_{a \\subseteq A, a \\neq \\varnothing} F(a)$. Here F(a) is calculated as the maximum among the distances between all pairs of computers from the set a. Formally, $F(a) = \\operatorname{max}_{i, j \\in a}|x_{i} - x_{j}|$. Since the required sum can be quite large Noora asks to find it modulo 10^9 + 7.\n\nThough, Leha is too tired. Consequently he is not able to solve this task. Help the hacker to attend a date.\n\n\n-----Input-----\n\nThe first line contains one integer n (1 ≤ n ≤ 3·10^5) denoting the number of hacked computers.\n\nThe second line contains n integers x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ 10^9) denoting the coordinates of hacked computers. It is guaranteed that all x_{i} are distinct.\n\n\n-----Output-----\n\nPrint a single integer — the required sum modulo 10^9 + 7.\n\n\n-----Examples-----\nInput\n2\n4 7\n\nOutput\n3\n\nInput\n3\n4 3 1\n\nOutput\n9\n\n\n\n-----Note-----\n\nThere are three non-empty subsets in the first sample test:$\\{4 \\}$, $\\{7 \\}$ and $\\{4,7 \\}$. The first and the second subset increase the sum by 0 and the third subset increases the sum by 7 - 4 = 3. In total the answer is 0 + 0 + 3 = 3.\n\nThere are seven non-empty subsets in the second sample test. Among them only the following subsets increase the answer: $\\{4,3 \\}$, $\\{4,1 \\}$, $\\{3,1 \\}$, $\\{4,3,1 \\}$. In total the sum is (4 - 3) + (4 - 1) + (3 - 1) + (4 - 1) = 9.",
        "solutions": "[\"3\\n# Copyright (C) 2017 Sayutin Dmitry.\\n#\\n# This program is free software; you can redistribute it and/or\\n# modify it under the terms of the GNU General Public License as\\n# published by the Free Software Foundation; version 3\\n#\\n# This program is distributed in the hope that it will be useful,\\n# but WITHOUT ANY WARRANTY; without even the implied warranty of\\n# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\\n# GNU General Public License for more details.\\n#\\n# You should have received a copy of the GNU General Public License\\n# along with this program; If not, see <http://www.gnu.org/licenses/>.\\n\\ndef main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\\n\", \"n = int(input())\\nx = sorted(map(int, input().split()))\\nmod = 10 ** 9 + 7\\nans, deg = 0, 1\\ndp = [0] * n\\nfor i in range(1, n):\\n    deg = (deg << 1) % mod\\n    dp[i] = (2 * dp[i - 1] + (x[i] - x[i - 1]) * (deg - 1)) % mod\\n    ans = (ans + dp[i]) % mod\\nprint(ans)\", \"import sys\\nimport math\\n\\nMOD = int(1e9 + 7)\\nline = lambda: list(map(int, input().split()))\\n\\ndef solve():\\n    n = int(input())\\n    x = [x for x in line()]\\n    x.sort()\\n    p2 = []\\n    p2.append(1)\\n    for i in range(1, n):\\n        p2.append(p2[i - 1] * 2 % MOD)\\n    ans = 0\\n    for i in range(n):\\n        ans += x[i] * (p2[i] - p2[n - i - 1] + MOD)\\n        ans %= MOD\\n    print(ans)\\n\\ndef main():\\n    solve()\\n    return\\n\\nmain()\\n\", \"import sys\\n\\nmod = 10**9 + 7\\n\\ndef solve():\\n    n = int(input())\\n    x = [int(i) for i in input().split()]\\n    x.sort()\\n\\n    ans = 0\\n\\n    p2 = [1] * (n + 1)\\n    for i in range(n):\\n        p2[i + 1] = (p2[i] * 2) % mod\\n\\n    for i in range(n - 1):\\n        length = x[i + 1] - x[i]\\n\\n        ans = (ans + length * (p2[i + 1] - 1) * (p2[n - 1 - i] - 1)) % mod\\n\\n    print(ans)\\n\\ndef __starting_point():\\n    solve()\\n__starting_point()\", \"n = int(input())\\nans = 0\\nMOD = int(10**9 + 7)\\na = [int(x) for x in input().split()]\\na.sort()\\npo = [1]\\nfor i in range(1,n):\\n    po.append(po[i-1]*2%MOD)\\n\\nfor i in range(n):\\n    ans += a[i]*(po[i] - po[n-i-1] + MOD)\\n    ans %= MOD\\n\\nprint(ans)\\n    \\n\", \"def main():\\n    mod = 10 ** 9 + 7\\n    pws = [None for i in range(4 * (10 ** 5))]\\n    pws[0] = 1\\n    for i in range(1, (4 * (10 ** 5))):\\n        pws[i] = 2 * pws[i - 1] % mod\\n    \\n    n = int(input())\\n    seq = list(map(int, input().split()))\\n\\n    seq.sort()\\n    \\n    ans = 0\\n    for i in range(n):\\n        ans += seq[i] * (pws[i] - 1)\\n        ans -= seq[i] * (pws[n - i - 1] - 1)\\n        ans = ans % mod\\n    print(ans)\\n\\nmain()\", \"n = int(input())\\na = sorted(list(map(int, input().split())))\\nbase = 10**9 + 7\\nd = [1]\\nfor i in range(n):\\n    d.append((2*d[-1]) % base)\\nans = 0\\nfor i in range(1, len(a)):\\n    diff = a[i] - a[i-1]\\n    add = diff*(d[i]-1)*(d[n - i]-1) % base\\n    ans += add\\n    ans = ans % base\\nprint(ans)\", \"n = int(input())\\nl = list(map(int, input().strip().split(' ')))\\np =1\\nl.sort()\\nlength = len(l)\\nans  = 0\\nfor i in range(len(l)) :\\n  ans += p*(l[i]-l[length-i-1])\\n  p*=2\\n  p  %=1000000007\\n  ans%=1000000007\\n\\nprint(int(ans))\", \"#!/usr/bin/pypy3\\n\\nfrom sys import stdin,stderr\\nfrom random import shuffle\\n\\ndef readInts(): return map(int,stdin.readline().strip().split())\\ndef print_err(*args,**kwargs): print(*args,file=stderr,**kwargs)\\n    \\ndef solve(vs):\\n    modval = 10**9+7\\n    return all_f(vs)\\n\\ndef expnm(a,b,m):\\n    if b==0: return 1\\n    if b%2 == 0: return expnm(a*a%m,b//2,m)\\n    return (a*expnm(a,b-1,m)%m)\\n\\ndef expnm2(a,b,m):\\n    #print(a,b)\\n    out = 1\\n    mult = a\\n    while b > 0:\\n        if b%2 == 0:\\n            mult *= mult\\n            b //= 2\\n        else:\\n            out *= mult\\n            b -= 1\\n        out %= m\\n    return out\\n\\ndef all_f_naive(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n-1):\\n        for x2ix in range(x1ix+1,n):\\n            c = x2ix-x1ix-1\\n            d = ns[x2ix]-ns[x1ix]\\n            out += d * (expnm(2,c,m))\\n            out %= m\\n    return out\\n\\ndef twos(n,m):\\n    v = 1\\n    out = []\\n    for _ in range(n+1):\\n        out.append(v)\\n        v <<= 1\\n        v %= m\\n    return out\\n\\ndef all_f(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    t = twos(n,m)\\n    for x1ix in range(n):\\n        v = ns[x1ix] * t[x1ix]\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * t[n-x1ix-1]\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f3(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef all_f2(ns):\\n    n = len(ns)\\n    ns.sort()\\n    out = 0\\n    m = 10**9+7\\n    for x1ix in range(n):\\n        v = ns[x1ix] * expnm2(2,x1ix,m)\\n        v %= m\\n        out += v\\n        out %= m\\n        v = ns[x1ix] * expnm2(2,n-x1ix-1,m)\\n        v %= m\\n        out -= v\\n        out %= m\\n    return out\\n\\ndef run():\\n    n, = readInts()    \\n    vs = list(readInts())\\n    print(solve(vs))\\n        \\nrun()\\n\", \"R, n, x, p, v = 10 ** 9 + 7, int(input()), sorted(map(int, input().split())), [1], 0\\nfor i in range(1, n):\\n    p.append(2 * p[-1] % R)\\nprint(sum((x[i] - x[i - 1]) * (p[i] - 1) * (p[n - i] - 1) for i in range(1, n)) % R)\", \"n=int(input())\\nR= lambda: map(int, input().split())\\nl= list(R())\\nl.sort()\\ns,m=0,1000000007\\nfor i in range(n):\\n    s= (s+l[i]*(pow(2,i,m)-pow(2,n-1-i,m)))%m\\nprint(s)\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\nkk = 1\\na = 0\\nfor i in range(len(x)):\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n    kk *= 2\\n    kk %= 1000000007\\n    a %= 1000000007\\nprint(a)\", \"n=int(input())\\nV=list(map (int,input().split()))\\nV.sort()\\nMod=int(1e9+7)\\nSum=0\\np=[1]\\nfor i in range(n):\\n    p.append(p[i]*2%Mod)\\nfor i in range(n-1):\\n    l=(p[i+1]+Mod-1)%Mod\\n    r=(p[n-i-1]-1+Mod)%Mod\\n    Sum=(Sum+(V[i+1]-V[i])*l*r)%Mod\\nprint(int(Sum))\\n\", \"n, x = int(input()), list(sorted(list(map(int, input().split()))))\\n\\nkk = 1\\n\\na = 0\\n\\nfor i in range(len(x)):\\n\\n    a += (x[i] - x[len(x) - i - 1]) * kk\\n\\n    kk *= 2\\n\\n    kk %= 1000000007\\n\\n    a %= 1000000007\\n\\nprint(a)\\n\\n\\n\\n# Made By Mostafa_Khaled\\n\", \"n=int(input())\\nl=[int(i) for i in input().split()]\\nsm=0 \\nMOD=10**9+7 \\nadd=0 \\nsub=0 \\nl.sort()\\np=3*(10**5)+9 \\npow2=[0]*p \\npow2[0]=1 \\nfor i in range(1,p):\\n    pow2[i]=(pow2[i-1]*2)%MOD\\nfor i in range(n):\\n    add+=l[i]*(pow2[i])\\n    sub+=l[i]*(pow2[n-i-1])\\nprint((add-sub+MOD)%MOD)\", \"n, ans, m, s = int(input()), 0, 1000000007, [0]\\na = sorted(map(int, input().split()))\\nfor i in range(1, n):\\n    s.append((2 * s[i - 1] + a[i - 1]) % m)\\n    ans += ((pow(2, i, m) - 1) * a[i] - s[i]) % m\\nprint(ans % m)\", \"N = int(3e5+3)\\nMOD = int(1e9+7)\\npow2 = [1] * N\\nfor i in range(1, N):\\n    pow2[i] = pow2[i-1] * 2 % MOD\\n\\nn = int(input())\\narr = list(map(int, input().split()))\\narr.sort()\\n\\ndef add(x, y):\\n    return ((x % MOD + MOD) + (y % MOD + MOD) + MOD) % MOD\\ndef substract(x, y):\\n    return ((x % MOD + MOD) - (y % MOD + MOD) + MOD) % MOD\\ndef multiple(x, y):\\n    return ((x % MOD) * (y % MOD)) % MOD\\n\\nres = 0\\nfor i in range(1, n):\\n    diff = arr[i] - arr[i-1]\\n    cnt = multiple(substract(pow2[i], 1), substract(pow2[n-i], 1))\\n    res = add(res, multiple(cnt, diff))\\nprint(res)\\n\", \"import sys\\ninput = sys.stdin.readline\\n\\nn = int(input())\\nx = list(map(int, input().split()))\\nx.sort()\\nans = 0\\nMOD = 10**9+7\\n\\nfor i in range(n-1):\\n    d = x[i+1]-x[i]\\n    l = pow(2, i+1, MOD)-1\\n    r = pow(2, n-i-1, MOD)-1\\n    ans += d*l*r\\n    ans %= MOD\\n\\nprint(ans)\", \"n = int(input())\\nar = list(map(int,input().split()))\\nar.sort()\\nMOD = 10**9 + 7\\nans = 0\\nfor i in range(n):\\n    bef = i  \\n    aft = n - i - 1\\n    if bef < aft:\\n        ans -= (ar[i]*(pow(2 , aft , MOD) - pow(2 , bef , MOD) + MOD) % MOD) \\n        ans %= MOD\\n    elif aft < bef:\\n        ans += (ar[i]*(pow(2 , bef , MOD) - pow(2 , aft , MOD) + MOD) % MOD)\\n        ans %= MOD    \\nprint(ans)\", \"n = int(input())\\narr = sorted(list(map(int, input().split())))\\nres = 0\\nmod = 1000000007\\nacc = 0\\narr = [arr[i+1] - arr[i] for i in range(n-1)]\\nn -= 1\\ns = sum(arr)\\nfor i in range((n+1)//2):\\n    acc += s\\n    res += acc * pow(2, n-i-1, mod) % mod\\n    if not i*2+1 == n:\\n        res += acc * pow(2, i, mod) % mod\\n    res %= mod\\n    s -= arr[i] + arr[n-i-1]\\nprint(res)\", \"n = int(input())\\nx = list(map(int, input().split()))\\n\\nx.sort()\\nans = 0\\nfor i in range(n):\\n    ans+=((pow(2,i,1000000007)-pow(2,n-i-1,1000000007))*x[i])%1000000007\\n    ans = ans%1000000007\\nprint(ans)\", \"n = int(input())\\narr = list(map(int,input().split()))\\narr.sort()\\nmod = 10**9 + 7\\n\\ns = 0\\nans = 0\\nm = 0\\nlAdd = 1\\n# print(arr)\\nfor i in range(n-2,-1,-1):\\n    # print('---------')\\n    # print(arr[i])\\n    m+=lAdd\\n    m%=mod\\n\\n    lAdd*=2\\n    lAdd%=mod\\n\\n    s *= 2\\n    s %= mod\\n    s += arr[i+1]\\n    s %= mod\\n\\n    # print(s,(((arr[i])*((1<<(n-i-1)) - 1))%mod))\\n    t = s-((m*arr[i])%mod)\\n    t+=mod\\n    t%=mod\\n    ans+=t\\n    ans%=mod\\nprint(ans%mod)\"]",
        "difficulty": "competition",
        "input": "5\n9 10 7 4 5\n",
        "output": "114\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/809/A"
    },
    {
        "id": 1223,
        "task_id": 4374,
        "test_case_id": 2,
        "question": "You are given a forest — an undirected graph with $n$ vertices such that each its connected component is a tree.\n\nThe diameter (aka \"longest shortest path\") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices.\n\nYou task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1 \\le n \\le 1000$, $0 \\le m \\le n - 1$) — the number of vertices of the graph and the number of edges, respectively.\n\nEach of the next $m$ lines contains two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the edges.\n\nIt is guaranteed that the given graph is a forest.\n\n\n-----Output-----\n\nIn the first line print the diameter of the resulting tree.\n\nEach of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the added edges.\n\nThe resulting graph should be a tree and its diameter should be minimal possible.\n\nFor $m = n - 1$ no edges are added, thus the output consists of a single integer — diameter of the given tree.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Examples-----\nInput\n4 2\n1 2\n2 3\n\nOutput\n2\n4 2\n\nInput\n2 0\n\nOutput\n1\n1 2\n\nInput\n3 2\n1 3\n2 3\n\nOutput\n2\n\n\n\n-----Note-----\n\nIn the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2.\n\nEdge (1, 2) is the only option you have for the second example. The diameter is 1.\n\nYou can't add any edges in the third example. The diameter is already 2.",
        "solutions": "[\"import math\\nn,m=map(int,input().split())\\nneigh=[]\\nfor i in range(n):\\n    neigh.append([])\\nfor i in range(m):\\n    a,b=map(int,input().split())\\n    neigh[a-1].append(b-1)\\n    neigh[b-1].append(a-1)\\nseen=set()\\nindex=[0]*n\\ndiams=[]\\ntrees=0\\nfor i in range(n):\\n    if i not in seen:\\n        trees+=1\\n        index[i]=trees\\n        seen.add(i)\\n        layer=[i]\\n        prev=None\\n        pars=[None]\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(child)\\n                        newpars.append(vert)\\n                        index[child]=trees\\n                        seen.add(child)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        far=prev[0]\\n        layer=[[far]]\\n        pars=[None]\\n        prev=None\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i][-1]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(layer[i]+[child])\\n                        newpars.append(vert)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        diam=prev[0]\\n        lent=len(diam)\\n        mid=diam[lent//2]\\n        diams.append((lent-1,mid))\\ndiams.sort(reverse=True)\\nposs=[diams[0][0]]\\nif len(diams)>1:\\n    poss.append(math.ceil(diams[0][0]/2)+1+math.ceil(diams[1][0]/2))\\nif len(diams)>2:\\n    poss.append(math.ceil(diams[1][0]/2)+2+math.ceil(diams[2][0]/2))\\nprint(max(poss))\\ncent=diams[0][1]\\nfor i in range(len(diams)-1):\\n    print(cent+1,diams[i+1][1]+1)\", \"import sys\\n\\n\\ndef dfs(v, d, prev, i):\\n    nonlocal mid\\n    nonlocal M\\n    M[v] = False\\n    way[d] = v\\n    if way[d + 1] == 0:\\n        mid[i] = way[d // 2]\\n    mx = (d, v)\\n    for x in E[v]:\\n        if x != prev:\\n            mx = max(mx, dfs(x, d + 1, v, i))\\n\\n    return mx\\n\\n\\nsys.setrecursionlimit(2000)\\nn, m = list(map(int, input().split()))\\n\\nE = [[] for i in range(n + 1)]\\nfor i in range(m):\\n    a, b = list(map(int, input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nmid = [0] * (n + 2)\\nc = 0\\nk = 0\\nway = [0] * (n + 2)\\n\\nM = [True] * (n + 1)\\nM[0] = False\\ni = -1\\nwhile True in M:\\n    i += 1\\n    idx = M.index(True)\\n    p1 = dfs(idx, 0, 0, i)[1]\\n    way = [0] * (n + 2)\\n    s, p2 = dfs(p1, 0, 0, i)\\n    if s > c:\\n        c = s\\n        k = i\\nr = []\\nfor j in range(0, i + 1):\\n    if j == k:\\n        continue\\n    r.append((mid[k], mid[j]))\\n    E[mid[k]].append(mid[j])\\n    E[mid[j]].append(mid[k])\\np1 = dfs(1, 0, 0, n + 1)[1]\\ns, p2 = dfs(p1, 0, 0, n + 1)\\nprint(s)\\nfor item in r:\\n    print(item[0], item[1])\\n\", \"import sys\\nsys.setrecursionlimit(1100)\\n\\ndef dfs1(u,pre):            #find the components\\n    vis[u] = True\\n    now.append(u)\\n    for v in to[u]:\\n        if v!=pre:dfs1(v,u)\\n\\ndef dfs2(u,pre):            #calulate the distance\\n    mxdist[u]=dist[u]\\n    for v in to[u]:\\n        if v!=pre:\\n            dist[v] = dist[u]+1\\n            dfs2(v,u)\\n            mxdist[u]=max(mxdist[u],mxdist[v])\\n\\ntry:\\n    lab = 1\\n    n, m = [int(x) for x in input().split()]\\n    to = [[] for i in range(n+10)]\\n    dist = [0 for i in range(n+10)]\\n    mxdist = [0 for  i in range(n+10)]\\n\\n    lab = 2\\n    for i in range(m):\\n        u,v = [int(x) for x in input().split()]\\n        to[u].append(v)\\n        to[v].append(u)\\n\\n    com=[]\\n    vis=[False for i in range(n+10)]\\n    for i in range(1,n+1):\\n        if vis[i]==False:\\n            now=[]\\n            dfs1(i,0)\\n            com.append(now)\\n\\n    lab = 3\\n    ct = []\\n    mx = 0\\n    for lis in com:\\n        tmp = []\\n        d = 0\\n        for root in lis:\\n            for u in lis:dist[u]=mxdist[u]=0\\n            dfs2(root,0)\\n            tmp.append((mxdist[root],root))\\n            d = max( d, sum( sorted([ mxdist[u] for u in to[root] ])[-2:] ) )\\n            #print(*[mxdist[u] for u in lis])\\n        mx = max(mx,d)\\n        #print('d =',d)\\n        for x in tmp:\\n            if x[0]==(d+1)//2:\\n                center = [x[1] for x in tmp if x[0]==(d+1)//2][0]\\n        ct.append( ((d+1)//2,center) )\\n\\n    #print(*ct)\\n\\n    lab = 4\\n    ct.sort(reverse=True)\\n    ans = []\\n    for i in range(1,len(ct)):\\n        mx = max(mx,ct[i][0]+1+ct[0][0])\\n        if i>1:mx = max(mx,ct[i][0]+2+ct[1][0])\\n        ans.append((ct[i][1],ct[0][1]))\\n    print(mx)\\n    for p in ans:\\n        print(*p)\\nexcept Exception as e:\\n    print('error after lable',lab,', type =',e)\\n\", \"import sys\\nsys.setrecursionlimit(10000)\\nclass Tree():\\n   def __init__(self, nodes):\\n     self.root = None\\n\\nclass Node():\\n  def __init__(self, val):\\n    self.parent = None\\n    self.val = val\\n    self.children = []\\n  def add(self, child):\\n    self.children.append(child)\\n  def get_max_dist(self, parent,dist):\\n     max_dist = dist\\n     for i in self.children:\\n        if i != parent:\\n           d = i.get_max_dist(self, dist+1)\\n           if max_dist < d:\\n              max_dist = d\\n     return max_dist\\n  def get_dist(self):\\n     return self.get_max_dist(self, 0)\\n  def get_count_ch(self, parent):\\n     count = len(self.children)\\n     for i in self.children:\\n        if i != parent:\\n           count += i.get_count_ch(self)-1\\n     return count\\n  def calc_child(self):\\n     return (self.get_count_ch(None),len(self.children), self.val)\\n     \\nclass Forest():\\n  def __init__(self, count):\\n    self.nodes = []  \\n    self.count = count\\n    self.set_nodes = set()\\n  def build(self):\\n    roots = []\\n    max_dist = []\\n    #diam = []\\n    list_root = []\\n    for i in self.nodes:\\n       tree =(list([x.get_dist() for x in i]))\\n       ma = max(tree)\\n       max_dist.append(ma)\\n       #diam.append(ma)\\n       m = i[tree.index(min(tree))]\\n       roots.append(m)#.val)\\n    if len(roots) > 1:\\n       ind = max_dist.index(max(max_dist))\\n       if len(roots) > 1 and ind != 0:\\n          roots[0], roots[ind] = roots[ind], roots[0]\\n    s = set()\\n    for i in self.nodes:\\n      for j in i:\\n        s.add(j.val)\\n    r = list(set(range(1, n+1))-s)\\n    \\n    if len(roots) > 0:# and len(diam) > 0:\\n       for i in range(1, len(roots)):\\n          self.add(roots[0].val, roots[i].val)\\n          list_root.append((roots[0].val, roots[i].val))\\n       #print(roots + r)\\n   \\n       for i in r:\\n         self.add(roots[0].val, i)\\n         list_root.append((roots[0].val, i))\\n    else:\\n       if len(r) == 1:\\n          print(0)\\n          return\\n       elif len(r) > 1:\\n          for i in range(1, len(r)):\\n             self.add(r[0], r[i])\\n             list_root.append((r[0], r[i]))\\n    distances = []\\n    for i in self.nodes[0]:\\n       dist =(i.get_dist())\\n       distances.append(dist)\\n    print(max(distances))\\n    for i in list_root:\\n       print(*i)\\n       \\n  def add(self, v, u):\\n    self.set_nodes.add(v)\\n    self.set_nodes.add(u)\\n    v_node, v_list = self.find(v)\\n    u_node, u_list = self.find(u)\\n    #print(v_node, u_node)\\n    if v_node == None and u_node == None:\\n      v_node = Node(v)\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      self.nodes.append([v_node, u_node])\\n    elif v_node != None and u_node != None and v_list != u_list:\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list += u_list\\n      self.nodes.remove(u_list)\\n    elif v_node == None and u_node != None:\\n      v_node = Node(v)\\n      u_node.add(v_node)\\n      v_node.add(u_node)\\n      u_list.append(v_node)\\n    elif v_node != None and u_node == None:\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list.append(u_node)\\n  def find(self, value):\\n    for i_list in self.nodes:\\n      for i in i_list:        \\n        if i.val == value:\\n          return i, i_list\\n    return None, None\\nn,m = list(map(int,input().split()))\\nf = Forest(n)\\nfor i in range(m):\\n  v,u = list(map(int,input().split()))\\n  f.add(v, u)\\n\\n#print(f.nodes)\\nf.build()\\n\"]",
        "difficulty": "introductory",
        "input": "2 0\n",
        "output": "1\n1 2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1092/E"
    },
    {
        "id": 1224,
        "task_id": 4374,
        "test_case_id": 3,
        "question": "You are given a forest — an undirected graph with $n$ vertices such that each its connected component is a tree.\n\nThe diameter (aka \"longest shortest path\") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices.\n\nYou task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1 \\le n \\le 1000$, $0 \\le m \\le n - 1$) — the number of vertices of the graph and the number of edges, respectively.\n\nEach of the next $m$ lines contains two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the edges.\n\nIt is guaranteed that the given graph is a forest.\n\n\n-----Output-----\n\nIn the first line print the diameter of the resulting tree.\n\nEach of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the added edges.\n\nThe resulting graph should be a tree and its diameter should be minimal possible.\n\nFor $m = n - 1$ no edges are added, thus the output consists of a single integer — diameter of the given tree.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Examples-----\nInput\n4 2\n1 2\n2 3\n\nOutput\n2\n4 2\n\nInput\n2 0\n\nOutput\n1\n1 2\n\nInput\n3 2\n1 3\n2 3\n\nOutput\n2\n\n\n\n-----Note-----\n\nIn the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2.\n\nEdge (1, 2) is the only option you have for the second example. The diameter is 1.\n\nYou can't add any edges in the third example. The diameter is already 2.",
        "solutions": "[\"import math\\nn,m=map(int,input().split())\\nneigh=[]\\nfor i in range(n):\\n    neigh.append([])\\nfor i in range(m):\\n    a,b=map(int,input().split())\\n    neigh[a-1].append(b-1)\\n    neigh[b-1].append(a-1)\\nseen=set()\\nindex=[0]*n\\ndiams=[]\\ntrees=0\\nfor i in range(n):\\n    if i not in seen:\\n        trees+=1\\n        index[i]=trees\\n        seen.add(i)\\n        layer=[i]\\n        prev=None\\n        pars=[None]\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(child)\\n                        newpars.append(vert)\\n                        index[child]=trees\\n                        seen.add(child)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        far=prev[0]\\n        layer=[[far]]\\n        pars=[None]\\n        prev=None\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i][-1]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(layer[i]+[child])\\n                        newpars.append(vert)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        diam=prev[0]\\n        lent=len(diam)\\n        mid=diam[lent//2]\\n        diams.append((lent-1,mid))\\ndiams.sort(reverse=True)\\nposs=[diams[0][0]]\\nif len(diams)>1:\\n    poss.append(math.ceil(diams[0][0]/2)+1+math.ceil(diams[1][0]/2))\\nif len(diams)>2:\\n    poss.append(math.ceil(diams[1][0]/2)+2+math.ceil(diams[2][0]/2))\\nprint(max(poss))\\ncent=diams[0][1]\\nfor i in range(len(diams)-1):\\n    print(cent+1,diams[i+1][1]+1)\", \"import sys\\n\\n\\ndef dfs(v, d, prev, i):\\n    nonlocal mid\\n    nonlocal M\\n    M[v] = False\\n    way[d] = v\\n    if way[d + 1] == 0:\\n        mid[i] = way[d // 2]\\n    mx = (d, v)\\n    for x in E[v]:\\n        if x != prev:\\n            mx = max(mx, dfs(x, d + 1, v, i))\\n\\n    return mx\\n\\n\\nsys.setrecursionlimit(2000)\\nn, m = list(map(int, input().split()))\\n\\nE = [[] for i in range(n + 1)]\\nfor i in range(m):\\n    a, b = list(map(int, input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nmid = [0] * (n + 2)\\nc = 0\\nk = 0\\nway = [0] * (n + 2)\\n\\nM = [True] * (n + 1)\\nM[0] = False\\ni = -1\\nwhile True in M:\\n    i += 1\\n    idx = M.index(True)\\n    p1 = dfs(idx, 0, 0, i)[1]\\n    way = [0] * (n + 2)\\n    s, p2 = dfs(p1, 0, 0, i)\\n    if s > c:\\n        c = s\\n        k = i\\nr = []\\nfor j in range(0, i + 1):\\n    if j == k:\\n        continue\\n    r.append((mid[k], mid[j]))\\n    E[mid[k]].append(mid[j])\\n    E[mid[j]].append(mid[k])\\np1 = dfs(1, 0, 0, n + 1)[1]\\ns, p2 = dfs(p1, 0, 0, n + 1)\\nprint(s)\\nfor item in r:\\n    print(item[0], item[1])\\n\", \"import sys\\nsys.setrecursionlimit(1100)\\n\\ndef dfs1(u,pre):            #find the components\\n    vis[u] = True\\n    now.append(u)\\n    for v in to[u]:\\n        if v!=pre:dfs1(v,u)\\n\\ndef dfs2(u,pre):            #calulate the distance\\n    mxdist[u]=dist[u]\\n    for v in to[u]:\\n        if v!=pre:\\n            dist[v] = dist[u]+1\\n            dfs2(v,u)\\n            mxdist[u]=max(mxdist[u],mxdist[v])\\n\\ntry:\\n    lab = 1\\n    n, m = [int(x) for x in input().split()]\\n    to = [[] for i in range(n+10)]\\n    dist = [0 for i in range(n+10)]\\n    mxdist = [0 for  i in range(n+10)]\\n\\n    lab = 2\\n    for i in range(m):\\n        u,v = [int(x) for x in input().split()]\\n        to[u].append(v)\\n        to[v].append(u)\\n\\n    com=[]\\n    vis=[False for i in range(n+10)]\\n    for i in range(1,n+1):\\n        if vis[i]==False:\\n            now=[]\\n            dfs1(i,0)\\n            com.append(now)\\n\\n    lab = 3\\n    ct = []\\n    mx = 0\\n    for lis in com:\\n        tmp = []\\n        d = 0\\n        for root in lis:\\n            for u in lis:dist[u]=mxdist[u]=0\\n            dfs2(root,0)\\n            tmp.append((mxdist[root],root))\\n            d = max( d, sum( sorted([ mxdist[u] for u in to[root] ])[-2:] ) )\\n            #print(*[mxdist[u] for u in lis])\\n        mx = max(mx,d)\\n        #print('d =',d)\\n        for x in tmp:\\n            if x[0]==(d+1)//2:\\n                center = [x[1] for x in tmp if x[0]==(d+1)//2][0]\\n        ct.append( ((d+1)//2,center) )\\n\\n    #print(*ct)\\n\\n    lab = 4\\n    ct.sort(reverse=True)\\n    ans = []\\n    for i in range(1,len(ct)):\\n        mx = max(mx,ct[i][0]+1+ct[0][0])\\n        if i>1:mx = max(mx,ct[i][0]+2+ct[1][0])\\n        ans.append((ct[i][1],ct[0][1]))\\n    print(mx)\\n    for p in ans:\\n        print(*p)\\nexcept Exception as e:\\n    print('error after lable',lab,', type =',e)\\n\", \"import sys\\nsys.setrecursionlimit(10000)\\nclass Tree():\\n   def __init__(self, nodes):\\n     self.root = None\\n\\nclass Node():\\n  def __init__(self, val):\\n    self.parent = None\\n    self.val = val\\n    self.children = []\\n  def add(self, child):\\n    self.children.append(child)\\n  def get_max_dist(self, parent,dist):\\n     max_dist = dist\\n     for i in self.children:\\n        if i != parent:\\n           d = i.get_max_dist(self, dist+1)\\n           if max_dist < d:\\n              max_dist = d\\n     return max_dist\\n  def get_dist(self):\\n     return self.get_max_dist(self, 0)\\n  def get_count_ch(self, parent):\\n     count = len(self.children)\\n     for i in self.children:\\n        if i != parent:\\n           count += i.get_count_ch(self)-1\\n     return count\\n  def calc_child(self):\\n     return (self.get_count_ch(None),len(self.children), self.val)\\n     \\nclass Forest():\\n  def __init__(self, count):\\n    self.nodes = []  \\n    self.count = count\\n    self.set_nodes = set()\\n  def build(self):\\n    roots = []\\n    max_dist = []\\n    #diam = []\\n    list_root = []\\n    for i in self.nodes:\\n       tree =(list([x.get_dist() for x in i]))\\n       ma = max(tree)\\n       max_dist.append(ma)\\n       #diam.append(ma)\\n       m = i[tree.index(min(tree))]\\n       roots.append(m)#.val)\\n    if len(roots) > 1:\\n       ind = max_dist.index(max(max_dist))\\n       if len(roots) > 1 and ind != 0:\\n          roots[0], roots[ind] = roots[ind], roots[0]\\n    s = set()\\n    for i in self.nodes:\\n      for j in i:\\n        s.add(j.val)\\n    r = list(set(range(1, n+1))-s)\\n    \\n    if len(roots) > 0:# and len(diam) > 0:\\n       for i in range(1, len(roots)):\\n          self.add(roots[0].val, roots[i].val)\\n          list_root.append((roots[0].val, roots[i].val))\\n       #print(roots + r)\\n   \\n       for i in r:\\n         self.add(roots[0].val, i)\\n         list_root.append((roots[0].val, i))\\n    else:\\n       if len(r) == 1:\\n          print(0)\\n          return\\n       elif len(r) > 1:\\n          for i in range(1, len(r)):\\n             self.add(r[0], r[i])\\n             list_root.append((r[0], r[i]))\\n    distances = []\\n    for i in self.nodes[0]:\\n       dist =(i.get_dist())\\n       distances.append(dist)\\n    print(max(distances))\\n    for i in list_root:\\n       print(*i)\\n       \\n  def add(self, v, u):\\n    self.set_nodes.add(v)\\n    self.set_nodes.add(u)\\n    v_node, v_list = self.find(v)\\n    u_node, u_list = self.find(u)\\n    #print(v_node, u_node)\\n    if v_node == None and u_node == None:\\n      v_node = Node(v)\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      self.nodes.append([v_node, u_node])\\n    elif v_node != None and u_node != None and v_list != u_list:\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list += u_list\\n      self.nodes.remove(u_list)\\n    elif v_node == None and u_node != None:\\n      v_node = Node(v)\\n      u_node.add(v_node)\\n      v_node.add(u_node)\\n      u_list.append(v_node)\\n    elif v_node != None and u_node == None:\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list.append(u_node)\\n  def find(self, value):\\n    for i_list in self.nodes:\\n      for i in i_list:        \\n        if i.val == value:\\n          return i, i_list\\n    return None, None\\nn,m = list(map(int,input().split()))\\nf = Forest(n)\\nfor i in range(m):\\n  v,u = list(map(int,input().split()))\\n  f.add(v, u)\\n\\n#print(f.nodes)\\nf.build()\\n\"]",
        "difficulty": "introductory",
        "input": "3 2\n1 3\n2 3\n",
        "output": "2\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1092/E"
    },
    {
        "id": 1225,
        "task_id": 4374,
        "test_case_id": 4,
        "question": "You are given a forest — an undirected graph with $n$ vertices such that each its connected component is a tree.\n\nThe diameter (aka \"longest shortest path\") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices.\n\nYou task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1 \\le n \\le 1000$, $0 \\le m \\le n - 1$) — the number of vertices of the graph and the number of edges, respectively.\n\nEach of the next $m$ lines contains two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the edges.\n\nIt is guaranteed that the given graph is a forest.\n\n\n-----Output-----\n\nIn the first line print the diameter of the resulting tree.\n\nEach of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the added edges.\n\nThe resulting graph should be a tree and its diameter should be minimal possible.\n\nFor $m = n - 1$ no edges are added, thus the output consists of a single integer — diameter of the given tree.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Examples-----\nInput\n4 2\n1 2\n2 3\n\nOutput\n2\n4 2\n\nInput\n2 0\n\nOutput\n1\n1 2\n\nInput\n3 2\n1 3\n2 3\n\nOutput\n2\n\n\n\n-----Note-----\n\nIn the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2.\n\nEdge (1, 2) is the only option you have for the second example. The diameter is 1.\n\nYou can't add any edges in the third example. The diameter is already 2.",
        "solutions": "[\"import math\\nn,m=map(int,input().split())\\nneigh=[]\\nfor i in range(n):\\n    neigh.append([])\\nfor i in range(m):\\n    a,b=map(int,input().split())\\n    neigh[a-1].append(b-1)\\n    neigh[b-1].append(a-1)\\nseen=set()\\nindex=[0]*n\\ndiams=[]\\ntrees=0\\nfor i in range(n):\\n    if i not in seen:\\n        trees+=1\\n        index[i]=trees\\n        seen.add(i)\\n        layer=[i]\\n        prev=None\\n        pars=[None]\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(child)\\n                        newpars.append(vert)\\n                        index[child]=trees\\n                        seen.add(child)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        far=prev[0]\\n        layer=[[far]]\\n        pars=[None]\\n        prev=None\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i][-1]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(layer[i]+[child])\\n                        newpars.append(vert)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        diam=prev[0]\\n        lent=len(diam)\\n        mid=diam[lent//2]\\n        diams.append((lent-1,mid))\\ndiams.sort(reverse=True)\\nposs=[diams[0][0]]\\nif len(diams)>1:\\n    poss.append(math.ceil(diams[0][0]/2)+1+math.ceil(diams[1][0]/2))\\nif len(diams)>2:\\n    poss.append(math.ceil(diams[1][0]/2)+2+math.ceil(diams[2][0]/2))\\nprint(max(poss))\\ncent=diams[0][1]\\nfor i in range(len(diams)-1):\\n    print(cent+1,diams[i+1][1]+1)\", \"import sys\\n\\n\\ndef dfs(v, d, prev, i):\\n    nonlocal mid\\n    nonlocal M\\n    M[v] = False\\n    way[d] = v\\n    if way[d + 1] == 0:\\n        mid[i] = way[d // 2]\\n    mx = (d, v)\\n    for x in E[v]:\\n        if x != prev:\\n            mx = max(mx, dfs(x, d + 1, v, i))\\n\\n    return mx\\n\\n\\nsys.setrecursionlimit(2000)\\nn, m = list(map(int, input().split()))\\n\\nE = [[] for i in range(n + 1)]\\nfor i in range(m):\\n    a, b = list(map(int, input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nmid = [0] * (n + 2)\\nc = 0\\nk = 0\\nway = [0] * (n + 2)\\n\\nM = [True] * (n + 1)\\nM[0] = False\\ni = -1\\nwhile True in M:\\n    i += 1\\n    idx = M.index(True)\\n    p1 = dfs(idx, 0, 0, i)[1]\\n    way = [0] * (n + 2)\\n    s, p2 = dfs(p1, 0, 0, i)\\n    if s > c:\\n        c = s\\n        k = i\\nr = []\\nfor j in range(0, i + 1):\\n    if j == k:\\n        continue\\n    r.append((mid[k], mid[j]))\\n    E[mid[k]].append(mid[j])\\n    E[mid[j]].append(mid[k])\\np1 = dfs(1, 0, 0, n + 1)[1]\\ns, p2 = dfs(p1, 0, 0, n + 1)\\nprint(s)\\nfor item in r:\\n    print(item[0], item[1])\\n\", \"import sys\\nsys.setrecursionlimit(1100)\\n\\ndef dfs1(u,pre):            #find the components\\n    vis[u] = True\\n    now.append(u)\\n    for v in to[u]:\\n        if v!=pre:dfs1(v,u)\\n\\ndef dfs2(u,pre):            #calulate the distance\\n    mxdist[u]=dist[u]\\n    for v in to[u]:\\n        if v!=pre:\\n            dist[v] = dist[u]+1\\n            dfs2(v,u)\\n            mxdist[u]=max(mxdist[u],mxdist[v])\\n\\ntry:\\n    lab = 1\\n    n, m = [int(x) for x in input().split()]\\n    to = [[] for i in range(n+10)]\\n    dist = [0 for i in range(n+10)]\\n    mxdist = [0 for  i in range(n+10)]\\n\\n    lab = 2\\n    for i in range(m):\\n        u,v = [int(x) for x in input().split()]\\n        to[u].append(v)\\n        to[v].append(u)\\n\\n    com=[]\\n    vis=[False for i in range(n+10)]\\n    for i in range(1,n+1):\\n        if vis[i]==False:\\n            now=[]\\n            dfs1(i,0)\\n            com.append(now)\\n\\n    lab = 3\\n    ct = []\\n    mx = 0\\n    for lis in com:\\n        tmp = []\\n        d = 0\\n        for root in lis:\\n            for u in lis:dist[u]=mxdist[u]=0\\n            dfs2(root,0)\\n            tmp.append((mxdist[root],root))\\n            d = max( d, sum( sorted([ mxdist[u] for u in to[root] ])[-2:] ) )\\n            #print(*[mxdist[u] for u in lis])\\n        mx = max(mx,d)\\n        #print('d =',d)\\n        for x in tmp:\\n            if x[0]==(d+1)//2:\\n                center = [x[1] for x in tmp if x[0]==(d+1)//2][0]\\n        ct.append( ((d+1)//2,center) )\\n\\n    #print(*ct)\\n\\n    lab = 4\\n    ct.sort(reverse=True)\\n    ans = []\\n    for i in range(1,len(ct)):\\n        mx = max(mx,ct[i][0]+1+ct[0][0])\\n        if i>1:mx = max(mx,ct[i][0]+2+ct[1][0])\\n        ans.append((ct[i][1],ct[0][1]))\\n    print(mx)\\n    for p in ans:\\n        print(*p)\\nexcept Exception as e:\\n    print('error after lable',lab,', type =',e)\\n\", \"import sys\\nsys.setrecursionlimit(10000)\\nclass Tree():\\n   def __init__(self, nodes):\\n     self.root = None\\n\\nclass Node():\\n  def __init__(self, val):\\n    self.parent = None\\n    self.val = val\\n    self.children = []\\n  def add(self, child):\\n    self.children.append(child)\\n  def get_max_dist(self, parent,dist):\\n     max_dist = dist\\n     for i in self.children:\\n        if i != parent:\\n           d = i.get_max_dist(self, dist+1)\\n           if max_dist < d:\\n              max_dist = d\\n     return max_dist\\n  def get_dist(self):\\n     return self.get_max_dist(self, 0)\\n  def get_count_ch(self, parent):\\n     count = len(self.children)\\n     for i in self.children:\\n        if i != parent:\\n           count += i.get_count_ch(self)-1\\n     return count\\n  def calc_child(self):\\n     return (self.get_count_ch(None),len(self.children), self.val)\\n     \\nclass Forest():\\n  def __init__(self, count):\\n    self.nodes = []  \\n    self.count = count\\n    self.set_nodes = set()\\n  def build(self):\\n    roots = []\\n    max_dist = []\\n    #diam = []\\n    list_root = []\\n    for i in self.nodes:\\n       tree =(list([x.get_dist() for x in i]))\\n       ma = max(tree)\\n       max_dist.append(ma)\\n       #diam.append(ma)\\n       m = i[tree.index(min(tree))]\\n       roots.append(m)#.val)\\n    if len(roots) > 1:\\n       ind = max_dist.index(max(max_dist))\\n       if len(roots) > 1 and ind != 0:\\n          roots[0], roots[ind] = roots[ind], roots[0]\\n    s = set()\\n    for i in self.nodes:\\n      for j in i:\\n        s.add(j.val)\\n    r = list(set(range(1, n+1))-s)\\n    \\n    if len(roots) > 0:# and len(diam) > 0:\\n       for i in range(1, len(roots)):\\n          self.add(roots[0].val, roots[i].val)\\n          list_root.append((roots[0].val, roots[i].val))\\n       #print(roots + r)\\n   \\n       for i in r:\\n         self.add(roots[0].val, i)\\n         list_root.append((roots[0].val, i))\\n    else:\\n       if len(r) == 1:\\n          print(0)\\n          return\\n       elif len(r) > 1:\\n          for i in range(1, len(r)):\\n             self.add(r[0], r[i])\\n             list_root.append((r[0], r[i]))\\n    distances = []\\n    for i in self.nodes[0]:\\n       dist =(i.get_dist())\\n       distances.append(dist)\\n    print(max(distances))\\n    for i in list_root:\\n       print(*i)\\n       \\n  def add(self, v, u):\\n    self.set_nodes.add(v)\\n    self.set_nodes.add(u)\\n    v_node, v_list = self.find(v)\\n    u_node, u_list = self.find(u)\\n    #print(v_node, u_node)\\n    if v_node == None and u_node == None:\\n      v_node = Node(v)\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      self.nodes.append([v_node, u_node])\\n    elif v_node != None and u_node != None and v_list != u_list:\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list += u_list\\n      self.nodes.remove(u_list)\\n    elif v_node == None and u_node != None:\\n      v_node = Node(v)\\n      u_node.add(v_node)\\n      v_node.add(u_node)\\n      u_list.append(v_node)\\n    elif v_node != None and u_node == None:\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list.append(u_node)\\n  def find(self, value):\\n    for i_list in self.nodes:\\n      for i in i_list:        \\n        if i.val == value:\\n          return i, i_list\\n    return None, None\\nn,m = list(map(int,input().split()))\\nf = Forest(n)\\nfor i in range(m):\\n  v,u = list(map(int,input().split()))\\n  f.add(v, u)\\n\\n#print(f.nodes)\\nf.build()\\n\"]",
        "difficulty": "introductory",
        "input": "5 4\n5 2\n3 2\n5 4\n3 1\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1092/E"
    },
    {
        "id": 1226,
        "task_id": 4374,
        "test_case_id": 6,
        "question": "You are given a forest — an undirected graph with $n$ vertices such that each its connected component is a tree.\n\nThe diameter (aka \"longest shortest path\") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices.\n\nYou task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1 \\le n \\le 1000$, $0 \\le m \\le n - 1$) — the number of vertices of the graph and the number of edges, respectively.\n\nEach of the next $m$ lines contains two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the edges.\n\nIt is guaranteed that the given graph is a forest.\n\n\n-----Output-----\n\nIn the first line print the diameter of the resulting tree.\n\nEach of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the added edges.\n\nThe resulting graph should be a tree and its diameter should be minimal possible.\n\nFor $m = n - 1$ no edges are added, thus the output consists of a single integer — diameter of the given tree.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Examples-----\nInput\n4 2\n1 2\n2 3\n\nOutput\n2\n4 2\n\nInput\n2 0\n\nOutput\n1\n1 2\n\nInput\n3 2\n1 3\n2 3\n\nOutput\n2\n\n\n\n-----Note-----\n\nIn the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2.\n\nEdge (1, 2) is the only option you have for the second example. The diameter is 1.\n\nYou can't add any edges in the third example. The diameter is already 2.",
        "solutions": "[\"import math\\nn,m=map(int,input().split())\\nneigh=[]\\nfor i in range(n):\\n    neigh.append([])\\nfor i in range(m):\\n    a,b=map(int,input().split())\\n    neigh[a-1].append(b-1)\\n    neigh[b-1].append(a-1)\\nseen=set()\\nindex=[0]*n\\ndiams=[]\\ntrees=0\\nfor i in range(n):\\n    if i not in seen:\\n        trees+=1\\n        index[i]=trees\\n        seen.add(i)\\n        layer=[i]\\n        prev=None\\n        pars=[None]\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(child)\\n                        newpars.append(vert)\\n                        index[child]=trees\\n                        seen.add(child)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        far=prev[0]\\n        layer=[[far]]\\n        pars=[None]\\n        prev=None\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i][-1]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(layer[i]+[child])\\n                        newpars.append(vert)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        diam=prev[0]\\n        lent=len(diam)\\n        mid=diam[lent//2]\\n        diams.append((lent-1,mid))\\ndiams.sort(reverse=True)\\nposs=[diams[0][0]]\\nif len(diams)>1:\\n    poss.append(math.ceil(diams[0][0]/2)+1+math.ceil(diams[1][0]/2))\\nif len(diams)>2:\\n    poss.append(math.ceil(diams[1][0]/2)+2+math.ceil(diams[2][0]/2))\\nprint(max(poss))\\ncent=diams[0][1]\\nfor i in range(len(diams)-1):\\n    print(cent+1,diams[i+1][1]+1)\", \"import sys\\n\\n\\ndef dfs(v, d, prev, i):\\n    nonlocal mid\\n    nonlocal M\\n    M[v] = False\\n    way[d] = v\\n    if way[d + 1] == 0:\\n        mid[i] = way[d // 2]\\n    mx = (d, v)\\n    for x in E[v]:\\n        if x != prev:\\n            mx = max(mx, dfs(x, d + 1, v, i))\\n\\n    return mx\\n\\n\\nsys.setrecursionlimit(2000)\\nn, m = list(map(int, input().split()))\\n\\nE = [[] for i in range(n + 1)]\\nfor i in range(m):\\n    a, b = list(map(int, input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nmid = [0] * (n + 2)\\nc = 0\\nk = 0\\nway = [0] * (n + 2)\\n\\nM = [True] * (n + 1)\\nM[0] = False\\ni = -1\\nwhile True in M:\\n    i += 1\\n    idx = M.index(True)\\n    p1 = dfs(idx, 0, 0, i)[1]\\n    way = [0] * (n + 2)\\n    s, p2 = dfs(p1, 0, 0, i)\\n    if s > c:\\n        c = s\\n        k = i\\nr = []\\nfor j in range(0, i + 1):\\n    if j == k:\\n        continue\\n    r.append((mid[k], mid[j]))\\n    E[mid[k]].append(mid[j])\\n    E[mid[j]].append(mid[k])\\np1 = dfs(1, 0, 0, n + 1)[1]\\ns, p2 = dfs(p1, 0, 0, n + 1)\\nprint(s)\\nfor item in r:\\n    print(item[0], item[1])\\n\", \"import sys\\nsys.setrecursionlimit(1100)\\n\\ndef dfs1(u,pre):            #find the components\\n    vis[u] = True\\n    now.append(u)\\n    for v in to[u]:\\n        if v!=pre:dfs1(v,u)\\n\\ndef dfs2(u,pre):            #calulate the distance\\n    mxdist[u]=dist[u]\\n    for v in to[u]:\\n        if v!=pre:\\n            dist[v] = dist[u]+1\\n            dfs2(v,u)\\n            mxdist[u]=max(mxdist[u],mxdist[v])\\n\\ntry:\\n    lab = 1\\n    n, m = [int(x) for x in input().split()]\\n    to = [[] for i in range(n+10)]\\n    dist = [0 for i in range(n+10)]\\n    mxdist = [0 for  i in range(n+10)]\\n\\n    lab = 2\\n    for i in range(m):\\n        u,v = [int(x) for x in input().split()]\\n        to[u].append(v)\\n        to[v].append(u)\\n\\n    com=[]\\n    vis=[False for i in range(n+10)]\\n    for i in range(1,n+1):\\n        if vis[i]==False:\\n            now=[]\\n            dfs1(i,0)\\n            com.append(now)\\n\\n    lab = 3\\n    ct = []\\n    mx = 0\\n    for lis in com:\\n        tmp = []\\n        d = 0\\n        for root in lis:\\n            for u in lis:dist[u]=mxdist[u]=0\\n            dfs2(root,0)\\n            tmp.append((mxdist[root],root))\\n            d = max( d, sum( sorted([ mxdist[u] for u in to[root] ])[-2:] ) )\\n            #print(*[mxdist[u] for u in lis])\\n        mx = max(mx,d)\\n        #print('d =',d)\\n        for x in tmp:\\n            if x[0]==(d+1)//2:\\n                center = [x[1] for x in tmp if x[0]==(d+1)//2][0]\\n        ct.append( ((d+1)//2,center) )\\n\\n    #print(*ct)\\n\\n    lab = 4\\n    ct.sort(reverse=True)\\n    ans = []\\n    for i in range(1,len(ct)):\\n        mx = max(mx,ct[i][0]+1+ct[0][0])\\n        if i>1:mx = max(mx,ct[i][0]+2+ct[1][0])\\n        ans.append((ct[i][1],ct[0][1]))\\n    print(mx)\\n    for p in ans:\\n        print(*p)\\nexcept Exception as e:\\n    print('error after lable',lab,', type =',e)\\n\", \"import sys\\nsys.setrecursionlimit(10000)\\nclass Tree():\\n   def __init__(self, nodes):\\n     self.root = None\\n\\nclass Node():\\n  def __init__(self, val):\\n    self.parent = None\\n    self.val = val\\n    self.children = []\\n  def add(self, child):\\n    self.children.append(child)\\n  def get_max_dist(self, parent,dist):\\n     max_dist = dist\\n     for i in self.children:\\n        if i != parent:\\n           d = i.get_max_dist(self, dist+1)\\n           if max_dist < d:\\n              max_dist = d\\n     return max_dist\\n  def get_dist(self):\\n     return self.get_max_dist(self, 0)\\n  def get_count_ch(self, parent):\\n     count = len(self.children)\\n     for i in self.children:\\n        if i != parent:\\n           count += i.get_count_ch(self)-1\\n     return count\\n  def calc_child(self):\\n     return (self.get_count_ch(None),len(self.children), self.val)\\n     \\nclass Forest():\\n  def __init__(self, count):\\n    self.nodes = []  \\n    self.count = count\\n    self.set_nodes = set()\\n  def build(self):\\n    roots = []\\n    max_dist = []\\n    #diam = []\\n    list_root = []\\n    for i in self.nodes:\\n       tree =(list([x.get_dist() for x in i]))\\n       ma = max(tree)\\n       max_dist.append(ma)\\n       #diam.append(ma)\\n       m = i[tree.index(min(tree))]\\n       roots.append(m)#.val)\\n    if len(roots) > 1:\\n       ind = max_dist.index(max(max_dist))\\n       if len(roots) > 1 and ind != 0:\\n          roots[0], roots[ind] = roots[ind], roots[0]\\n    s = set()\\n    for i in self.nodes:\\n      for j in i:\\n        s.add(j.val)\\n    r = list(set(range(1, n+1))-s)\\n    \\n    if len(roots) > 0:# and len(diam) > 0:\\n       for i in range(1, len(roots)):\\n          self.add(roots[0].val, roots[i].val)\\n          list_root.append((roots[0].val, roots[i].val))\\n       #print(roots + r)\\n   \\n       for i in r:\\n         self.add(roots[0].val, i)\\n         list_root.append((roots[0].val, i))\\n    else:\\n       if len(r) == 1:\\n          print(0)\\n          return\\n       elif len(r) > 1:\\n          for i in range(1, len(r)):\\n             self.add(r[0], r[i])\\n             list_root.append((r[0], r[i]))\\n    distances = []\\n    for i in self.nodes[0]:\\n       dist =(i.get_dist())\\n       distances.append(dist)\\n    print(max(distances))\\n    for i in list_root:\\n       print(*i)\\n       \\n  def add(self, v, u):\\n    self.set_nodes.add(v)\\n    self.set_nodes.add(u)\\n    v_node, v_list = self.find(v)\\n    u_node, u_list = self.find(u)\\n    #print(v_node, u_node)\\n    if v_node == None and u_node == None:\\n      v_node = Node(v)\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      self.nodes.append([v_node, u_node])\\n    elif v_node != None and u_node != None and v_list != u_list:\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list += u_list\\n      self.nodes.remove(u_list)\\n    elif v_node == None and u_node != None:\\n      v_node = Node(v)\\n      u_node.add(v_node)\\n      v_node.add(u_node)\\n      u_list.append(v_node)\\n    elif v_node != None and u_node == None:\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list.append(u_node)\\n  def find(self, value):\\n    for i_list in self.nodes:\\n      for i in i_list:        \\n        if i.val == value:\\n          return i, i_list\\n    return None, None\\nn,m = list(map(int,input().split()))\\nf = Forest(n)\\nfor i in range(m):\\n  v,u = list(map(int,input().split()))\\n  f.add(v, u)\\n\\n#print(f.nodes)\\nf.build()\\n\"]",
        "difficulty": "introductory",
        "input": "5 0\n",
        "output": "2\n1 5\n2 5\n3 5\n4 5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1092/E"
    },
    {
        "id": 1227,
        "task_id": 4374,
        "test_case_id": 8,
        "question": "You are given a forest — an undirected graph with $n$ vertices such that each its connected component is a tree.\n\nThe diameter (aka \"longest shortest path\") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices.\n\nYou task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1 \\le n \\le 1000$, $0 \\le m \\le n - 1$) — the number of vertices of the graph and the number of edges, respectively.\n\nEach of the next $m$ lines contains two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the edges.\n\nIt is guaranteed that the given graph is a forest.\n\n\n-----Output-----\n\nIn the first line print the diameter of the resulting tree.\n\nEach of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the added edges.\n\nThe resulting graph should be a tree and its diameter should be minimal possible.\n\nFor $m = n - 1$ no edges are added, thus the output consists of a single integer — diameter of the given tree.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Examples-----\nInput\n4 2\n1 2\n2 3\n\nOutput\n2\n4 2\n\nInput\n2 0\n\nOutput\n1\n1 2\n\nInput\n3 2\n1 3\n2 3\n\nOutput\n2\n\n\n\n-----Note-----\n\nIn the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2.\n\nEdge (1, 2) is the only option you have for the second example. The diameter is 1.\n\nYou can't add any edges in the third example. The diameter is already 2.",
        "solutions": "[\"import math\\nn,m=map(int,input().split())\\nneigh=[]\\nfor i in range(n):\\n    neigh.append([])\\nfor i in range(m):\\n    a,b=map(int,input().split())\\n    neigh[a-1].append(b-1)\\n    neigh[b-1].append(a-1)\\nseen=set()\\nindex=[0]*n\\ndiams=[]\\ntrees=0\\nfor i in range(n):\\n    if i not in seen:\\n        trees+=1\\n        index[i]=trees\\n        seen.add(i)\\n        layer=[i]\\n        prev=None\\n        pars=[None]\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(child)\\n                        newpars.append(vert)\\n                        index[child]=trees\\n                        seen.add(child)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        far=prev[0]\\n        layer=[[far]]\\n        pars=[None]\\n        prev=None\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i][-1]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(layer[i]+[child])\\n                        newpars.append(vert)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        diam=prev[0]\\n        lent=len(diam)\\n        mid=diam[lent//2]\\n        diams.append((lent-1,mid))\\ndiams.sort(reverse=True)\\nposs=[diams[0][0]]\\nif len(diams)>1:\\n    poss.append(math.ceil(diams[0][0]/2)+1+math.ceil(diams[1][0]/2))\\nif len(diams)>2:\\n    poss.append(math.ceil(diams[1][0]/2)+2+math.ceil(diams[2][0]/2))\\nprint(max(poss))\\ncent=diams[0][1]\\nfor i in range(len(diams)-1):\\n    print(cent+1,diams[i+1][1]+1)\", \"import sys\\n\\n\\ndef dfs(v, d, prev, i):\\n    nonlocal mid\\n    nonlocal M\\n    M[v] = False\\n    way[d] = v\\n    if way[d + 1] == 0:\\n        mid[i] = way[d // 2]\\n    mx = (d, v)\\n    for x in E[v]:\\n        if x != prev:\\n            mx = max(mx, dfs(x, d + 1, v, i))\\n\\n    return mx\\n\\n\\nsys.setrecursionlimit(2000)\\nn, m = list(map(int, input().split()))\\n\\nE = [[] for i in range(n + 1)]\\nfor i in range(m):\\n    a, b = list(map(int, input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nmid = [0] * (n + 2)\\nc = 0\\nk = 0\\nway = [0] * (n + 2)\\n\\nM = [True] * (n + 1)\\nM[0] = False\\ni = -1\\nwhile True in M:\\n    i += 1\\n    idx = M.index(True)\\n    p1 = dfs(idx, 0, 0, i)[1]\\n    way = [0] * (n + 2)\\n    s, p2 = dfs(p1, 0, 0, i)\\n    if s > c:\\n        c = s\\n        k = i\\nr = []\\nfor j in range(0, i + 1):\\n    if j == k:\\n        continue\\n    r.append((mid[k], mid[j]))\\n    E[mid[k]].append(mid[j])\\n    E[mid[j]].append(mid[k])\\np1 = dfs(1, 0, 0, n + 1)[1]\\ns, p2 = dfs(p1, 0, 0, n + 1)\\nprint(s)\\nfor item in r:\\n    print(item[0], item[1])\\n\", \"import sys\\nsys.setrecursionlimit(1100)\\n\\ndef dfs1(u,pre):            #find the components\\n    vis[u] = True\\n    now.append(u)\\n    for v in to[u]:\\n        if v!=pre:dfs1(v,u)\\n\\ndef dfs2(u,pre):            #calulate the distance\\n    mxdist[u]=dist[u]\\n    for v in to[u]:\\n        if v!=pre:\\n            dist[v] = dist[u]+1\\n            dfs2(v,u)\\n            mxdist[u]=max(mxdist[u],mxdist[v])\\n\\ntry:\\n    lab = 1\\n    n, m = [int(x) for x in input().split()]\\n    to = [[] for i in range(n+10)]\\n    dist = [0 for i in range(n+10)]\\n    mxdist = [0 for  i in range(n+10)]\\n\\n    lab = 2\\n    for i in range(m):\\n        u,v = [int(x) for x in input().split()]\\n        to[u].append(v)\\n        to[v].append(u)\\n\\n    com=[]\\n    vis=[False for i in range(n+10)]\\n    for i in range(1,n+1):\\n        if vis[i]==False:\\n            now=[]\\n            dfs1(i,0)\\n            com.append(now)\\n\\n    lab = 3\\n    ct = []\\n    mx = 0\\n    for lis in com:\\n        tmp = []\\n        d = 0\\n        for root in lis:\\n            for u in lis:dist[u]=mxdist[u]=0\\n            dfs2(root,0)\\n            tmp.append((mxdist[root],root))\\n            d = max( d, sum( sorted([ mxdist[u] for u in to[root] ])[-2:] ) )\\n            #print(*[mxdist[u] for u in lis])\\n        mx = max(mx,d)\\n        #print('d =',d)\\n        for x in tmp:\\n            if x[0]==(d+1)//2:\\n                center = [x[1] for x in tmp if x[0]==(d+1)//2][0]\\n        ct.append( ((d+1)//2,center) )\\n\\n    #print(*ct)\\n\\n    lab = 4\\n    ct.sort(reverse=True)\\n    ans = []\\n    for i in range(1,len(ct)):\\n        mx = max(mx,ct[i][0]+1+ct[0][0])\\n        if i>1:mx = max(mx,ct[i][0]+2+ct[1][0])\\n        ans.append((ct[i][1],ct[0][1]))\\n    print(mx)\\n    for p in ans:\\n        print(*p)\\nexcept Exception as e:\\n    print('error after lable',lab,', type =',e)\\n\", \"import sys\\nsys.setrecursionlimit(10000)\\nclass Tree():\\n   def __init__(self, nodes):\\n     self.root = None\\n\\nclass Node():\\n  def __init__(self, val):\\n    self.parent = None\\n    self.val = val\\n    self.children = []\\n  def add(self, child):\\n    self.children.append(child)\\n  def get_max_dist(self, parent,dist):\\n     max_dist = dist\\n     for i in self.children:\\n        if i != parent:\\n           d = i.get_max_dist(self, dist+1)\\n           if max_dist < d:\\n              max_dist = d\\n     return max_dist\\n  def get_dist(self):\\n     return self.get_max_dist(self, 0)\\n  def get_count_ch(self, parent):\\n     count = len(self.children)\\n     for i in self.children:\\n        if i != parent:\\n           count += i.get_count_ch(self)-1\\n     return count\\n  def calc_child(self):\\n     return (self.get_count_ch(None),len(self.children), self.val)\\n     \\nclass Forest():\\n  def __init__(self, count):\\n    self.nodes = []  \\n    self.count = count\\n    self.set_nodes = set()\\n  def build(self):\\n    roots = []\\n    max_dist = []\\n    #diam = []\\n    list_root = []\\n    for i in self.nodes:\\n       tree =(list([x.get_dist() for x in i]))\\n       ma = max(tree)\\n       max_dist.append(ma)\\n       #diam.append(ma)\\n       m = i[tree.index(min(tree))]\\n       roots.append(m)#.val)\\n    if len(roots) > 1:\\n       ind = max_dist.index(max(max_dist))\\n       if len(roots) > 1 and ind != 0:\\n          roots[0], roots[ind] = roots[ind], roots[0]\\n    s = set()\\n    for i in self.nodes:\\n      for j in i:\\n        s.add(j.val)\\n    r = list(set(range(1, n+1))-s)\\n    \\n    if len(roots) > 0:# and len(diam) > 0:\\n       for i in range(1, len(roots)):\\n          self.add(roots[0].val, roots[i].val)\\n          list_root.append((roots[0].val, roots[i].val))\\n       #print(roots + r)\\n   \\n       for i in r:\\n         self.add(roots[0].val, i)\\n         list_root.append((roots[0].val, i))\\n    else:\\n       if len(r) == 1:\\n          print(0)\\n          return\\n       elif len(r) > 1:\\n          for i in range(1, len(r)):\\n             self.add(r[0], r[i])\\n             list_root.append((r[0], r[i]))\\n    distances = []\\n    for i in self.nodes[0]:\\n       dist =(i.get_dist())\\n       distances.append(dist)\\n    print(max(distances))\\n    for i in list_root:\\n       print(*i)\\n       \\n  def add(self, v, u):\\n    self.set_nodes.add(v)\\n    self.set_nodes.add(u)\\n    v_node, v_list = self.find(v)\\n    u_node, u_list = self.find(u)\\n    #print(v_node, u_node)\\n    if v_node == None and u_node == None:\\n      v_node = Node(v)\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      self.nodes.append([v_node, u_node])\\n    elif v_node != None and u_node != None and v_list != u_list:\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list += u_list\\n      self.nodes.remove(u_list)\\n    elif v_node == None and u_node != None:\\n      v_node = Node(v)\\n      u_node.add(v_node)\\n      v_node.add(u_node)\\n      u_list.append(v_node)\\n    elif v_node != None and u_node == None:\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list.append(u_node)\\n  def find(self, value):\\n    for i_list in self.nodes:\\n      for i in i_list:        \\n        if i.val == value:\\n          return i, i_list\\n    return None, None\\nn,m = list(map(int,input().split()))\\nf = Forest(n)\\nfor i in range(m):\\n  v,u = list(map(int,input().split()))\\n  f.add(v, u)\\n\\n#print(f.nodes)\\nf.build()\\n\"]",
        "difficulty": "introductory",
        "input": "1 0\n",
        "output": "0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1092/E"
    },
    {
        "id": 1228,
        "task_id": 4374,
        "test_case_id": 9,
        "question": "You are given a forest — an undirected graph with $n$ vertices such that each its connected component is a tree.\n\nThe diameter (aka \"longest shortest path\") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices.\n\nYou task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Input-----\n\nThe first line contains two integers $n$ and $m$ ($1 \\le n \\le 1000$, $0 \\le m \\le n - 1$) — the number of vertices of the graph and the number of edges, respectively.\n\nEach of the next $m$ lines contains two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the edges.\n\nIt is guaranteed that the given graph is a forest.\n\n\n-----Output-----\n\nIn the first line print the diameter of the resulting tree.\n\nEach of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \\le v, u \\le n$, $v \\ne u$) — the descriptions of the added edges.\n\nThe resulting graph should be a tree and its diameter should be minimal possible.\n\nFor $m = n - 1$ no edges are added, thus the output consists of a single integer — diameter of the given tree.\n\nIf there are multiple correct answers, print any of them.\n\n\n-----Examples-----\nInput\n4 2\n1 2\n2 3\n\nOutput\n2\n4 2\n\nInput\n2 0\n\nOutput\n1\n1 2\n\nInput\n3 2\n1 3\n2 3\n\nOutput\n2\n\n\n\n-----Note-----\n\nIn the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2.\n\nEdge (1, 2) is the only option you have for the second example. The diameter is 1.\n\nYou can't add any edges in the third example. The diameter is already 2.",
        "solutions": "[\"import math\\nn,m=map(int,input().split())\\nneigh=[]\\nfor i in range(n):\\n    neigh.append([])\\nfor i in range(m):\\n    a,b=map(int,input().split())\\n    neigh[a-1].append(b-1)\\n    neigh[b-1].append(a-1)\\nseen=set()\\nindex=[0]*n\\ndiams=[]\\ntrees=0\\nfor i in range(n):\\n    if i not in seen:\\n        trees+=1\\n        index[i]=trees\\n        seen.add(i)\\n        layer=[i]\\n        prev=None\\n        pars=[None]\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(child)\\n                        newpars.append(vert)\\n                        index[child]=trees\\n                        seen.add(child)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        far=prev[0]\\n        layer=[[far]]\\n        pars=[None]\\n        prev=None\\n        while layer!=[]:\\n            newlayer=[]\\n            newpars=[]\\n            for i in range(len(layer)):\\n                vert=layer[i][-1]\\n                par=pars[i]\\n                for child in neigh[vert]:\\n                    if child!=par:\\n                        newlayer.append(layer[i]+[child])\\n                        newpars.append(vert)\\n            prev=layer\\n            layer=newlayer\\n            pars=newpars\\n        diam=prev[0]\\n        lent=len(diam)\\n        mid=diam[lent//2]\\n        diams.append((lent-1,mid))\\ndiams.sort(reverse=True)\\nposs=[diams[0][0]]\\nif len(diams)>1:\\n    poss.append(math.ceil(diams[0][0]/2)+1+math.ceil(diams[1][0]/2))\\nif len(diams)>2:\\n    poss.append(math.ceil(diams[1][0]/2)+2+math.ceil(diams[2][0]/2))\\nprint(max(poss))\\ncent=diams[0][1]\\nfor i in range(len(diams)-1):\\n    print(cent+1,diams[i+1][1]+1)\", \"import sys\\n\\n\\ndef dfs(v, d, prev, i):\\n    nonlocal mid\\n    nonlocal M\\n    M[v] = False\\n    way[d] = v\\n    if way[d + 1] == 0:\\n        mid[i] = way[d // 2]\\n    mx = (d, v)\\n    for x in E[v]:\\n        if x != prev:\\n            mx = max(mx, dfs(x, d + 1, v, i))\\n\\n    return mx\\n\\n\\nsys.setrecursionlimit(2000)\\nn, m = list(map(int, input().split()))\\n\\nE = [[] for i in range(n + 1)]\\nfor i in range(m):\\n    a, b = list(map(int, input().split()))\\n    E[a].append(b)\\n    E[b].append(a)\\n\\nmid = [0] * (n + 2)\\nc = 0\\nk = 0\\nway = [0] * (n + 2)\\n\\nM = [True] * (n + 1)\\nM[0] = False\\ni = -1\\nwhile True in M:\\n    i += 1\\n    idx = M.index(True)\\n    p1 = dfs(idx, 0, 0, i)[1]\\n    way = [0] * (n + 2)\\n    s, p2 = dfs(p1, 0, 0, i)\\n    if s > c:\\n        c = s\\n        k = i\\nr = []\\nfor j in range(0, i + 1):\\n    if j == k:\\n        continue\\n    r.append((mid[k], mid[j]))\\n    E[mid[k]].append(mid[j])\\n    E[mid[j]].append(mid[k])\\np1 = dfs(1, 0, 0, n + 1)[1]\\ns, p2 = dfs(p1, 0, 0, n + 1)\\nprint(s)\\nfor item in r:\\n    print(item[0], item[1])\\n\", \"import sys\\nsys.setrecursionlimit(1100)\\n\\ndef dfs1(u,pre):            #find the components\\n    vis[u] = True\\n    now.append(u)\\n    for v in to[u]:\\n        if v!=pre:dfs1(v,u)\\n\\ndef dfs2(u,pre):            #calulate the distance\\n    mxdist[u]=dist[u]\\n    for v in to[u]:\\n        if v!=pre:\\n            dist[v] = dist[u]+1\\n            dfs2(v,u)\\n            mxdist[u]=max(mxdist[u],mxdist[v])\\n\\ntry:\\n    lab = 1\\n    n, m = [int(x) for x in input().split()]\\n    to = [[] for i in range(n+10)]\\n    dist = [0 for i in range(n+10)]\\n    mxdist = [0 for  i in range(n+10)]\\n\\n    lab = 2\\n    for i in range(m):\\n        u,v = [int(x) for x in input().split()]\\n        to[u].append(v)\\n        to[v].append(u)\\n\\n    com=[]\\n    vis=[False for i in range(n+10)]\\n    for i in range(1,n+1):\\n        if vis[i]==False:\\n            now=[]\\n            dfs1(i,0)\\n            com.append(now)\\n\\n    lab = 3\\n    ct = []\\n    mx = 0\\n    for lis in com:\\n        tmp = []\\n        d = 0\\n        for root in lis:\\n            for u in lis:dist[u]=mxdist[u]=0\\n            dfs2(root,0)\\n            tmp.append((mxdist[root],root))\\n            d = max( d, sum( sorted([ mxdist[u] for u in to[root] ])[-2:] ) )\\n            #print(*[mxdist[u] for u in lis])\\n        mx = max(mx,d)\\n        #print('d =',d)\\n        for x in tmp:\\n            if x[0]==(d+1)//2:\\n                center = [x[1] for x in tmp if x[0]==(d+1)//2][0]\\n        ct.append( ((d+1)//2,center) )\\n\\n    #print(*ct)\\n\\n    lab = 4\\n    ct.sort(reverse=True)\\n    ans = []\\n    for i in range(1,len(ct)):\\n        mx = max(mx,ct[i][0]+1+ct[0][0])\\n        if i>1:mx = max(mx,ct[i][0]+2+ct[1][0])\\n        ans.append((ct[i][1],ct[0][1]))\\n    print(mx)\\n    for p in ans:\\n        print(*p)\\nexcept Exception as e:\\n    print('error after lable',lab,', type =',e)\\n\", \"import sys\\nsys.setrecursionlimit(10000)\\nclass Tree():\\n   def __init__(self, nodes):\\n     self.root = None\\n\\nclass Node():\\n  def __init__(self, val):\\n    self.parent = None\\n    self.val = val\\n    self.children = []\\n  def add(self, child):\\n    self.children.append(child)\\n  def get_max_dist(self, parent,dist):\\n     max_dist = dist\\n     for i in self.children:\\n        if i != parent:\\n           d = i.get_max_dist(self, dist+1)\\n           if max_dist < d:\\n              max_dist = d\\n     return max_dist\\n  def get_dist(self):\\n     return self.get_max_dist(self, 0)\\n  def get_count_ch(self, parent):\\n     count = len(self.children)\\n     for i in self.children:\\n        if i != parent:\\n           count += i.get_count_ch(self)-1\\n     return count\\n  def calc_child(self):\\n     return (self.get_count_ch(None),len(self.children), self.val)\\n     \\nclass Forest():\\n  def __init__(self, count):\\n    self.nodes = []  \\n    self.count = count\\n    self.set_nodes = set()\\n  def build(self):\\n    roots = []\\n    max_dist = []\\n    #diam = []\\n    list_root = []\\n    for i in self.nodes:\\n       tree =(list([x.get_dist() for x in i]))\\n       ma = max(tree)\\n       max_dist.append(ma)\\n       #diam.append(ma)\\n       m = i[tree.index(min(tree))]\\n       roots.append(m)#.val)\\n    if len(roots) > 1:\\n       ind = max_dist.index(max(max_dist))\\n       if len(roots) > 1 and ind != 0:\\n          roots[0], roots[ind] = roots[ind], roots[0]\\n    s = set()\\n    for i in self.nodes:\\n      for j in i:\\n        s.add(j.val)\\n    r = list(set(range(1, n+1))-s)\\n    \\n    if len(roots) > 0:# and len(diam) > 0:\\n       for i in range(1, len(roots)):\\n          self.add(roots[0].val, roots[i].val)\\n          list_root.append((roots[0].val, roots[i].val))\\n       #print(roots + r)\\n   \\n       for i in r:\\n         self.add(roots[0].val, i)\\n         list_root.append((roots[0].val, i))\\n    else:\\n       if len(r) == 1:\\n          print(0)\\n          return\\n       elif len(r) > 1:\\n          for i in range(1, len(r)):\\n             self.add(r[0], r[i])\\n             list_root.append((r[0], r[i]))\\n    distances = []\\n    for i in self.nodes[0]:\\n       dist =(i.get_dist())\\n       distances.append(dist)\\n    print(max(distances))\\n    for i in list_root:\\n       print(*i)\\n       \\n  def add(self, v, u):\\n    self.set_nodes.add(v)\\n    self.set_nodes.add(u)\\n    v_node, v_list = self.find(v)\\n    u_node, u_list = self.find(u)\\n    #print(v_node, u_node)\\n    if v_node == None and u_node == None:\\n      v_node = Node(v)\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      self.nodes.append([v_node, u_node])\\n    elif v_node != None and u_node != None and v_list != u_list:\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list += u_list\\n      self.nodes.remove(u_list)\\n    elif v_node == None and u_node != None:\\n      v_node = Node(v)\\n      u_node.add(v_node)\\n      v_node.add(u_node)\\n      u_list.append(v_node)\\n    elif v_node != None and u_node == None:\\n      u_node = Node(u)\\n      v_node.add(u_node)\\n      u_node.add(v_node)\\n      v_list.append(u_node)\\n  def find(self, value):\\n    for i_list in self.nodes:\\n      for i in i_list:        \\n        if i.val == value:\\n          return i, i_list\\n    return None, None\\nn,m = list(map(int,input().split()))\\nf = Forest(n)\\nfor i in range(m):\\n  v,u = list(map(int,input().split()))\\n  f.add(v, u)\\n\\n#print(f.nodes)\\nf.build()\\n\"]",
        "difficulty": "introductory",
        "input": "33 0\n",
        "output": "2\n1 33\n2 33\n3 33\n4 33\n5 33\n6 33\n7 33\n8 33\n9 33\n10 33\n11 33\n12 33\n13 33\n14 33\n15 33\n16 33\n17 33\n18 33\n19 33\n20 33\n21 33\n22 33\n23 33\n24 33\n25 33\n26 33\n27 33\n28 33\n29 33\n30 33\n31 33\n32 33\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1092/E"
    },
    {
        "id": 1229,
        "task_id": 4375,
        "test_case_id": 1,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n",
        "output": "11\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1230,
        "task_id": 4375,
        "test_case_id": 2,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n",
        "output": "4\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1231,
        "task_id": 4375,
        "test_case_id": 3,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "14 2\n3 2 3 1 2 3 4 2 3 1 4 2 1 3\n12 10\n2 7\n13 12\n14 11\n10 5\n1 2\n11 4\n4 9\n14 8\n12 4\n7 9\n13 6\n10 3\n",
        "output": "15\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1232,
        "task_id": 4375,
        "test_case_id": 4,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "5 3\n1 5 2 4 3\n5 2\n3 2\n1 4\n2 1\n",
        "output": "5\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1233,
        "task_id": 4375,
        "test_case_id": 5,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "14 4\n6 4 1 2 3 3 5 2 2 7 3 6 6 1\n13 9\n3 12\n10 11\n14 13\n6 7\n5 2\n8 12\n10 14\n2 14\n6 8\n1 9\n4 13\n8 13\n",
        "output": "14\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1234,
        "task_id": 4375,
        "test_case_id": 6,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "9 2\n4 4 4 4 3 4 4 5 3\n6 5\n2 3\n7 5\n2 4\n9 8\n1 3\n3 9\n9 6\n",
        "output": "17\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1235,
        "task_id": 4375,
        "test_case_id": 7,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "10 2\n4 1 4 4 2 3 2 1 1 4\n1 7\n1 3\n8 3\n3 6\n4 1\n1 2\n9 7\n8 10\n1 5\n",
        "output": "12\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1236,
        "task_id": 4375,
        "test_case_id": 8,
        "question": "You are given a tree, which consists of $n$ vertices. Recall that a tree is a connected undirected graph without cycles.  [Image] Example of a tree. \n\nVertices are numbered from $1$ to $n$. All vertices have weights, the weight of the vertex $v$ is $a_v$.\n\nRecall that the distance between two vertices in the tree is the number of edges on a simple path between them.\n\nYour task is to find the subset of vertices with the maximum total weight (the weight of the subset is the sum of weights of all vertices in it) such that there is no pair of vertices with the distance $k$ or less between them in this subset.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $k$ ($1 \\le n, k \\le 200$) — the number of vertices in the tree and the distance restriction, respectively.\n\nThe second line of the input contains $n$ integers $a_1, a_2, \\dots, a_n$ ($1 \\le a_i \\le 10^5$), where $a_i$ is the weight of the vertex $i$.\n\nThe next $n - 1$ lines contain edges of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$).\n\nIt is guaranteed that the given edges form a tree.\n\n\n-----Output-----\n\nPrint one integer — the maximum total weight of the subset in which all pairs of vertices have distance more than $k$.\n\n\n-----Examples-----\nInput\n5 1\n1 2 3 4 5\n1 2\n2 3\n3 4\n3 5\n\nOutput\n11\n\nInput\n7 2\n2 1 2 1 2 1 1\n6 4\n1 5\n3 1\n2 3\n7 5\n7 4\n\nOutput\n4",
        "solutions": "[\"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = map(int, input().split())\\n\\na = list(map(int, input().split()))\\ng = [[] for i in range(n)]\\nfor i in range(n - 1):\\n\\tv, u = map(int, input().split())\\n\\tv -= 1\\n\\tu -= 1\\n\\tg[v].append(u)\\n\\tg[u].append(v)\\n\\ndp = [[] for i in range(n)]\\nd = [1 for i in range(n)]\\n\\ndef dfs(v, p = -1):\\n\\tdp[v].append(a[v])\\n\\tfor u in g[v]:\\n\\t\\tif u == p:\\n\\t\\t\\tcontinue\\n\\t\\tdfs(u, v)\\n\\t\\ttmp = [-10**18 for i in range(max(d[v], d[u] + 1))]\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tfor j in range(max(0, k - i), d[u]):\\n\\t\\t\\t\\ttmp[min(i, j + 1)] = max(tmp[min(i, j + 1)], dp[v][i] + dp[u][j])\\n\\t\\tfor i in range(d[u]):\\n\\t\\t\\ttmp[i + 1] = max(tmp[i + 1], dp[u][i])\\n\\t\\tfor i in range(d[v]):\\n\\t\\t\\tdp[v][i] = max(dp[v][i], tmp[i])\\n\\t\\tdp[v] += tmp[d[v]:]\\n\\t\\td[v] = max(d[v], d[u] + 1)\\n\\t\\tfor i in range(d[v] - 1, 0, -1):\\n\\t\\t\\tdp[v][i - 1] = max(dp[v][i - 1], dp[v][i])\\n\\ndfs(0)\\nprint(max(dp[0]))\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n \\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n \\n \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n \\nprint(dfs(0)[0])\", \"n, k = list(map(int, input().split()))\\na = list(map(int, input().split()))\\ng = {}\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n\\nfor _ in range(1, n):\\n\\tu, v = [int(x) - 1 for x in input().split()]\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\n\\nprint(dfs(0)[0])\\n\", \"n, k = map(int, input().split())\\na = list(map(int, input().split()))\\ng = {}\\n\\n\\ndef dfs(v, p=-1):\\n\\tc = [dfs(child, v) for child in g.get(v, set()) - {p}]\\n\\tc.sort(key=len, reverse=True)\\n\\tr = []\\n\\ti = 0\\n\\twhile c:\\n\\t\\tif i >= len(c[-1]):\\n\\t\\t\\tc.pop()\\n\\t\\telse:\\n\\t\\t\\to = max(i, k - i - 1)\\n\\t\\t\\ts = q = 0\\n\\t\\t\\tfor x in c:\\n\\t\\t\\t\\tif len(x) <= o:\\n\\t\\t\\t\\t\\tq = max(q, x[i])\\n\\t\\t\\t\\telse:\\n\\t\\t\\t\\t\\ts += x[o]\\n\\t\\t\\t\\t\\tq = max(q, x[i] - x[o])\\n\\t\\t\\tr.append(q + s)\\n\\t\\t\\ti += 1\\n\\tr.append(0)\\n\\tfor i in range(len(r) - 1, 0, -1):\\n\\t\\tr[i - 1] = max(r[i - 1], r[i])\\n\\twhile len(r) > 1 and r[-2] == 0:\\n\\t\\tr.pop()\\n\\to = (r[k] if k < len(r) else 0) + a[v]\\n\\tr.insert(0, max(o, r[0]))\\n\\treturn r\\n\\n    \\nfor _ in range(1, n):\\n\\tu, v = map(lambda x: int(x) - 1, input().split())\\n\\tg.setdefault(u, set()).add(v)\\n\\tg.setdefault(v, set()).add(u)\\nprint(dfs(0)[0])\"]",
        "difficulty": "introductory",
        "input": "14 2\n1 2 2 1 1 2 2 1 1 1 2 2 1 2\n2 9\n7 1\n4 1\n11 4\n1 10\n2 6\n4 12\n6 14\n3 1\n1 2\n1 13\n8 1\n5 3\n",
        "output": "8\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1249/F"
    },
    {
        "id": 1237,
        "task_id": 4522,
        "test_case_id": 1,
        "question": "You are given a weighted tree consisting of $n$ vertices. Recall that a tree is a connected graph without cycles. Vertices $u_i$ and $v_i$ are connected by an edge with weight $w_i$.\n\nYou are given $m$ queries. The $i$-th query is given as an integer $q_i$. In this query you need to calculate the number of pairs of vertices $(u, v)$ ($u < v$) such that the maximum weight of an edge on a simple path between $u$ and $v$ doesn't exceed $q_i$.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^5$) — the number of vertices in the tree and the number of queries.\n\nEach of the next $n - 1$ lines describes an edge of the tree. Edge $i$ is denoted by three integers $u_i$, $v_i$ and $w_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$) and the weight of the edge ($1 \\le w_i \\le 2 \\cdot 10^5$). It is guaranteed that the given edges form a tree.\n\nThe last line of the input contains $m$ integers $q_1, q_2, \\dots, q_m$ ($1 \\le q_i \\le 2 \\cdot 10^5$), where $q_i$ is the maximum weight of an edge in the $i$-th query.\n\n\n-----Output-----\n\nPrint $m$ integers — the answers to the queries. The $i$-th value should be equal to the number of pairs of vertices $(u, v)$ ($u < v$) such that the maximum weight of an edge on a simple path between $u$ and $v$ doesn't exceed $q_i$.\n\nQueries are numbered from $1$ to $m$ in the order of the input.\n\n\n-----Examples-----\nInput\n7 5\n1 2 1\n3 2 3\n2 4 1\n4 5 2\n5 7 4\n3 6 2\n5 2 3 4 1\n\nOutput\n21 7 15 21 3 \n\nInput\n1 2\n1 2\n\nOutput\n0 0 \n\nInput\n3 3\n1 2 1\n2 3 2\n1 3 2\n\nOutput\n1 3 3 \n\n\n\n-----Note-----\n\nThe picture shows the tree from the first example: [Image]",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmaxN = 2 * (10 ** 5) + 10\\nedges = [[] for i in range(0, maxN)]\\nque = [[] for _ in range(0, maxN)]\\nans = [0] * m\\nsz = [1 for _ in range(0, n)]\\np = [i for i in range(0, n)]\\ntotal_sum = 0\\n\\ndef get(u):\\n    if p[u] == u:\\n        return u\\n    p[u] = get(p[u])\\n    return p[u]\\n\\n\\ndef unite(u, v):\\n    u = get(u)\\n    v = get(v)\\n    if u == v:\\n        return\\n    nonlocal total_sum\\n    total_sum -= (sz[u] * (sz[u] - 1)) // 2\\n    total_sum -= (sz[v] * (sz[v] - 1)) // 2\\n    total_sum += ((sz[u] + sz[v]) * (sz[u] + sz[v] - 1)) // 2\\n    if sz[u] < sz[v]:\\n        p[u] = v\\n        sz[v] += sz[u]\\n    else:\\n        p[v] = u\\n        sz[u] += sz[v]\\n\\n\\nfor i in range(1, n):\\n    u, v, w = list(map(int, input().split()))\\n    u -= 1\\n    v -= 1\\n    edges[w].append((u, v))\\nques = list(map(int, input().split()))\\n\\nfor i in range(0, m):\\n    que[ques[i]].append(i)\\nfor i in range(0, maxN):\\n    for u, v in edges[i]:\\n        unite(u, v)\\n    for id in que[i]:\\n        ans[id] = total_sum\\nprint(\\\" \\\".join(str(x) for x in ans))\\n\\n\\n\\n\\n\\n\", \"from bisect import bisect\\nfrom math import inf\\n\\n\\ndef union_init(s):\\n    d = [i for i in range(s)]\\n    size = [1 for i in range(s)]\\n    return (d, size)\\n\\n\\ndef union_query(d, size, n):\\n    if d[n] != n:\\n        d[n] = union_query(d, size, d[n])\\n    return d[n]\\n\\n\\ndef union_merge(d, size, x, y):\\n    xRoot = union_query(d, size, x)\\n    yRoot = union_query(d, size, y)\\n\\n    if xRoot == yRoot:\\n        return\\n    if size[xRoot] < size[yRoot]:\\n        xRoot, yRoot = yRoot, xRoot\\n    d[yRoot] = xRoot\\n    size[xRoot] = size[xRoot] + size[yRoot]\\n\\n\\ndef sizeComponent(d, size, x):\\n    root = union_query(d, size, x)\\n    return size[root]\\n\\n\\nnm = input().split()\\nn = int(nm[0])\\nm = int(nm[1])\\n\\nd, size = union_init(n)\\npairs = [(0, 0)]\\nedges = []\\nfor _ in range(n - 1):\\n    edge = input().split()\\n    u = int(edge[0]) - 1\\n    v = int(edge[1]) - 1\\n    w = int(edge[2])\\n    edges.append((w, u, v))\\n\\nedges.sort()\\ntotalP = 0\\nfor e in edges:\\n    u = e[1]\\n    v = e[2]\\n    uSize = sizeComponent(d, size, u)\\n    vSize = sizeComponent(d, size, v)\\n    totalP += uSize * vSize\\n    union_merge(d, size, u, v)\\n    pairs.append((e[0], totalP))\\n\\nms = [int(mi) for mi in input().split()]\\nanswer = [0] * m\\nfor i, mi in enumerate(ms):\\n    start = bisect(pairs, (mi, inf)) - 1\\n    answer[i] = pairs[start][1]\\nprint(*answer)\\n\", \"'''input\\n3 3\\n1 2 1\\n2 3 2\\n1 3 2\\n'''\\nfrom sys import stdin\\nfrom copy import deepcopy\\nfrom collections import deque, defaultdict\\n\\n\\ndef find_parent(n):\\n\\tnonlocal parent\\n\\tnode = n\\n\\twhile parent[node] != node:\\n\\t\\tnode = parent[node]\\n\\tparent[n] = node\\n\\treturn parent[n]\\n\\n\\ndef combination(num):\\n\\treturn (num * (num - 1)) // 2\\n\\n# main starts\\nn, m = list(map(int, stdin.readline().split()))\\ngraph = defaultdict(list)\\nedges = []\\nfor _ in range(n - 1):\\n\\tedges.append(list(map(int, stdin.readline().split())))\\n\\nqueries = list(map(int, stdin.readline().split()))\\n\\nedges.sort(key = lambda x:x[2], reverse = True)\\nparent = dict()\\ncount = dict()\\nfor i in range(1, n + 1):\\n\\tparent[i] = i\\n\\tcount[i] = 1\\n\\nans = [0] * (200001)\\nwhile len(edges) > 0:\\n\\tu, v, w = edges.pop()\\n\\n\\tif u > v:\\n\\t\\tu, v = v, u\\n\\n\\tpv = find_parent(v)\\n\\tpu = find_parent(u)\\n\\tans[w] -= (combination(count[pu]) + combination(count[pv]))\\n\\tans[w] += combination(count[pu] + count[pv])\\n\\tcount[pu] += count[pv]\\n\\tcount[pv] = 0\\n\\tparent[pv] = parent[pu]\\n\\t\\t\\n\\t# print(u, v, w)\\n\\t# print('parent', parent)\\n\\t# print('count', count)\\n\\t# print(ans)\\n\\n\\nfor i in range(1, len(ans)):\\n\\tans[i] = ans[i - 1] + ans[i]\\n# print(ans[:10])\\n\\nfor i in queries:\\n\\tprint(ans[i], end = ' ')\\n\\n\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn,m=list(map(int,input().split()))\\npar=[-1]*n\\ndef c(x):\\n  return x*(x-1)//2\\n\\ndef find(x):\\n  if par[x]<0:\\n    return x\\n  else:\\n    par[x]=find(par[x])\\n    return par[x]\\n\\ndef unite(x,y):\\n  x=find(x)\\n  y=find(y)\\n  if x==y:\\n    return False\\n  else:\\n    if par[x]>par[y]:\\n      x,y=y,x\\n    par[x]+=par[y]\\n    par[y]=x\\n    return True\\n\\ndef same(x,y):\\n  return find(x)==find(y)\\n\\ndef size(x):\\n  return -par[find(x)]\\nimport collections\\nEdge=collections.defaultdict(list)\\nfor _ in range(n-1):\\n  u,v,w=list(map(int,input().split()))\\n  Edge[w].append((u,v))\\nQ=[int(i) for i in input().split()]\\nq=max(Q)\\nAns=[0]*(q+1)\\nans=0\\nfor i in range(1,q+1):\\n  for u,v in Edge[i]:\\n    if same(u-1,v-1):\\n      continue\\n    else:\\n      uu,vv=size(u-1),size(v-1)\\n      unite(u-1,v-1)\\n      uv=size(u-1)\\n      ans+=c(uv)-c(uu)-c(vv)\\n  Ans[i]=ans\\nAns2=[]\\nfor q in Q:\\n  Ans2.append(Ans[q])\\nprint(*Ans2)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn,m=list(map(int,input().split()))\\npar=[-1]*n\\ndef c(x):\\n  return x*(x-1)//2\\n\\ndef find(x):\\n  if par[x]<0:\\n    return x\\n  else:\\n    par[x]=find(par[x])\\n    return par[x]\\n\\ndef unite(x,y):\\n  x=find(x)\\n  y=find(y)\\n  if x==y:\\n    return False\\n  else:\\n    if par[x]>par[y]:\\n      x,y=y,x\\n    par[x]+=par[y]\\n    par[y]=x\\n    return True\\n\\ndef same(x,y):\\n  return find(x)==find(y)\\n\\ndef size(x):\\n  return -par[find(x)]\\nimport collections\\nEdge=collections.defaultdict(list)\\nfor _ in range(n-1):\\n  u,v,w=list(map(int,input().split()))\\n  Edge[w].append((u,v))\\nQ=[int(i) for i in input().split()]\\nq=max(Q)\\nAns=[0]*(q+1)\\nans=0\\nfor i in range(1,q+1):\\n  for u,v in Edge[i]:\\n    if same(u-1,v-1):\\n      continue\\n    else:\\n      uu,vv=size(u-1),size(v-1)\\n      unite(u-1,v-1)\\n      uv=size(u-1)\\n      ans+=c(uv)-c(uu)-c(vv)\\n  Ans[i]=ans\\nAns2=[]\\nfor q in Q:\\n  Ans2.append(Ans[q])\\nprint(*Ans2)\\n\\n\", \"# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\n# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\n# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\nimport sys\\nfrom sys import stdin, stdout\\nfrom collections import defaultdict\\nfrom collections import deque\\nimport math\\nimport copy\\n\\n#T = int(input())\\n#N = int(input())\\n#s1 = input()\\n#s2 = input()\\nN,E = [int(x) for x in stdin.readline().split()]\\n#arr = [int(x) for x in stdin.readline().split()]\\n\\ns = 0\\nsize = [1]*N\\n\\nf = [x for x in range(N)]\\n\\ndef getf(v):\\n    stack = []\\n    while f[v] != v:\\n       stack.append(v)\\n       v = f[v]\\n    for idx in stack:\\n        f[idx] = v\\n    return v\\n\\ndef merge(f, v, u):\\n    nonlocal s\\n    t1 = getf(v)\\n    t2 = getf(u)\\n    if t1 != t2:\\n        A = size[t1]\\n        B = size[t2]\\n        s -= A*(A-1)//2\\n        s -= B*(B-1)//2\\n        size[t1] = size[t1] + size[t2]\\n        C = size[t1]\\n        s += C*(C-1)//2\\n        f[t2] = t1\\n\\n\\nw_edge = {}\\nfor i in range(N-1):\\n    u,v,w = [int(x) for x in stdin.readline().split()]\\n    if u>v:\\n        u,v = v,u\\n\\n    if w not in w_edge:\\n        w_edge[w] = []\\n\\n    w_edge[w].append((u,v))\\n\\nq = [0]*200000\\nfor i in range(200000):\\n    if i+1 in w_edge:\\n        for e in w_edge[i+1]:\\n            u,v = e\\n            merge(f,u-1,v-1)\\n        q[i] = s\\n    else:\\n        if i!=0:\\n            q[i] = q[i-1]\\n        else:\\n            q[i] = 0\\n\\narr = [int(x) for x in stdin.readline().split()]\\nans = [0]*E\\nfor i in range(E):\\n    ans[i] = q[arr[i]-1]\\n\\nprint(*ans)\\n\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2000)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2100)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2500)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2800)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2900)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = list(map(int, readline().split()))\\nE = []\\nfor i in range(N-1):\\n    u, v, w = list(map(int, readline().split()))\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(3000)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = list(range(N))\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\\n\", \"n,m=[int(x) for x in input().split()]\\nlink=[0]*(n+1)\\nsize=[0]*(n+1)\\narr=[0]*m\\nres=0\\nfor i in range(1,n+1):\\n    link[i]=i\\n    size[i]=1\\ndef find(x):\\n    while x!=link[x]:\\n        x=link[x]\\n    return x\\ndef comb(n):\\n    return n*(n-1)//2\\ndef unite(a,b,res):\\n    a=find(a)\\n    b=find(b)\\n    if size[a]<size[b]:\\n        a,b=b,a\\n    res=res-comb(size[a])-comb(size[b])+comb(size[a]+size[b])\\n    size[a]+=size[b]\\n    link[b]=a\\n    return res\\nedges=[]\\nask=[]\\nfor i in range(n-1):\\n    x,y,z=[int(x) for x in input().split()]\\n    edges.append((z,x,y))\\nedges.sort()\\nx=0\\nedges.append((10**100,1,1))\\nask=sorted(zip([int(x) for x in input().split()],list(range(m))))\\nfor i in range(m):\\n    while edges[x][0]<=ask[i][0]:\\n        res=unite(edges[x][1],edges[x][2],res)\\n        x+=1\\n    arr[ask[i][1]]=res\\nprint(*arr)\\n    \\n\", \"from sys import setrecursionlimit as SRL, stdin\\n\\nSRL(10 ** 7)\\nrd = stdin.readline\\nrrd = lambda: map(int, rd().strip().split())\\n\\nfa = [i for i in range(200005)]\\ns = [1] * 200005\\n\\n\\ndef find(x):\\n    t = []\\n    while fa[x] != x:\\n        t.append(x)\\n        x = fa[x]\\n    for i in t:\\n        fa[i] = x\\n    return fa[x]\\n\\n\\nans = [0] * 200005\\nn, q = rrd()\\n\\nw = []\\nfor i in range(n - 1):\\n    x, y, z = rrd()\\n    w.append([z, x, y])\\n\\nw.sort(key=lambda x: x[0])\\nfor x in w:\\n    u = find(x[1])\\n    v = find(x[2])\\n\\n    ans[x[0]] += s[u] * s[v]\\n    fa[u] = v\\n    s[v] += s[u]\\n\\nfor i in range(1, 200001):\\n    ans[i] += ans[i - 1]\\n\\nq = list(rrd())\\n\\nfor x in q:\\n    print(ans[x], end=' ')\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\ndef find(a):\\n    if par[a]==a:\\n        return a\\n    par[a]=find(par[a])\\n    return par[a]\\nn,m=list(map(int,input().split()))\\ned=[]\\npar=[i for i in range(n)]\\nsize=[1 for i in range(n)]\\nfor _ in range(n-1):\\n    a,b,c=list(map(int,input().split()))\\n    ed.append([a-1,b-1,c])\\ned.sort(key=lambda x:x[2])\\nit=list(map(int,input().split()))\\nit=[[i,j,0] for j,i in enumerate(it)]\\nit.sort()\\nind=0\\ntot=0\\nj=0\\n#print(it)\\nss={}\\nfor i in it[:]:\\n    while ind<n-1:\\n        if ed[ind][2]<=i[0]:\\n            a=find(ed[ind][0])\\n            b=find(ed[ind][1])\\n            if a!=b:\\n                tot+=size[a]*size[b]\\n              #  print(a,b,j,tot)\\n                if size[a]>=size[b]:\\n\\n                    par[b]=a\\n                    size[a]+=size[b]\\n                    size[b]=0\\n                else:\\n                    par[a]=b\\n                    size[b]+=size[a]\\n                    size[a]=0\\n            ind+=1\\n            \\n        else:\\n            break\\n    it[j][2]=tot\\n    #ss[it[j][1]]=tot\\n    j+=1\\n\\nit.sort(key=lambda x:x[1])\\naa=[i[2] for i in it]\\n\\n#for i in range(len(it)):\\n #   print(ss[i],end=\\\" \\\")\\nprint(*aa)\\n        \\n    \\n    \\n        \\n    \\n\", \"from operator import itemgetter\\nimport sys\\ninput = sys.stdin.readline\\n\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.parent = [-1] * n\\n        self.cnt = n\\n\\n    def root(self, x):\\n        if self.parent[x] < 0:\\n            return x\\n        else:\\n            self.parent[x] = self.root(self.parent[x])\\n            return self.parent[x]\\n\\n    def merge(self, x, y):\\n        x = self.root(x)\\n        y = self.root(y)\\n        if x != y:\\n            if self.parent[x] > self.parent[y]:\\n                x, y = y, x\\n            self.parent[x] += self.parent[y]\\n            self.parent[y] = x\\n            self.cnt -= 1\\n\\n    def is_same(self, x, y):\\n        return self.root(x) == self.root(y)\\n\\n    def get_size(self, x):\\n        return -self.parent[self.root(x)]\\n\\n    def get_cnt(self):\\n        return self.cnt\\n\\n\\nn, m = map(int, input().split())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\nq = list(map(int, input().split()))\\ninfo = sorted(info, key = itemgetter(2))\\nans = [0] * (2*10**5)\\nuf = UnionFind(2*10**5)\\ninfo_i = 0\\nfor i in range(2*10**5):\\n    if i-1 >= 0:\\n        ans[i] = ans[i-1]\\n    while True:\\n        if info_i >= n - 1:\\n           break\\n        if info[info_i][2] == i + 1:\\n            a, b, _ = info[info_i]\\n            a -= 1\\n            b -= 1\\n            num_a = uf.get_size(a)\\n            num_b = uf.get_size(b)\\n            num_ab = num_a + num_b\\n            comb_num_a = (num_a*(num_a-1)) // 2\\n            comb_num_b = (num_b*(num_b-1)) // 2\\n            comb_num_ab = (num_ab*(num_ab-1)) // 2\\n            ans[i] += comb_num_ab - (comb_num_a + comb_num_b)\\n            uf.merge(a, b)\\n            info_i += 1\\n        else:\\n            break\\n\\nres = [0] * len(q)\\nfor i, j in enumerate(q):\\n    res[i] = ans[j - 1]\\nprint(*res)\", \"def find_ancestor(i, father):\\n    if father[i] == i:\\n        return i\\n    father[i] = find_ancestor(father[i], father)\\n    return father[i]\\n\\ndef connect(i, j, father, n_child):\\n    i_anc = find_ancestor(i, father)\\n    j_anc = find_ancestor(j, father)\\n    if n_child[i_anc] > n_child[j_anc]:\\n        n_child[i_anc] += n_child[j_anc]\\n        father[j_anc] = i_anc\\n    else:\\n        n_child[j_anc] += n_child[i_anc]\\n        father[i_anc] = j_anc\\n\\nn, m = list(map(int, input().split()))\\nedges = []\\nfather = [i for i in range(n)]\\nn_child = [1]*n\\n\\nfor i in range(n-1):\\n    i, j, w = list(map(int, input().split()))\\n    edges.append((i-1, j-1, w))\\n\\nedges.sort(key=lambda x: -x[2])\\nqueries = list(map(int, input().split()))\\n\\ns_queries = sorted(queries)\\n\\n# final map the index to the query\\nans = {}\\n\\nw_limit = []\\nans_cum = 0\\nfor query in s_queries:\\n    while len(edges) and edges[-1][2] <= query:\\n        i, j, w = edges[-1]\\n        edges.pop()\\n        i_anc = find_ancestor(i, father)\\n        j_anc = find_ancestor(j, father)\\n        # it's tree father may not be same\\n        ans_cum += n_child[i_anc] * n_child[j_anc]\\n        connect(i, j, father, n_child)\\n    ans[query] = ans_cum\\n\\nprint(\\\" \\\".join(list(map(str, [ans[query] for query in queries]))))\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        if self.par[x] == x:\\n            return x\\n        else:\\n            self.par[x] = self.find(self.par[x])\\n            return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nedges.sort()\\nA = [0] * (2*10**5+7)\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        A[c] = A[prevc]\\n    sz1 = uf.get_size(a)\\n    A[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    A[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    A[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nA = list(accumulate(A, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = A[q]\\nprint(*ans)\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\nfrom operator import itemgetter\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        if self.par[x] == x:\\n            return x\\n        else:\\n            self.par[x] = self.find(self.par[x])\\n            return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nMAX = 2 * 10**5 + 7\\nedges.sort(key=itemgetter(0))\\nC = [0] * MAX\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        C[c] = C[prevc]\\n    sz1 = uf.get_size(a)\\n    C[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    C[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    C[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nC = list(accumulate(C, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = C[q]\\nprint(*ans)\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\nfrom operator import itemgetter\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        t = []\\n        while self.par[x] != x:\\n            t.append(x)\\n            x = self.par[x]\\n        for i in t:\\n            self.par[i] = x\\n        return self.par[x]\\n        # if self.par[x] == x:\\n        #     return x\\n        # else:\\n        #     self.par[x] = self.find(self.par[x])\\n        #     return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nMAX = 2 * 10**5 + 7\\nedges.sort(key=itemgetter(0))\\nC = [0] * MAX\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        C[c] = C[prevc]\\n    sz1 = uf.get_size(a)\\n    C[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    C[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    C[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nC = list(accumulate(C, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = C[q]\\nprint(*ans)\\n\", \"class UnionFind():\\n    def __init__(self, n):\\n        self.n = n\\n        self.root = [-1]*(n+1)\\n        self.rnk = [0]*(n+1)\\n\\n    def Find_Root(self, x):\\n        if(self.root[x] < 0):\\n            return x\\n        else:\\n            self.root[x] = self.Find_Root(self.root[x])\\n            return self.root[x]\\n    \\n    def Unite(self, x, y):\\n        x = self.Find_Root(x)\\n        y = self.Find_Root(y)\\n        if(x == y):\\n            return \\n        elif(self.rnk[x] > self.rnk[y]):\\n            self.root[x] += self.root[y]\\n            self.root[y] = x\\n\\n        else:\\n            self.root[y] += self.root[x]\\n            self.root[x] = y\\n            if(self.rnk[x] == self.rnk[y]):\\n                self.rnk[y] += 1\\n    \\n    def isSameGroup(self, x, y):\\n        return self.Find_Root(x) == self.Find_Root(y)\\n\\n    def Count(self, x):\\n        return -self.root[self.Find_Root(x)]\\n\\n\\nimport sys\\ninput = sys.stdin.readline\\nfrom bisect import bisect_right\\n\\nN, M = map(int, input().split())\\nuni = UnionFind(N+1)\\nEdges = {}\\nfor _ in range(N-1):\\n    a, b, w = map(int, input().split())\\n    if not w in Edges:\\n        Edges[w] = [(a, b)]\\n    else:\\n        Edges[w].append((a, b))\\n\\nQuery = list(map(int, input().split()))\\n\\nWeights = sorted(list(Edges.keys()))\\n\\nScore = [0]\\nscore = 0\\nfor w in Weights:\\n    for a, b in Edges[w]:\\n        c1 = uni.Count(a)\\n        c2 = uni.Count(b)\\n        c = c1 + c2\\n        score += c*(c-1)//2 - c1*(c1-1)//2 - c2*(c2-1)//2\\n        uni.Unite(a, b)\\n    Score.append(score)\\n\\nans = []\\nfor q in Query:\\n    ind = bisect_right(Weights, q)\\n    ans.append(Score[ind])\\n\\nprint(*ans)\"]",
        "difficulty": "introductory",
        "input": "7 5\n1 2 1\n3 2 3\n2 4 1\n4 5 2\n5 7 4\n3 6 2\n5 2 3 4 1\n",
        "output": "21 7 15 21 3 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1213/G"
    },
    {
        "id": 1238,
        "task_id": 4522,
        "test_case_id": 2,
        "question": "You are given a weighted tree consisting of $n$ vertices. Recall that a tree is a connected graph without cycles. Vertices $u_i$ and $v_i$ are connected by an edge with weight $w_i$.\n\nYou are given $m$ queries. The $i$-th query is given as an integer $q_i$. In this query you need to calculate the number of pairs of vertices $(u, v)$ ($u < v$) such that the maximum weight of an edge on a simple path between $u$ and $v$ doesn't exceed $q_i$.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^5$) — the number of vertices in the tree and the number of queries.\n\nEach of the next $n - 1$ lines describes an edge of the tree. Edge $i$ is denoted by three integers $u_i$, $v_i$ and $w_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$) and the weight of the edge ($1 \\le w_i \\le 2 \\cdot 10^5$). It is guaranteed that the given edges form a tree.\n\nThe last line of the input contains $m$ integers $q_1, q_2, \\dots, q_m$ ($1 \\le q_i \\le 2 \\cdot 10^5$), where $q_i$ is the maximum weight of an edge in the $i$-th query.\n\n\n-----Output-----\n\nPrint $m$ integers — the answers to the queries. The $i$-th value should be equal to the number of pairs of vertices $(u, v)$ ($u < v$) such that the maximum weight of an edge on a simple path between $u$ and $v$ doesn't exceed $q_i$.\n\nQueries are numbered from $1$ to $m$ in the order of the input.\n\n\n-----Examples-----\nInput\n7 5\n1 2 1\n3 2 3\n2 4 1\n4 5 2\n5 7 4\n3 6 2\n5 2 3 4 1\n\nOutput\n21 7 15 21 3 \n\nInput\n1 2\n1 2\n\nOutput\n0 0 \n\nInput\n3 3\n1 2 1\n2 3 2\n1 3 2\n\nOutput\n1 3 3 \n\n\n\n-----Note-----\n\nThe picture shows the tree from the first example: [Image]",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmaxN = 2 * (10 ** 5) + 10\\nedges = [[] for i in range(0, maxN)]\\nque = [[] for _ in range(0, maxN)]\\nans = [0] * m\\nsz = [1 for _ in range(0, n)]\\np = [i for i in range(0, n)]\\ntotal_sum = 0\\n\\ndef get(u):\\n    if p[u] == u:\\n        return u\\n    p[u] = get(p[u])\\n    return p[u]\\n\\n\\ndef unite(u, v):\\n    u = get(u)\\n    v = get(v)\\n    if u == v:\\n        return\\n    nonlocal total_sum\\n    total_sum -= (sz[u] * (sz[u] - 1)) // 2\\n    total_sum -= (sz[v] * (sz[v] - 1)) // 2\\n    total_sum += ((sz[u] + sz[v]) * (sz[u] + sz[v] - 1)) // 2\\n    if sz[u] < sz[v]:\\n        p[u] = v\\n        sz[v] += sz[u]\\n    else:\\n        p[v] = u\\n        sz[u] += sz[v]\\n\\n\\nfor i in range(1, n):\\n    u, v, w = list(map(int, input().split()))\\n    u -= 1\\n    v -= 1\\n    edges[w].append((u, v))\\nques = list(map(int, input().split()))\\n\\nfor i in range(0, m):\\n    que[ques[i]].append(i)\\nfor i in range(0, maxN):\\n    for u, v in edges[i]:\\n        unite(u, v)\\n    for id in que[i]:\\n        ans[id] = total_sum\\nprint(\\\" \\\".join(str(x) for x in ans))\\n\\n\\n\\n\\n\\n\", \"from bisect import bisect\\nfrom math import inf\\n\\n\\ndef union_init(s):\\n    d = [i for i in range(s)]\\n    size = [1 for i in range(s)]\\n    return (d, size)\\n\\n\\ndef union_query(d, size, n):\\n    if d[n] != n:\\n        d[n] = union_query(d, size, d[n])\\n    return d[n]\\n\\n\\ndef union_merge(d, size, x, y):\\n    xRoot = union_query(d, size, x)\\n    yRoot = union_query(d, size, y)\\n\\n    if xRoot == yRoot:\\n        return\\n    if size[xRoot] < size[yRoot]:\\n        xRoot, yRoot = yRoot, xRoot\\n    d[yRoot] = xRoot\\n    size[xRoot] = size[xRoot] + size[yRoot]\\n\\n\\ndef sizeComponent(d, size, x):\\n    root = union_query(d, size, x)\\n    return size[root]\\n\\n\\nnm = input().split()\\nn = int(nm[0])\\nm = int(nm[1])\\n\\nd, size = union_init(n)\\npairs = [(0, 0)]\\nedges = []\\nfor _ in range(n - 1):\\n    edge = input().split()\\n    u = int(edge[0]) - 1\\n    v = int(edge[1]) - 1\\n    w = int(edge[2])\\n    edges.append((w, u, v))\\n\\nedges.sort()\\ntotalP = 0\\nfor e in edges:\\n    u = e[1]\\n    v = e[2]\\n    uSize = sizeComponent(d, size, u)\\n    vSize = sizeComponent(d, size, v)\\n    totalP += uSize * vSize\\n    union_merge(d, size, u, v)\\n    pairs.append((e[0], totalP))\\n\\nms = [int(mi) for mi in input().split()]\\nanswer = [0] * m\\nfor i, mi in enumerate(ms):\\n    start = bisect(pairs, (mi, inf)) - 1\\n    answer[i] = pairs[start][1]\\nprint(*answer)\\n\", \"'''input\\n3 3\\n1 2 1\\n2 3 2\\n1 3 2\\n'''\\nfrom sys import stdin\\nfrom copy import deepcopy\\nfrom collections import deque, defaultdict\\n\\n\\ndef find_parent(n):\\n\\tnonlocal parent\\n\\tnode = n\\n\\twhile parent[node] != node:\\n\\t\\tnode = parent[node]\\n\\tparent[n] = node\\n\\treturn parent[n]\\n\\n\\ndef combination(num):\\n\\treturn (num * (num - 1)) // 2\\n\\n# main starts\\nn, m = list(map(int, stdin.readline().split()))\\ngraph = defaultdict(list)\\nedges = []\\nfor _ in range(n - 1):\\n\\tedges.append(list(map(int, stdin.readline().split())))\\n\\nqueries = list(map(int, stdin.readline().split()))\\n\\nedges.sort(key = lambda x:x[2], reverse = True)\\nparent = dict()\\ncount = dict()\\nfor i in range(1, n + 1):\\n\\tparent[i] = i\\n\\tcount[i] = 1\\n\\nans = [0] * (200001)\\nwhile len(edges) > 0:\\n\\tu, v, w = edges.pop()\\n\\n\\tif u > v:\\n\\t\\tu, v = v, u\\n\\n\\tpv = find_parent(v)\\n\\tpu = find_parent(u)\\n\\tans[w] -= (combination(count[pu]) + combination(count[pv]))\\n\\tans[w] += combination(count[pu] + count[pv])\\n\\tcount[pu] += count[pv]\\n\\tcount[pv] = 0\\n\\tparent[pv] = parent[pu]\\n\\t\\t\\n\\t# print(u, v, w)\\n\\t# print('parent', parent)\\n\\t# print('count', count)\\n\\t# print(ans)\\n\\n\\nfor i in range(1, len(ans)):\\n\\tans[i] = ans[i - 1] + ans[i]\\n# print(ans[:10])\\n\\nfor i in queries:\\n\\tprint(ans[i], end = ' ')\\n\\n\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn,m=list(map(int,input().split()))\\npar=[-1]*n\\ndef c(x):\\n  return x*(x-1)//2\\n\\ndef find(x):\\n  if par[x]<0:\\n    return x\\n  else:\\n    par[x]=find(par[x])\\n    return par[x]\\n\\ndef unite(x,y):\\n  x=find(x)\\n  y=find(y)\\n  if x==y:\\n    return False\\n  else:\\n    if par[x]>par[y]:\\n      x,y=y,x\\n    par[x]+=par[y]\\n    par[y]=x\\n    return True\\n\\ndef same(x,y):\\n  return find(x)==find(y)\\n\\ndef size(x):\\n  return -par[find(x)]\\nimport collections\\nEdge=collections.defaultdict(list)\\nfor _ in range(n-1):\\n  u,v,w=list(map(int,input().split()))\\n  Edge[w].append((u,v))\\nQ=[int(i) for i in input().split()]\\nq=max(Q)\\nAns=[0]*(q+1)\\nans=0\\nfor i in range(1,q+1):\\n  for u,v in Edge[i]:\\n    if same(u-1,v-1):\\n      continue\\n    else:\\n      uu,vv=size(u-1),size(v-1)\\n      unite(u-1,v-1)\\n      uv=size(u-1)\\n      ans+=c(uv)-c(uu)-c(vv)\\n  Ans[i]=ans\\nAns2=[]\\nfor q in Q:\\n  Ans2.append(Ans[q])\\nprint(*Ans2)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn,m=list(map(int,input().split()))\\npar=[-1]*n\\ndef c(x):\\n  return x*(x-1)//2\\n\\ndef find(x):\\n  if par[x]<0:\\n    return x\\n  else:\\n    par[x]=find(par[x])\\n    return par[x]\\n\\ndef unite(x,y):\\n  x=find(x)\\n  y=find(y)\\n  if x==y:\\n    return False\\n  else:\\n    if par[x]>par[y]:\\n      x,y=y,x\\n    par[x]+=par[y]\\n    par[y]=x\\n    return True\\n\\ndef same(x,y):\\n  return find(x)==find(y)\\n\\ndef size(x):\\n  return -par[find(x)]\\nimport collections\\nEdge=collections.defaultdict(list)\\nfor _ in range(n-1):\\n  u,v,w=list(map(int,input().split()))\\n  Edge[w].append((u,v))\\nQ=[int(i) for i in input().split()]\\nq=max(Q)\\nAns=[0]*(q+1)\\nans=0\\nfor i in range(1,q+1):\\n  for u,v in Edge[i]:\\n    if same(u-1,v-1):\\n      continue\\n    else:\\n      uu,vv=size(u-1),size(v-1)\\n      unite(u-1,v-1)\\n      uv=size(u-1)\\n      ans+=c(uv)-c(uu)-c(vv)\\n  Ans[i]=ans\\nAns2=[]\\nfor q in Q:\\n  Ans2.append(Ans[q])\\nprint(*Ans2)\\n\\n\", \"# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\n# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\n# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\nimport sys\\nfrom sys import stdin, stdout\\nfrom collections import defaultdict\\nfrom collections import deque\\nimport math\\nimport copy\\n\\n#T = int(input())\\n#N = int(input())\\n#s1 = input()\\n#s2 = input()\\nN,E = [int(x) for x in stdin.readline().split()]\\n#arr = [int(x) for x in stdin.readline().split()]\\n\\ns = 0\\nsize = [1]*N\\n\\nf = [x for x in range(N)]\\n\\ndef getf(v):\\n    stack = []\\n    while f[v] != v:\\n       stack.append(v)\\n       v = f[v]\\n    for idx in stack:\\n        f[idx] = v\\n    return v\\n\\ndef merge(f, v, u):\\n    nonlocal s\\n    t1 = getf(v)\\n    t2 = getf(u)\\n    if t1 != t2:\\n        A = size[t1]\\n        B = size[t2]\\n        s -= A*(A-1)//2\\n        s -= B*(B-1)//2\\n        size[t1] = size[t1] + size[t2]\\n        C = size[t1]\\n        s += C*(C-1)//2\\n        f[t2] = t1\\n\\n\\nw_edge = {}\\nfor i in range(N-1):\\n    u,v,w = [int(x) for x in stdin.readline().split()]\\n    if u>v:\\n        u,v = v,u\\n\\n    if w not in w_edge:\\n        w_edge[w] = []\\n\\n    w_edge[w].append((u,v))\\n\\nq = [0]*200000\\nfor i in range(200000):\\n    if i+1 in w_edge:\\n        for e in w_edge[i+1]:\\n            u,v = e\\n            merge(f,u-1,v-1)\\n        q[i] = s\\n    else:\\n        if i!=0:\\n            q[i] = q[i-1]\\n        else:\\n            q[i] = 0\\n\\narr = [int(x) for x in stdin.readline().split()]\\nans = [0]*E\\nfor i in range(E):\\n    ans[i] = q[arr[i]-1]\\n\\nprint(*ans)\\n\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2000)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2100)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2500)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2800)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2900)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = list(map(int, readline().split()))\\nE = []\\nfor i in range(N-1):\\n    u, v, w = list(map(int, readline().split()))\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(3000)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = list(range(N))\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\\n\", \"n,m=[int(x) for x in input().split()]\\nlink=[0]*(n+1)\\nsize=[0]*(n+1)\\narr=[0]*m\\nres=0\\nfor i in range(1,n+1):\\n    link[i]=i\\n    size[i]=1\\ndef find(x):\\n    while x!=link[x]:\\n        x=link[x]\\n    return x\\ndef comb(n):\\n    return n*(n-1)//2\\ndef unite(a,b,res):\\n    a=find(a)\\n    b=find(b)\\n    if size[a]<size[b]:\\n        a,b=b,a\\n    res=res-comb(size[a])-comb(size[b])+comb(size[a]+size[b])\\n    size[a]+=size[b]\\n    link[b]=a\\n    return res\\nedges=[]\\nask=[]\\nfor i in range(n-1):\\n    x,y,z=[int(x) for x in input().split()]\\n    edges.append((z,x,y))\\nedges.sort()\\nx=0\\nedges.append((10**100,1,1))\\nask=sorted(zip([int(x) for x in input().split()],list(range(m))))\\nfor i in range(m):\\n    while edges[x][0]<=ask[i][0]:\\n        res=unite(edges[x][1],edges[x][2],res)\\n        x+=1\\n    arr[ask[i][1]]=res\\nprint(*arr)\\n    \\n\", \"from sys import setrecursionlimit as SRL, stdin\\n\\nSRL(10 ** 7)\\nrd = stdin.readline\\nrrd = lambda: map(int, rd().strip().split())\\n\\nfa = [i for i in range(200005)]\\ns = [1] * 200005\\n\\n\\ndef find(x):\\n    t = []\\n    while fa[x] != x:\\n        t.append(x)\\n        x = fa[x]\\n    for i in t:\\n        fa[i] = x\\n    return fa[x]\\n\\n\\nans = [0] * 200005\\nn, q = rrd()\\n\\nw = []\\nfor i in range(n - 1):\\n    x, y, z = rrd()\\n    w.append([z, x, y])\\n\\nw.sort(key=lambda x: x[0])\\nfor x in w:\\n    u = find(x[1])\\n    v = find(x[2])\\n\\n    ans[x[0]] += s[u] * s[v]\\n    fa[u] = v\\n    s[v] += s[u]\\n\\nfor i in range(1, 200001):\\n    ans[i] += ans[i - 1]\\n\\nq = list(rrd())\\n\\nfor x in q:\\n    print(ans[x], end=' ')\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\ndef find(a):\\n    if par[a]==a:\\n        return a\\n    par[a]=find(par[a])\\n    return par[a]\\nn,m=list(map(int,input().split()))\\ned=[]\\npar=[i for i in range(n)]\\nsize=[1 for i in range(n)]\\nfor _ in range(n-1):\\n    a,b,c=list(map(int,input().split()))\\n    ed.append([a-1,b-1,c])\\ned.sort(key=lambda x:x[2])\\nit=list(map(int,input().split()))\\nit=[[i,j,0] for j,i in enumerate(it)]\\nit.sort()\\nind=0\\ntot=0\\nj=0\\n#print(it)\\nss={}\\nfor i in it[:]:\\n    while ind<n-1:\\n        if ed[ind][2]<=i[0]:\\n            a=find(ed[ind][0])\\n            b=find(ed[ind][1])\\n            if a!=b:\\n                tot+=size[a]*size[b]\\n              #  print(a,b,j,tot)\\n                if size[a]>=size[b]:\\n\\n                    par[b]=a\\n                    size[a]+=size[b]\\n                    size[b]=0\\n                else:\\n                    par[a]=b\\n                    size[b]+=size[a]\\n                    size[a]=0\\n            ind+=1\\n            \\n        else:\\n            break\\n    it[j][2]=tot\\n    #ss[it[j][1]]=tot\\n    j+=1\\n\\nit.sort(key=lambda x:x[1])\\naa=[i[2] for i in it]\\n\\n#for i in range(len(it)):\\n #   print(ss[i],end=\\\" \\\")\\nprint(*aa)\\n        \\n    \\n    \\n        \\n    \\n\", \"from operator import itemgetter\\nimport sys\\ninput = sys.stdin.readline\\n\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.parent = [-1] * n\\n        self.cnt = n\\n\\n    def root(self, x):\\n        if self.parent[x] < 0:\\n            return x\\n        else:\\n            self.parent[x] = self.root(self.parent[x])\\n            return self.parent[x]\\n\\n    def merge(self, x, y):\\n        x = self.root(x)\\n        y = self.root(y)\\n        if x != y:\\n            if self.parent[x] > self.parent[y]:\\n                x, y = y, x\\n            self.parent[x] += self.parent[y]\\n            self.parent[y] = x\\n            self.cnt -= 1\\n\\n    def is_same(self, x, y):\\n        return self.root(x) == self.root(y)\\n\\n    def get_size(self, x):\\n        return -self.parent[self.root(x)]\\n\\n    def get_cnt(self):\\n        return self.cnt\\n\\n\\nn, m = map(int, input().split())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\nq = list(map(int, input().split()))\\ninfo = sorted(info, key = itemgetter(2))\\nans = [0] * (2*10**5)\\nuf = UnionFind(2*10**5)\\ninfo_i = 0\\nfor i in range(2*10**5):\\n    if i-1 >= 0:\\n        ans[i] = ans[i-1]\\n    while True:\\n        if info_i >= n - 1:\\n           break\\n        if info[info_i][2] == i + 1:\\n            a, b, _ = info[info_i]\\n            a -= 1\\n            b -= 1\\n            num_a = uf.get_size(a)\\n            num_b = uf.get_size(b)\\n            num_ab = num_a + num_b\\n            comb_num_a = (num_a*(num_a-1)) // 2\\n            comb_num_b = (num_b*(num_b-1)) // 2\\n            comb_num_ab = (num_ab*(num_ab-1)) // 2\\n            ans[i] += comb_num_ab - (comb_num_a + comb_num_b)\\n            uf.merge(a, b)\\n            info_i += 1\\n        else:\\n            break\\n\\nres = [0] * len(q)\\nfor i, j in enumerate(q):\\n    res[i] = ans[j - 1]\\nprint(*res)\", \"def find_ancestor(i, father):\\n    if father[i] == i:\\n        return i\\n    father[i] = find_ancestor(father[i], father)\\n    return father[i]\\n\\ndef connect(i, j, father, n_child):\\n    i_anc = find_ancestor(i, father)\\n    j_anc = find_ancestor(j, father)\\n    if n_child[i_anc] > n_child[j_anc]:\\n        n_child[i_anc] += n_child[j_anc]\\n        father[j_anc] = i_anc\\n    else:\\n        n_child[j_anc] += n_child[i_anc]\\n        father[i_anc] = j_anc\\n\\nn, m = list(map(int, input().split()))\\nedges = []\\nfather = [i for i in range(n)]\\nn_child = [1]*n\\n\\nfor i in range(n-1):\\n    i, j, w = list(map(int, input().split()))\\n    edges.append((i-1, j-1, w))\\n\\nedges.sort(key=lambda x: -x[2])\\nqueries = list(map(int, input().split()))\\n\\ns_queries = sorted(queries)\\n\\n# final map the index to the query\\nans = {}\\n\\nw_limit = []\\nans_cum = 0\\nfor query in s_queries:\\n    while len(edges) and edges[-1][2] <= query:\\n        i, j, w = edges[-1]\\n        edges.pop()\\n        i_anc = find_ancestor(i, father)\\n        j_anc = find_ancestor(j, father)\\n        # it's tree father may not be same\\n        ans_cum += n_child[i_anc] * n_child[j_anc]\\n        connect(i, j, father, n_child)\\n    ans[query] = ans_cum\\n\\nprint(\\\" \\\".join(list(map(str, [ans[query] for query in queries]))))\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        if self.par[x] == x:\\n            return x\\n        else:\\n            self.par[x] = self.find(self.par[x])\\n            return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nedges.sort()\\nA = [0] * (2*10**5+7)\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        A[c] = A[prevc]\\n    sz1 = uf.get_size(a)\\n    A[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    A[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    A[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nA = list(accumulate(A, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = A[q]\\nprint(*ans)\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\nfrom operator import itemgetter\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        if self.par[x] == x:\\n            return x\\n        else:\\n            self.par[x] = self.find(self.par[x])\\n            return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nMAX = 2 * 10**5 + 7\\nedges.sort(key=itemgetter(0))\\nC = [0] * MAX\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        C[c] = C[prevc]\\n    sz1 = uf.get_size(a)\\n    C[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    C[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    C[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nC = list(accumulate(C, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = C[q]\\nprint(*ans)\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\nfrom operator import itemgetter\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        t = []\\n        while self.par[x] != x:\\n            t.append(x)\\n            x = self.par[x]\\n        for i in t:\\n            self.par[i] = x\\n        return self.par[x]\\n        # if self.par[x] == x:\\n        #     return x\\n        # else:\\n        #     self.par[x] = self.find(self.par[x])\\n        #     return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nMAX = 2 * 10**5 + 7\\nedges.sort(key=itemgetter(0))\\nC = [0] * MAX\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        C[c] = C[prevc]\\n    sz1 = uf.get_size(a)\\n    C[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    C[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    C[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nC = list(accumulate(C, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = C[q]\\nprint(*ans)\\n\", \"class UnionFind():\\n    def __init__(self, n):\\n        self.n = n\\n        self.root = [-1]*(n+1)\\n        self.rnk = [0]*(n+1)\\n\\n    def Find_Root(self, x):\\n        if(self.root[x] < 0):\\n            return x\\n        else:\\n            self.root[x] = self.Find_Root(self.root[x])\\n            return self.root[x]\\n    \\n    def Unite(self, x, y):\\n        x = self.Find_Root(x)\\n        y = self.Find_Root(y)\\n        if(x == y):\\n            return \\n        elif(self.rnk[x] > self.rnk[y]):\\n            self.root[x] += self.root[y]\\n            self.root[y] = x\\n\\n        else:\\n            self.root[y] += self.root[x]\\n            self.root[x] = y\\n            if(self.rnk[x] == self.rnk[y]):\\n                self.rnk[y] += 1\\n    \\n    def isSameGroup(self, x, y):\\n        return self.Find_Root(x) == self.Find_Root(y)\\n\\n    def Count(self, x):\\n        return -self.root[self.Find_Root(x)]\\n\\n\\nimport sys\\ninput = sys.stdin.readline\\nfrom bisect import bisect_right\\n\\nN, M = map(int, input().split())\\nuni = UnionFind(N+1)\\nEdges = {}\\nfor _ in range(N-1):\\n    a, b, w = map(int, input().split())\\n    if not w in Edges:\\n        Edges[w] = [(a, b)]\\n    else:\\n        Edges[w].append((a, b))\\n\\nQuery = list(map(int, input().split()))\\n\\nWeights = sorted(list(Edges.keys()))\\n\\nScore = [0]\\nscore = 0\\nfor w in Weights:\\n    for a, b in Edges[w]:\\n        c1 = uni.Count(a)\\n        c2 = uni.Count(b)\\n        c = c1 + c2\\n        score += c*(c-1)//2 - c1*(c1-1)//2 - c2*(c2-1)//2\\n        uni.Unite(a, b)\\n    Score.append(score)\\n\\nans = []\\nfor q in Query:\\n    ind = bisect_right(Weights, q)\\n    ans.append(Score[ind])\\n\\nprint(*ans)\"]",
        "difficulty": "introductory",
        "input": "1 2\n1 2\n",
        "output": "0 0 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1213/G"
    },
    {
        "id": 1239,
        "task_id": 4522,
        "test_case_id": 3,
        "question": "You are given a weighted tree consisting of $n$ vertices. Recall that a tree is a connected graph without cycles. Vertices $u_i$ and $v_i$ are connected by an edge with weight $w_i$.\n\nYou are given $m$ queries. The $i$-th query is given as an integer $q_i$. In this query you need to calculate the number of pairs of vertices $(u, v)$ ($u < v$) such that the maximum weight of an edge on a simple path between $u$ and $v$ doesn't exceed $q_i$.\n\n\n-----Input-----\n\nThe first line of the input contains two integers $n$ and $m$ ($1 \\le n, m \\le 2 \\cdot 10^5$) — the number of vertices in the tree and the number of queries.\n\nEach of the next $n - 1$ lines describes an edge of the tree. Edge $i$ is denoted by three integers $u_i$, $v_i$ and $w_i$ — the labels of vertices it connects ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$) and the weight of the edge ($1 \\le w_i \\le 2 \\cdot 10^5$). It is guaranteed that the given edges form a tree.\n\nThe last line of the input contains $m$ integers $q_1, q_2, \\dots, q_m$ ($1 \\le q_i \\le 2 \\cdot 10^5$), where $q_i$ is the maximum weight of an edge in the $i$-th query.\n\n\n-----Output-----\n\nPrint $m$ integers — the answers to the queries. The $i$-th value should be equal to the number of pairs of vertices $(u, v)$ ($u < v$) such that the maximum weight of an edge on a simple path between $u$ and $v$ doesn't exceed $q_i$.\n\nQueries are numbered from $1$ to $m$ in the order of the input.\n\n\n-----Examples-----\nInput\n7 5\n1 2 1\n3 2 3\n2 4 1\n4 5 2\n5 7 4\n3 6 2\n5 2 3 4 1\n\nOutput\n21 7 15 21 3 \n\nInput\n1 2\n1 2\n\nOutput\n0 0 \n\nInput\n3 3\n1 2 1\n2 3 2\n1 3 2\n\nOutput\n1 3 3 \n\n\n\n-----Note-----\n\nThe picture shows the tree from the first example: [Image]",
        "solutions": "[\"n, m = list(map(int, input().split()))\\nmaxN = 2 * (10 ** 5) + 10\\nedges = [[] for i in range(0, maxN)]\\nque = [[] for _ in range(0, maxN)]\\nans = [0] * m\\nsz = [1 for _ in range(0, n)]\\np = [i for i in range(0, n)]\\ntotal_sum = 0\\n\\ndef get(u):\\n    if p[u] == u:\\n        return u\\n    p[u] = get(p[u])\\n    return p[u]\\n\\n\\ndef unite(u, v):\\n    u = get(u)\\n    v = get(v)\\n    if u == v:\\n        return\\n    nonlocal total_sum\\n    total_sum -= (sz[u] * (sz[u] - 1)) // 2\\n    total_sum -= (sz[v] * (sz[v] - 1)) // 2\\n    total_sum += ((sz[u] + sz[v]) * (sz[u] + sz[v] - 1)) // 2\\n    if sz[u] < sz[v]:\\n        p[u] = v\\n        sz[v] += sz[u]\\n    else:\\n        p[v] = u\\n        sz[u] += sz[v]\\n\\n\\nfor i in range(1, n):\\n    u, v, w = list(map(int, input().split()))\\n    u -= 1\\n    v -= 1\\n    edges[w].append((u, v))\\nques = list(map(int, input().split()))\\n\\nfor i in range(0, m):\\n    que[ques[i]].append(i)\\nfor i in range(0, maxN):\\n    for u, v in edges[i]:\\n        unite(u, v)\\n    for id in que[i]:\\n        ans[id] = total_sum\\nprint(\\\" \\\".join(str(x) for x in ans))\\n\\n\\n\\n\\n\\n\", \"from bisect import bisect\\nfrom math import inf\\n\\n\\ndef union_init(s):\\n    d = [i for i in range(s)]\\n    size = [1 for i in range(s)]\\n    return (d, size)\\n\\n\\ndef union_query(d, size, n):\\n    if d[n] != n:\\n        d[n] = union_query(d, size, d[n])\\n    return d[n]\\n\\n\\ndef union_merge(d, size, x, y):\\n    xRoot = union_query(d, size, x)\\n    yRoot = union_query(d, size, y)\\n\\n    if xRoot == yRoot:\\n        return\\n    if size[xRoot] < size[yRoot]:\\n        xRoot, yRoot = yRoot, xRoot\\n    d[yRoot] = xRoot\\n    size[xRoot] = size[xRoot] + size[yRoot]\\n\\n\\ndef sizeComponent(d, size, x):\\n    root = union_query(d, size, x)\\n    return size[root]\\n\\n\\nnm = input().split()\\nn = int(nm[0])\\nm = int(nm[1])\\n\\nd, size = union_init(n)\\npairs = [(0, 0)]\\nedges = []\\nfor _ in range(n - 1):\\n    edge = input().split()\\n    u = int(edge[0]) - 1\\n    v = int(edge[1]) - 1\\n    w = int(edge[2])\\n    edges.append((w, u, v))\\n\\nedges.sort()\\ntotalP = 0\\nfor e in edges:\\n    u = e[1]\\n    v = e[2]\\n    uSize = sizeComponent(d, size, u)\\n    vSize = sizeComponent(d, size, v)\\n    totalP += uSize * vSize\\n    union_merge(d, size, u, v)\\n    pairs.append((e[0], totalP))\\n\\nms = [int(mi) for mi in input().split()]\\nanswer = [0] * m\\nfor i, mi in enumerate(ms):\\n    start = bisect(pairs, (mi, inf)) - 1\\n    answer[i] = pairs[start][1]\\nprint(*answer)\\n\", \"'''input\\n3 3\\n1 2 1\\n2 3 2\\n1 3 2\\n'''\\nfrom sys import stdin\\nfrom copy import deepcopy\\nfrom collections import deque, defaultdict\\n\\n\\ndef find_parent(n):\\n\\tnonlocal parent\\n\\tnode = n\\n\\twhile parent[node] != node:\\n\\t\\tnode = parent[node]\\n\\tparent[n] = node\\n\\treturn parent[n]\\n\\n\\ndef combination(num):\\n\\treturn (num * (num - 1)) // 2\\n\\n# main starts\\nn, m = list(map(int, stdin.readline().split()))\\ngraph = defaultdict(list)\\nedges = []\\nfor _ in range(n - 1):\\n\\tedges.append(list(map(int, stdin.readline().split())))\\n\\nqueries = list(map(int, stdin.readline().split()))\\n\\nedges.sort(key = lambda x:x[2], reverse = True)\\nparent = dict()\\ncount = dict()\\nfor i in range(1, n + 1):\\n\\tparent[i] = i\\n\\tcount[i] = 1\\n\\nans = [0] * (200001)\\nwhile len(edges) > 0:\\n\\tu, v, w = edges.pop()\\n\\n\\tif u > v:\\n\\t\\tu, v = v, u\\n\\n\\tpv = find_parent(v)\\n\\tpu = find_parent(u)\\n\\tans[w] -= (combination(count[pu]) + combination(count[pv]))\\n\\tans[w] += combination(count[pu] + count[pv])\\n\\tcount[pu] += count[pv]\\n\\tcount[pv] = 0\\n\\tparent[pv] = parent[pu]\\n\\t\\t\\n\\t# print(u, v, w)\\n\\t# print('parent', parent)\\n\\t# print('count', count)\\n\\t# print(ans)\\n\\n\\nfor i in range(1, len(ans)):\\n\\tans[i] = ans[i - 1] + ans[i]\\n# print(ans[:10])\\n\\nfor i in queries:\\n\\tprint(ans[i], end = ' ')\\n\\n\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn,m=list(map(int,input().split()))\\npar=[-1]*n\\ndef c(x):\\n  return x*(x-1)//2\\n\\ndef find(x):\\n  if par[x]<0:\\n    return x\\n  else:\\n    par[x]=find(par[x])\\n    return par[x]\\n\\ndef unite(x,y):\\n  x=find(x)\\n  y=find(y)\\n  if x==y:\\n    return False\\n  else:\\n    if par[x]>par[y]:\\n      x,y=y,x\\n    par[x]+=par[y]\\n    par[y]=x\\n    return True\\n\\ndef same(x,y):\\n  return find(x)==find(y)\\n\\ndef size(x):\\n  return -par[find(x)]\\nimport collections\\nEdge=collections.defaultdict(list)\\nfor _ in range(n-1):\\n  u,v,w=list(map(int,input().split()))\\n  Edge[w].append((u,v))\\nQ=[int(i) for i in input().split()]\\nq=max(Q)\\nAns=[0]*(q+1)\\nans=0\\nfor i in range(1,q+1):\\n  for u,v in Edge[i]:\\n    if same(u-1,v-1):\\n      continue\\n    else:\\n      uu,vv=size(u-1),size(v-1)\\n      unite(u-1,v-1)\\n      uv=size(u-1)\\n      ans+=c(uv)-c(uu)-c(vv)\\n  Ans[i]=ans\\nAns2=[]\\nfor q in Q:\\n  Ans2.append(Ans[q])\\nprint(*Ans2)\\n\\n\", \"import sys\\ninput = sys.stdin.readline\\nn,m=list(map(int,input().split()))\\npar=[-1]*n\\ndef c(x):\\n  return x*(x-1)//2\\n\\ndef find(x):\\n  if par[x]<0:\\n    return x\\n  else:\\n    par[x]=find(par[x])\\n    return par[x]\\n\\ndef unite(x,y):\\n  x=find(x)\\n  y=find(y)\\n  if x==y:\\n    return False\\n  else:\\n    if par[x]>par[y]:\\n      x,y=y,x\\n    par[x]+=par[y]\\n    par[y]=x\\n    return True\\n\\ndef same(x,y):\\n  return find(x)==find(y)\\n\\ndef size(x):\\n  return -par[find(x)]\\nimport collections\\nEdge=collections.defaultdict(list)\\nfor _ in range(n-1):\\n  u,v,w=list(map(int,input().split()))\\n  Edge[w].append((u,v))\\nQ=[int(i) for i in input().split()]\\nq=max(Q)\\nAns=[0]*(q+1)\\nans=0\\nfor i in range(1,q+1):\\n  for u,v in Edge[i]:\\n    if same(u-1,v-1):\\n      continue\\n    else:\\n      uu,vv=size(u-1),size(v-1)\\n      unite(u-1,v-1)\\n      uv=size(u-1)\\n      ans+=c(uv)-c(uu)-c(vv)\\n  Ans[i]=ans\\nAns2=[]\\nfor q in Q:\\n  Ans2.append(Ans[q])\\nprint(*Ans2)\\n\\n\", \"# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\n# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\n# TAIWAN NUMBER ONE!!!!!!!!!!!!!!!!!!!\\nimport sys\\nfrom sys import stdin, stdout\\nfrom collections import defaultdict\\nfrom collections import deque\\nimport math\\nimport copy\\n\\n#T = int(input())\\n#N = int(input())\\n#s1 = input()\\n#s2 = input()\\nN,E = [int(x) for x in stdin.readline().split()]\\n#arr = [int(x) for x in stdin.readline().split()]\\n\\ns = 0\\nsize = [1]*N\\n\\nf = [x for x in range(N)]\\n\\ndef getf(v):\\n    stack = []\\n    while f[v] != v:\\n       stack.append(v)\\n       v = f[v]\\n    for idx in stack:\\n        f[idx] = v\\n    return v\\n\\ndef merge(f, v, u):\\n    nonlocal s\\n    t1 = getf(v)\\n    t2 = getf(u)\\n    if t1 != t2:\\n        A = size[t1]\\n        B = size[t2]\\n        s -= A*(A-1)//2\\n        s -= B*(B-1)//2\\n        size[t1] = size[t1] + size[t2]\\n        C = size[t1]\\n        s += C*(C-1)//2\\n        f[t2] = t1\\n\\n\\nw_edge = {}\\nfor i in range(N-1):\\n    u,v,w = [int(x) for x in stdin.readline().split()]\\n    if u>v:\\n        u,v = v,u\\n\\n    if w not in w_edge:\\n        w_edge[w] = []\\n\\n    w_edge[w].append((u,v))\\n\\nq = [0]*200000\\nfor i in range(200000):\\n    if i+1 in w_edge:\\n        for e in w_edge[i+1]:\\n            u,v = e\\n            merge(f,u-1,v-1)\\n        q[i] = s\\n    else:\\n        if i!=0:\\n            q[i] = q[i-1]\\n        else:\\n            q[i] = 0\\n\\narr = [int(x) for x in stdin.readline().split()]\\nans = [0]*E\\nfor i in range(E):\\n    ans[i] = q[arr[i]-1]\\n\\nprint(*ans)\\n\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2000)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2100)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2500)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2800)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = map(int, readline().split())\\nE = []\\nfor i in range(N-1):\\n    u, v, w = map(int, readline().split())\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(2900)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = range(N)\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\", \"from collections import defaultdict\\nimport sys\\nsys.setrecursionlimit(10**6)\\n\\nreadline = sys.stdin.readline\\nN, M = list(map(int, readline().split()))\\nE = []\\nfor i in range(N-1):\\n    u, v, w = list(map(int, readline().split()))\\n    E.append((w, u-1, v-1))\\nE.sort()\\nQ = set()\\nMP = defaultdict(list)\\nfor i, q in enumerate(map(int, readline().split())):\\n    MP[q].append(i)\\n    Q.add(q)\\nQ = list(Q)\\nQ.sort()\\n\\ndef fact(N):\\n    return N*fact(N-1) % 100 if N > 1 else 1\\nfact(3000)\\n\\ndef root(x):\\n    if x == p[x]:\\n        return x\\n    y = x\\n    while y != p[y]:\\n        y = p[y]\\n    while x != y:\\n        p[x], x = y, p[x]\\n    return y\\n*p, = list(range(N))\\nsz = [1]*N\\nc = 0\\ndef unite(x, y):\\n    nonlocal c\\n    px = root(x); py = root(y)\\n    if px == py:\\n        return 0\\n    c += sz[px] * sz[py]\\n    if sz[px] < sz[py]:\\n        p[py] = px\\n        sz[px] += sz[py]\\n    else:\\n        p[px] = py\\n        sz[py] += sz[px]\\n    return 1\\n\\nk = 0\\nans = [N*(N-1)//2]*M\\nL = len(Q)\\nfor w, u, v in E:\\n    while k < L and Q[k] < w:\\n        e = Q[k]\\n        for i in MP[e]:\\n            ans[i] = c\\n        k += 1\\n    unite(u, v)\\nsys.stdout.write(\\\" \\\".join(map(str, ans)))\\nsys.stdout.write(\\\"\\\\n\\\")\\n\", \"n,m=[int(x) for x in input().split()]\\nlink=[0]*(n+1)\\nsize=[0]*(n+1)\\narr=[0]*m\\nres=0\\nfor i in range(1,n+1):\\n    link[i]=i\\n    size[i]=1\\ndef find(x):\\n    while x!=link[x]:\\n        x=link[x]\\n    return x\\ndef comb(n):\\n    return n*(n-1)//2\\ndef unite(a,b,res):\\n    a=find(a)\\n    b=find(b)\\n    if size[a]<size[b]:\\n        a,b=b,a\\n    res=res-comb(size[a])-comb(size[b])+comb(size[a]+size[b])\\n    size[a]+=size[b]\\n    link[b]=a\\n    return res\\nedges=[]\\nask=[]\\nfor i in range(n-1):\\n    x,y,z=[int(x) for x in input().split()]\\n    edges.append((z,x,y))\\nedges.sort()\\nx=0\\nedges.append((10**100,1,1))\\nask=sorted(zip([int(x) for x in input().split()],list(range(m))))\\nfor i in range(m):\\n    while edges[x][0]<=ask[i][0]:\\n        res=unite(edges[x][1],edges[x][2],res)\\n        x+=1\\n    arr[ask[i][1]]=res\\nprint(*arr)\\n    \\n\", \"from sys import setrecursionlimit as SRL, stdin\\n\\nSRL(10 ** 7)\\nrd = stdin.readline\\nrrd = lambda: map(int, rd().strip().split())\\n\\nfa = [i for i in range(200005)]\\ns = [1] * 200005\\n\\n\\ndef find(x):\\n    t = []\\n    while fa[x] != x:\\n        t.append(x)\\n        x = fa[x]\\n    for i in t:\\n        fa[i] = x\\n    return fa[x]\\n\\n\\nans = [0] * 200005\\nn, q = rrd()\\n\\nw = []\\nfor i in range(n - 1):\\n    x, y, z = rrd()\\n    w.append([z, x, y])\\n\\nw.sort(key=lambda x: x[0])\\nfor x in w:\\n    u = find(x[1])\\n    v = find(x[2])\\n\\n    ans[x[0]] += s[u] * s[v]\\n    fa[u] = v\\n    s[v] += s[u]\\n\\nfor i in range(1, 200001):\\n    ans[i] += ans[i - 1]\\n\\nq = list(rrd())\\n\\nfor x in q:\\n    print(ans[x], end=' ')\\n\", \"import sys\\nsys.setrecursionlimit(10**9)\\ndef find(a):\\n    if par[a]==a:\\n        return a\\n    par[a]=find(par[a])\\n    return par[a]\\nn,m=list(map(int,input().split()))\\ned=[]\\npar=[i for i in range(n)]\\nsize=[1 for i in range(n)]\\nfor _ in range(n-1):\\n    a,b,c=list(map(int,input().split()))\\n    ed.append([a-1,b-1,c])\\ned.sort(key=lambda x:x[2])\\nit=list(map(int,input().split()))\\nit=[[i,j,0] for j,i in enumerate(it)]\\nit.sort()\\nind=0\\ntot=0\\nj=0\\n#print(it)\\nss={}\\nfor i in it[:]:\\n    while ind<n-1:\\n        if ed[ind][2]<=i[0]:\\n            a=find(ed[ind][0])\\n            b=find(ed[ind][1])\\n            if a!=b:\\n                tot+=size[a]*size[b]\\n              #  print(a,b,j,tot)\\n                if size[a]>=size[b]:\\n\\n                    par[b]=a\\n                    size[a]+=size[b]\\n                    size[b]=0\\n                else:\\n                    par[a]=b\\n                    size[b]+=size[a]\\n                    size[a]=0\\n            ind+=1\\n            \\n        else:\\n            break\\n    it[j][2]=tot\\n    #ss[it[j][1]]=tot\\n    j+=1\\n\\nit.sort(key=lambda x:x[1])\\naa=[i[2] for i in it]\\n\\n#for i in range(len(it)):\\n #   print(ss[i],end=\\\" \\\")\\nprint(*aa)\\n        \\n    \\n    \\n        \\n    \\n\", \"from operator import itemgetter\\nimport sys\\ninput = sys.stdin.readline\\n\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.parent = [-1] * n\\n        self.cnt = n\\n\\n    def root(self, x):\\n        if self.parent[x] < 0:\\n            return x\\n        else:\\n            self.parent[x] = self.root(self.parent[x])\\n            return self.parent[x]\\n\\n    def merge(self, x, y):\\n        x = self.root(x)\\n        y = self.root(y)\\n        if x != y:\\n            if self.parent[x] > self.parent[y]:\\n                x, y = y, x\\n            self.parent[x] += self.parent[y]\\n            self.parent[y] = x\\n            self.cnt -= 1\\n\\n    def is_same(self, x, y):\\n        return self.root(x) == self.root(y)\\n\\n    def get_size(self, x):\\n        return -self.parent[self.root(x)]\\n\\n    def get_cnt(self):\\n        return self.cnt\\n\\n\\nn, m = map(int, input().split())\\ninfo = [list(map(int, input().split())) for i in range(n - 1)]\\nq = list(map(int, input().split()))\\ninfo = sorted(info, key = itemgetter(2))\\nans = [0] * (2*10**5)\\nuf = UnionFind(2*10**5)\\ninfo_i = 0\\nfor i in range(2*10**5):\\n    if i-1 >= 0:\\n        ans[i] = ans[i-1]\\n    while True:\\n        if info_i >= n - 1:\\n           break\\n        if info[info_i][2] == i + 1:\\n            a, b, _ = info[info_i]\\n            a -= 1\\n            b -= 1\\n            num_a = uf.get_size(a)\\n            num_b = uf.get_size(b)\\n            num_ab = num_a + num_b\\n            comb_num_a = (num_a*(num_a-1)) // 2\\n            comb_num_b = (num_b*(num_b-1)) // 2\\n            comb_num_ab = (num_ab*(num_ab-1)) // 2\\n            ans[i] += comb_num_ab - (comb_num_a + comb_num_b)\\n            uf.merge(a, b)\\n            info_i += 1\\n        else:\\n            break\\n\\nres = [0] * len(q)\\nfor i, j in enumerate(q):\\n    res[i] = ans[j - 1]\\nprint(*res)\", \"def find_ancestor(i, father):\\n    if father[i] == i:\\n        return i\\n    father[i] = find_ancestor(father[i], father)\\n    return father[i]\\n\\ndef connect(i, j, father, n_child):\\n    i_anc = find_ancestor(i, father)\\n    j_anc = find_ancestor(j, father)\\n    if n_child[i_anc] > n_child[j_anc]:\\n        n_child[i_anc] += n_child[j_anc]\\n        father[j_anc] = i_anc\\n    else:\\n        n_child[j_anc] += n_child[i_anc]\\n        father[i_anc] = j_anc\\n\\nn, m = list(map(int, input().split()))\\nedges = []\\nfather = [i for i in range(n)]\\nn_child = [1]*n\\n\\nfor i in range(n-1):\\n    i, j, w = list(map(int, input().split()))\\n    edges.append((i-1, j-1, w))\\n\\nedges.sort(key=lambda x: -x[2])\\nqueries = list(map(int, input().split()))\\n\\ns_queries = sorted(queries)\\n\\n# final map the index to the query\\nans = {}\\n\\nw_limit = []\\nans_cum = 0\\nfor query in s_queries:\\n    while len(edges) and edges[-1][2] <= query:\\n        i, j, w = edges[-1]\\n        edges.pop()\\n        i_anc = find_ancestor(i, father)\\n        j_anc = find_ancestor(j, father)\\n        # it's tree father may not be same\\n        ans_cum += n_child[i_anc] * n_child[j_anc]\\n        connect(i, j, father, n_child)\\n    ans[query] = ans_cum\\n\\nprint(\\\" \\\".join(list(map(str, [ans[query] for query in queries]))))\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        if self.par[x] == x:\\n            return x\\n        else:\\n            self.par[x] = self.find(self.par[x])\\n            return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nedges.sort()\\nA = [0] * (2*10**5+7)\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        A[c] = A[prevc]\\n    sz1 = uf.get_size(a)\\n    A[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    A[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    A[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nA = list(accumulate(A, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = A[q]\\nprint(*ans)\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\nfrom operator import itemgetter\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        if self.par[x] == x:\\n            return x\\n        else:\\n            self.par[x] = self.find(self.par[x])\\n            return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nMAX = 2 * 10**5 + 7\\nedges.sort(key=itemgetter(0))\\nC = [0] * MAX\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        C[c] = C[prevc]\\n    sz1 = uf.get_size(a)\\n    C[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    C[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    C[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nC = list(accumulate(C, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = C[q]\\nprint(*ans)\\n\", \"# -*- coding: utf-8 -*-\\n\\nimport sys\\nfrom itertools import accumulate\\nfrom operator import itemgetter\\n\\ndef input(): return sys.stdin.readline().strip()\\ndef list2d(a, b, c): return [[c] * b for i in range(a)]\\ndef list3d(a, b, c, d): return [[[d] * c for j in range(b)] for i in range(a)]\\ndef list4d(a, b, c, d, e): return [[[[e] * d for j in range(c)] for j in range(b)] for i in range(a)]\\ndef ceil(x, y=1): return int(-(-x // y))\\ndef INT(): return int(input())\\ndef MAP(): return list(map(int, input().split()))\\ndef LIST(N=None): return list(MAP()) if N is None else [INT() for i in range(N)]\\ndef Yes(): print('Yes')\\ndef No(): print('No')\\ndef YES(): print('YES')\\ndef NO(): print('NO')\\nINF = 10 ** 18\\nMOD = 10 ** 9 + 7\\n\\nclass UnionFind:\\n    def __init__(self, n):\\n        self.n = n\\n        self.par = [i for i in range(n+1)]\\n        self.rank = [0] * (n+1)\\n        self.size = [1] * (n+1)\\n        self.tree = [True] * (n+1)\\n\\n    def find(self, x):\\n        t = []\\n        while self.par[x] != x:\\n            t.append(x)\\n            x = self.par[x]\\n        for i in t:\\n            self.par[i] = x\\n        return self.par[x]\\n        # if self.par[x] == x:\\n        #     return x\\n        # else:\\n        #     self.par[x] = self.find(self.par[x])\\n        #     return self.par[x]\\n\\n    def union(self, x, y):\\n        x = self.find(x)\\n        y = self.find(y)\\n\\n        if x == y:\\n            self.tree[x] = False\\n            return\\n        if not self.tree[x] or not self.tree[y]:\\n            self.tree[x] = self.tree[y] = False\\n\\n        if self.rank[x] < self.rank[y]:\\n            self.par[x] = y\\n            self.size[y] += self.size[x]\\n        else:\\n            self.par[y] = x\\n            self.size[x] += self.size[y]\\n            if self.rank[x] == self.rank[y]:\\n                self.rank[x] += 1\\n\\n    def is_same(self, x, y):\\n        return self.find(x) == self.find(y)\\n\\n    def get_size(self, x=None):\\n        if x is not None:\\n            return self.size[self.find(x)]\\n        else:\\n            res = set()\\n            for i in range(self.n+1):\\n                res.add(self.find(i))\\n            return len(res) - 1\\n    \\n    def is_tree(self, x):\\n        return self.tree[self.find(x)]\\n\\nN, M = MAP()\\nedges = []\\nfor i in range(N-1):\\n    a, b, c = MAP()\\n    a -= 1; b -= 1\\n    edges.append((c, a, b))\\n\\nif N == 1:\\n    LIST()\\n    ans = [0] * M\\n    print(*ans)\\n    return\\n\\nMAX = 2 * 10**5 + 7\\nedges.sort(key=itemgetter(0))\\nC = [0] * MAX\\nuf = UnionFind(N)\\nprevc = edges[0][0]\\nfor c, a, b in edges:\\n    if prevc != c:\\n        C[c] = C[prevc]\\n    sz1 = uf.get_size(a)\\n    C[c] -= sz1 * (sz1-1) // 2\\n    sz2 = uf.get_size(b)\\n    C[c] -= sz2 * (sz2-1) // 2\\n    uf.union(a, b)\\n    sz = sz1 + sz2\\n    C[c] += sz * (sz-1) // 2\\n    prevc = c\\n\\nC = list(accumulate(C, max))\\nQ = LIST()\\nans = [0] * M\\nfor i, q in enumerate(Q):\\n    ans[i] = C[q]\\nprint(*ans)\\n\", \"class UnionFind():\\n    def __init__(self, n):\\n        self.n = n\\n        self.root = [-1]*(n+1)\\n        self.rnk = [0]*(n+1)\\n\\n    def Find_Root(self, x):\\n        if(self.root[x] < 0):\\n            return x\\n        else:\\n            self.root[x] = self.Find_Root(self.root[x])\\n            return self.root[x]\\n    \\n    def Unite(self, x, y):\\n        x = self.Find_Root(x)\\n        y = self.Find_Root(y)\\n        if(x == y):\\n            return \\n        elif(self.rnk[x] > self.rnk[y]):\\n            self.root[x] += self.root[y]\\n            self.root[y] = x\\n\\n        else:\\n            self.root[y] += self.root[x]\\n            self.root[x] = y\\n            if(self.rnk[x] == self.rnk[y]):\\n                self.rnk[y] += 1\\n    \\n    def isSameGroup(self, x, y):\\n        return self.Find_Root(x) == self.Find_Root(y)\\n\\n    def Count(self, x):\\n        return -self.root[self.Find_Root(x)]\\n\\n\\nimport sys\\ninput = sys.stdin.readline\\nfrom bisect import bisect_right\\n\\nN, M = map(int, input().split())\\nuni = UnionFind(N+1)\\nEdges = {}\\nfor _ in range(N-1):\\n    a, b, w = map(int, input().split())\\n    if not w in Edges:\\n        Edges[w] = [(a, b)]\\n    else:\\n        Edges[w].append((a, b))\\n\\nQuery = list(map(int, input().split()))\\n\\nWeights = sorted(list(Edges.keys()))\\n\\nScore = [0]\\nscore = 0\\nfor w in Weights:\\n    for a, b in Edges[w]:\\n        c1 = uni.Count(a)\\n        c2 = uni.Count(b)\\n        c = c1 + c2\\n        score += c*(c-1)//2 - c1*(c1-1)//2 - c2*(c2-1)//2\\n        uni.Unite(a, b)\\n    Score.append(score)\\n\\nans = []\\nfor q in Query:\\n    ind = bisect_right(Weights, q)\\n    ans.append(Score[ind])\\n\\nprint(*ans)\"]",
        "difficulty": "introductory",
        "input": "3 3\n1 2 1\n2 3 2\n1 3 2\n",
        "output": "1 3 3 \n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1213/G"
    },
    {
        "id": 1240,
        "task_id": 4861,
        "test_case_id": 1,
        "question": "The famous Arora-Mitchell approximation algorithm for the Euclidean Travelling Salesman Problem (Euclidean TSP) was discovered independently by Sanjeev Arora and Joseph S. B. Mitchell in 1998. It can approximate the value of an optimal TSP tour in $d$ dimensions within a factor of $1+1/c$ in running time \\[ n (\\log n)^{O((c\\sqrt {d})^{d-1})}, \\]\n\nwhere $n$ is the number of nodes in the tour.\n\nMiroslava works for a computer security company and it is time to renew a shared cryptographic key in many data centres across Europe. To do this, Miroslava is going to rent a private jet and deliver the key to employees waiting at all major European airports. She wants to be back as soon as possible.\n\nMiroslava’s company has a computer that is able to execute $p$ billions of operations per second. Since we can approximate Europe by a two-dimensional plane, we assume that the Arora-Mitchell algorithm runs for exactly\\[ \\frac{n (\\log _2 n)^{c\\sqrt {2}}}{p \\cdot 10^9} \\]\n\nseconds on this computer to produce the exact $(1+1/c)$-approximation of the optimal tour.\n\nMiroslava noticed that $c$ is a parameter of the algorithm that can be used to her advantage, but one also needs to be very careful when choosing the right value. If she sets $c$ too low, the algorithm will finish very fast but the time she spends flying around Europe will be too long. On the other hand, setting it too high will force her to wait for an answer from the computer, while she could be flying instead.\n\nMiroslava used to work in a different company and from there she knows that the optimal tour of all major European airports is $s$ meters long, but she wasn’t ranked high enough in the company to know the actual tour. Given the speed $v$ of the private jet in meters per second, Miroslava needs $s(1 + 1 / c) / v$ seconds to complete the tour produced by the algorithm run with parameter $c$. For the sake of simplicity, we assume that Miroslava can land, leave a copy of the private key and take off from each airport in an instant.\n\nHow long does it take Miroslava to first run the algorithm and then distribute all the keys, assuming that she chooses the optimal parameter $c$?\n\n-----Input-----\nThe input consists of one line with four numbers:\n - an integer $n$ ($4 \\le n \\le 1000000$), the number of airports;\n - a real number $p$ ($0.001 \\le p \\le 5000$), the number of billions of operations the computer can execute per second;\n - a real number $s$ ($10^6 \\le s \\le 10^9$), the length of the optimal tour of all European airports in meters;\n - a real number $v$ ($50 \\le v \\le 900$), the speed of the private jet in meters per second.\n\nAll real numbers will have at most 10 digits after the decimal point.\n\n-----Output-----\nOutput one line with the shortest possible time $t$ in seconds to distribute the keys and the value of the parameter $c$ Miroslava should use to achieve time $t$. Your answer should have an absolute or relative error of at most $10^{-6}$.\n\n-----Examples-----\nSample Input:\n10 8.9 40075000 272.1\nSample Output:\n157079.04857106 15.598261092309",
        "solutions": "",
        "difficulty": "introductory",
        "input": "10 8.9 40075000 272.1\n",
        "output": "157079.04857106 15.598261092309\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/euclideantsp"
    },
    {
        "id": 1241,
        "task_id": 4861,
        "test_case_id": 2,
        "question": "The famous Arora-Mitchell approximation algorithm for the Euclidean Travelling Salesman Problem (Euclidean TSP) was discovered independently by Sanjeev Arora and Joseph S. B. Mitchell in 1998. It can approximate the value of an optimal TSP tour in $d$ dimensions within a factor of $1+1/c$ in running time \\[ n (\\log n)^{O((c\\sqrt {d})^{d-1})}, \\]\n\nwhere $n$ is the number of nodes in the tour.\n\nMiroslava works for a computer security company and it is time to renew a shared cryptographic key in many data centres across Europe. To do this, Miroslava is going to rent a private jet and deliver the key to employees waiting at all major European airports. She wants to be back as soon as possible.\n\nMiroslava’s company has a computer that is able to execute $p$ billions of operations per second. Since we can approximate Europe by a two-dimensional plane, we assume that the Arora-Mitchell algorithm runs for exactly\\[ \\frac{n (\\log _2 n)^{c\\sqrt {2}}}{p \\cdot 10^9} \\]\n\nseconds on this computer to produce the exact $(1+1/c)$-approximation of the optimal tour.\n\nMiroslava noticed that $c$ is a parameter of the algorithm that can be used to her advantage, but one also needs to be very careful when choosing the right value. If she sets $c$ too low, the algorithm will finish very fast but the time she spends flying around Europe will be too long. On the other hand, setting it too high will force her to wait for an answer from the computer, while she could be flying instead.\n\nMiroslava used to work in a different company and from there she knows that the optimal tour of all major European airports is $s$ meters long, but she wasn’t ranked high enough in the company to know the actual tour. Given the speed $v$ of the private jet in meters per second, Miroslava needs $s(1 + 1 / c) / v$ seconds to complete the tour produced by the algorithm run with parameter $c$. For the sake of simplicity, we assume that Miroslava can land, leave a copy of the private key and take off from each airport in an instant.\n\nHow long does it take Miroslava to first run the algorithm and then distribute all the keys, assuming that she chooses the optimal parameter $c$?\n\n-----Input-----\nThe input consists of one line with four numbers:\n - an integer $n$ ($4 \\le n \\le 1000000$), the number of airports;\n - a real number $p$ ($0.001 \\le p \\le 5000$), the number of billions of operations the computer can execute per second;\n - a real number $s$ ($10^6 \\le s \\le 10^9$), the length of the optimal tour of all European airports in meters;\n - a real number $v$ ($50 \\le v \\le 900$), the speed of the private jet in meters per second.\n\nAll real numbers will have at most 10 digits after the decimal point.\n\n-----Output-----\nOutput one line with the shortest possible time $t$ in seconds to distribute the keys and the value of the parameter $c$ Miroslava should use to achieve time $t$. Your answer should have an absolute or relative error of at most $10^{-6}$.\n\n-----Examples-----\nSample Input:\n10 8.9 40075000 272.1\nSample Output:\n157079.04857106 15.598261092309",
        "solutions": "",
        "difficulty": "introductory",
        "input": "47 4.2 1337102.4 256\n",
        "output": "5836.2936298227 8.9113418228146\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://open.kattis.com/problems/euclideantsp"
    },
    {
        "id": 1242,
        "task_id": 4479,
        "test_case_id": 1,
        "question": "Given an array A of integers, we must modify the array in the following way: we choose an i and replace A[i] with -A[i], and we repeat this process K times in total.  (We may choose the same index i multiple times.)\nReturn the largest possible sum of the array after modifying it in this way.\n \nExample 1:\nInput: A = [4,2,3], K = 1\nOutput: 5\nExplanation: Choose indices (1,) and A becomes [4,-2,3].\n\n\nExample 2:\nInput: A = [3,-1,0,2], K = 3\nOutput: 6\nExplanation: Choose indices (1, 2, 2) and A becomes [3,1,0,2].\n\n\nExample 3:\nInput: A = [2,-3,-1,5,-4], K = 2\nOutput: 13\nExplanation: Choose indices (1, 4) and A becomes [2,3,-1,5,4].\n\n\n\n \nNote:\n\n1 <= A.length <= 10000\n1 <= K <= 10000\n-100 <= A[i] <= 100",
        "solutions": "[\"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        \\n        i = 0\\n        while A[i] < 0 and K > 0:\\n            A[i] *= -1\\n            i += 1\\n            K -= 1\\n            \\n        if K % 2 == 1 and 0 not in A:\\n            return sum(A) - 2*min(A)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        i=0\\n        while(i<len(A) and A[i]<0 and i<K):\\n            A[i]=-A[i]\\n            i+=1\\n        \\n        return sum(A)-(K - i) % 2 * min(A) * 2\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        S=0\\n        negs=[]\\n        posmin=100\\n        negmax=-100\\n        for i in range(len(A)):\\n            S+=A[i]\\n            if A[i]<0:\\n                negs.append(A[i])\\n                negmax=max(negmax,A[i])\\n            else:\\n                posmin=min(posmin,A[i])\\n        if K is 0:\\n            return(S)\\n        negs.sort()\\n        if K <= len(negs):\\n            return(S-2*sum(negs[:K]))\\n        elif (K - len(negs))%2 is 0:\\n            return(S-2*sum(negs[:K]))\\n        elif posmin<-negmax:\\n            return(S-2*sum(negs[:K])-2*posmin)\\n        else:\\n            return(S-2*sum(negs[:K])+2*negmax)\", \"class Solution(object):\\n    def largestSumAfterKNegations(self, A, K):\\n        lst = sorted(A)\\n        count = K\\n        i = 0\\n        while count>0:\\n            if lst[i] <= lst[i + 1]:\\n                lst[i] = lst[i] * -1\\n                count -= 1\\n            else:\\n                i += 1\\n        return sum(lst)\\n\", \"import heapq\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for _ in range(K):\\n            heapq.heapreplace(A, -A[0])\\n        return sum(A)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        p = 0\\n        while A[p] < 0:\\n            A[p] = - A[p]\\n            p += 1\\n            K -= 1\\n            if K == 0:\\n                 return sum(A)\\n        if K % 2 == 0:\\n            return sum(A)\\n        else:\\n            return sum(A) - 2 * min(A[p], A[p-1])\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        #heapify is in place algo so take linear time\\n        for x in range(K):\\n            #if x %2 :\\n                #heappreplace(A, -A[0])\\n            #heappush(A, -heapq.heappop(A))\\n            heapreplace(A, -A[0])\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], k: int) -> int:\\n        d = collections.defaultdict(int)\\n        for x in A:\\n            d[x] += 1\\n        \\n        res = 0\\n        for i in range(-100, 0):\\n            if i in d and k > 0:\\n                x = min(k, d[i])\\n                k -= x\\n                d[i] -= x\\n                d[-i] += x\\n            \\n        res = 0\\n        if k % 2 == 1:\\n            j = 0\\n            while j not in d or d[j] <= 0:\\n                j += 1\\n            print(j)\\n            d[j] -= 1\\n            d[-j] += 1\\n        for i in range(-100, 101):\\n            res += d[i] * i\\n        return res\\n        \\n            \\n                \\n                    \\n            \\n\\n                \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        res = 0\\n        A.sort()\\n        i = 0\\n        while i < len(A) and i < K and A[i] < 0:\\n            A[i] = -A[i]\\n            i += 1\\n\\n        return sum(A) - (K - i) % 2 * 2 * min(A[i-1:i] + A[i:i+1])\\n            \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n          A.sort()\\n          A[0] = -A[0]\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        result = 0\\n        count = 0\\n        A.sort()\\n       \\n        if A[0] > 0:\\n            if K%2 == 1:\\n                A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            return(result)\\n        elif A[0] == 0:\\n            for i in A:\\n                result += i\\n            return(result)\\n        else:\\n            for j in range (len(A)):\\n                if A[j] >= 0:\\n                    break\\n                else:\\n                    count += 1\\n            ra = K%count\\n            if K < count:\\n                for m in range(K):\\n                    A[m] = -A[m]\\n            else:\\n                for m in range(count):\\n                    A[m] = -A[m]\\n                K -= count\\n                A.sort()\\n                if K%2 == 1:\\n                    A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            \\n            return(result)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for i in range(K):\\n            m = heapq.heappop(A)\\n            heapq.heappush(A, -m)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        newA = sorted(A)\\n        print(newA)\\n        count = 0\\n        for i in range(K):\\n            newA[0] = -newA[0]\\n            newA = sorted(newA)\\n        print(newA)\\n        return sum(newA)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        B = sorted(A)\\n        for i in range(K):\\n            B[0] = -B[0]\\n            B = sorted(B)\\n        return sum(B)\", \"class Solution:\\n    def largestSumAfterKNegations(self, arr: List[int], k: int) -> int:\\n        ra = k\\n        arr.sort()\\n        for _ in range(ra):\\n            for j in range(len(arr)):\\n                if arr[j] < 0:\\n                    arr[j] = abs(arr[j])\\n                    k -= 1\\n                    arr.sort()\\n                    break\\n                elif arr[j] == 0:\\n                    return sum(arr)\\n                else:\\n                    if k % 2 == 0:\\n                        return sum(arr)\\n                    else:\\n                        arr.sort()\\n                        arr[0] = -arr[0]\\n                        return sum(arr)\\n\\n        return sum(arr)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K > 0:\\n            t = A.index(min(A))\\n            if A[t] < 0:\\n                A[t] = abs(A[t])\\n                K -= 1\\n                continue\\n            if A[t] == 0:\\n                K = 0\\n                continue\\n            if A[t] > 0:\\n                if K % 2 == 0:\\n                    K = 0\\n                    continue\\n                else:\\n                    A[t] = -A[t]\\n                    K = 0\\n                    continue\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for x in range(K):\\n            minNum=A.index(min(A))\\n            A[minNum]=A[minNum]*(-1)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            variable = min(A)\\n            A.remove(variable)\\n            A.append(variable*-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n        \\n            A.remove(val)\\n            A.append(-1 * val)\\n                                \\n        return sum(A)\\n                \\n               \\n\", \"# 5:47 -> 5:50 slowwww\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K:\\n            m = min(A)\\n            m_i = A.index(m)\\n            A[m_i] = -m\\n            K -= 1\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n\\n        for i in range(K):\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        a1 =A\\n        sum1 =0\\n        while K > 0 :\\n            mina = min(a1)\\n            a1.remove(mina)\\n            a1.append(mina * (-1))\\n            K -=1\\n            \\n    \\n        \\n        return sum(a1)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > A[1]:\\n                A.append(A.pop(0))\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        cur = sum(A)\\n        for i in range(K):\\n           # max_ = sum(A)\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        max_sum = -999999\\n        for _ in range(K):\\n            A = sorted(A)\\n            min_index = A.index(min(A))\\n            \\n            A[min_index] = -A[min_index]\\n\\n            max_ = sum(A)\\n            # print(max_)\\n            # max_sum = max(max_sum, max_)\\n        return max_\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        while K > 0:\\n            min_index = A.index(min(A))\\n            A[min_index] = - A[min_index]\\n            K = K - 1\\n        \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n            \\n            #if val <= 0:\\n            A.remove(val)\\n            A.append(-1 * val)\\n                \\n           \\n                \\n        return sum(A)\\n                \\n               \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K != 0:\\n            A[A.index(min(A))] = -min(A)\\n            K -= 1\\n        return(sum(A))\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        c = 0\\n        \\n        while c < K:\\n            c += 1\\n            min_ = min(A)\\n            index = A.index(min_)\\n            A[index] = -1 * min_\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K>0:\\n            K-=1\\n            A[A.index(min(A))]=-min(A)\\n        return sum(A)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        ## find the min element and its index and negate it K times\\n        \\n        ## Complexity = O(kn)\\n        \\n#         def findMin(A):\\n#             minValue = float('inf')\\n#             minIndex = 0\\n\\n#             for i, ele in enumerate(A):\\n#                 if ele < minValue:\\n#                     minValue = ele\\n#                     minIndex = i\\n            \\n#             return minIndex\\n\\n#         for k in range(K):\\n#             minIndex = findMin(A)\\n#             A[minIndex] = -A[minIndex]\\n            \\n        \\n        ### using min heap\\n        \\n        heapq.heapify(A)\\n        for k in range(K):\\n            A[0] = -A[0]\\n            heapq.heapify(A)\\n        \\n        return sum(A)\\n    \\n    ### let's do with heap, no need to pop just negate that element\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K) :\\n            check = min(A)\\n            t = A.index(check)\\n            A[t] = -A[t]\\n        \\n        return sum(A)\\n            \\n        \\n        \\n        \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A[A.index(min(A))] *= -1\\n            \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        if not A: return sum(A)\\n        \\n        if K == 0: return sum(A)\\n        \\n        \\n        for i in range(K):\\n    \\n            A[A.index(min(A))] = - A[A.index(min(A))]\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if not A:\\n            return sum(A)\\n        if K==0:\\n            return sum(A)\\n        for i in range(K):\\n            A[A.index(min(A))] = -A[A.index(min(A))]\\n        return sum(A)\\n\", \"# 5:47 ->\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if K == 0:\\n            return sum(A)\\n        m = min(A)\\n        m_i = A.index(m)\\n        A[m_i] = -m\\n        return self.largestSumAfterKNegations(A, K - 1)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for item in A :\\n            if item < 0 and K > 1:\\n                A[A.index(item)] = item * (-1)\\n                K-= 1\\n            elif item > 0 and K >= 2:\\n                K-=2\\n        else:\\n            A.sort()\\n            while K != 0:\\n                K -=1\\n                A[0] = A[0] * (-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > 0:\\n                A.sort()\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A.sort()\\n            A[0] = -A[0]\\n        \\n        return sum(A)\", \"\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        ans = sum(A)\\n        A.sort()\\n        \\n        for i in range(len(A)):\\n            if not K or A[i]==0:\\n                break\\n            if A[i]<0:\\n                ans -= 2*A[i]\\n                K-=1\\n            else:\\n                if K%2:\\n                    if i and A[i]>abs(A[i-1]):\\n                        ans += 2*A[i-1]\\n                    else:\\n                        ans -= 2*A[i]\\n                break\\n                    \\n        return ans\"]",
        "difficulty": "introductory",
        "input": [
            [
                -2,
                3,
                4
            ],
            1
        ],
        "output": 9,
        "halu_type": "Identification Hallucination",
        "fn_name": "largestSumAfterKNegations",
        "starter_code": "\nclass Solution:\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\n        ",
        "url": "https://leetcode.com/problems/maximize-sum-of-array-after-k-negations/"
    },
    {
        "id": 1243,
        "task_id": 4479,
        "test_case_id": 2,
        "question": "Given an array A of integers, we must modify the array in the following way: we choose an i and replace A[i] with -A[i], and we repeat this process K times in total.  (We may choose the same index i multiple times.)\nReturn the largest possible sum of the array after modifying it in this way.\n \nExample 1:\nInput: A = [4,2,3], K = 1\nOutput: 5\nExplanation: Choose indices (1,) and A becomes [4,-2,3].\n\n\nExample 2:\nInput: A = [3,-1,0,2], K = 3\nOutput: 6\nExplanation: Choose indices (1, 2, 2) and A becomes [3,1,0,2].\n\n\nExample 3:\nInput: A = [2,-3,-1,5,-4], K = 2\nOutput: 13\nExplanation: Choose indices (1, 4) and A becomes [2,3,-1,5,4].\n\n\n\n \nNote:\n\n1 <= A.length <= 10000\n1 <= K <= 10000\n-100 <= A[i] <= 100",
        "solutions": "[\"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        \\n        i = 0\\n        while A[i] < 0 and K > 0:\\n            A[i] *= -1\\n            i += 1\\n            K -= 1\\n            \\n        if K % 2 == 1 and 0 not in A:\\n            return sum(A) - 2*min(A)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        i=0\\n        while(i<len(A) and A[i]<0 and i<K):\\n            A[i]=-A[i]\\n            i+=1\\n        \\n        return sum(A)-(K - i) % 2 * min(A) * 2\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        S=0\\n        negs=[]\\n        posmin=100\\n        negmax=-100\\n        for i in range(len(A)):\\n            S+=A[i]\\n            if A[i]<0:\\n                negs.append(A[i])\\n                negmax=max(negmax,A[i])\\n            else:\\n                posmin=min(posmin,A[i])\\n        if K is 0:\\n            return(S)\\n        negs.sort()\\n        if K <= len(negs):\\n            return(S-2*sum(negs[:K]))\\n        elif (K - len(negs))%2 is 0:\\n            return(S-2*sum(negs[:K]))\\n        elif posmin<-negmax:\\n            return(S-2*sum(negs[:K])-2*posmin)\\n        else:\\n            return(S-2*sum(negs[:K])+2*negmax)\", \"class Solution(object):\\n    def largestSumAfterKNegations(self, A, K):\\n        lst = sorted(A)\\n        count = K\\n        i = 0\\n        while count>0:\\n            if lst[i] <= lst[i + 1]:\\n                lst[i] = lst[i] * -1\\n                count -= 1\\n            else:\\n                i += 1\\n        return sum(lst)\\n\", \"import heapq\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for _ in range(K):\\n            heapq.heapreplace(A, -A[0])\\n        return sum(A)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        p = 0\\n        while A[p] < 0:\\n            A[p] = - A[p]\\n            p += 1\\n            K -= 1\\n            if K == 0:\\n                 return sum(A)\\n        if K % 2 == 0:\\n            return sum(A)\\n        else:\\n            return sum(A) - 2 * min(A[p], A[p-1])\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        #heapify is in place algo so take linear time\\n        for x in range(K):\\n            #if x %2 :\\n                #heappreplace(A, -A[0])\\n            #heappush(A, -heapq.heappop(A))\\n            heapreplace(A, -A[0])\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], k: int) -> int:\\n        d = collections.defaultdict(int)\\n        for x in A:\\n            d[x] += 1\\n        \\n        res = 0\\n        for i in range(-100, 0):\\n            if i in d and k > 0:\\n                x = min(k, d[i])\\n                k -= x\\n                d[i] -= x\\n                d[-i] += x\\n            \\n        res = 0\\n        if k % 2 == 1:\\n            j = 0\\n            while j not in d or d[j] <= 0:\\n                j += 1\\n            print(j)\\n            d[j] -= 1\\n            d[-j] += 1\\n        for i in range(-100, 101):\\n            res += d[i] * i\\n        return res\\n        \\n            \\n                \\n                    \\n            \\n\\n                \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        res = 0\\n        A.sort()\\n        i = 0\\n        while i < len(A) and i < K and A[i] < 0:\\n            A[i] = -A[i]\\n            i += 1\\n\\n        return sum(A) - (K - i) % 2 * 2 * min(A[i-1:i] + A[i:i+1])\\n            \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n          A.sort()\\n          A[0] = -A[0]\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        result = 0\\n        count = 0\\n        A.sort()\\n       \\n        if A[0] > 0:\\n            if K%2 == 1:\\n                A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            return(result)\\n        elif A[0] == 0:\\n            for i in A:\\n                result += i\\n            return(result)\\n        else:\\n            for j in range (len(A)):\\n                if A[j] >= 0:\\n                    break\\n                else:\\n                    count += 1\\n            ra = K%count\\n            if K < count:\\n                for m in range(K):\\n                    A[m] = -A[m]\\n            else:\\n                for m in range(count):\\n                    A[m] = -A[m]\\n                K -= count\\n                A.sort()\\n                if K%2 == 1:\\n                    A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            \\n            return(result)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for i in range(K):\\n            m = heapq.heappop(A)\\n            heapq.heappush(A, -m)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        newA = sorted(A)\\n        print(newA)\\n        count = 0\\n        for i in range(K):\\n            newA[0] = -newA[0]\\n            newA = sorted(newA)\\n        print(newA)\\n        return sum(newA)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        B = sorted(A)\\n        for i in range(K):\\n            B[0] = -B[0]\\n            B = sorted(B)\\n        return sum(B)\", \"class Solution:\\n    def largestSumAfterKNegations(self, arr: List[int], k: int) -> int:\\n        ra = k\\n        arr.sort()\\n        for _ in range(ra):\\n            for j in range(len(arr)):\\n                if arr[j] < 0:\\n                    arr[j] = abs(arr[j])\\n                    k -= 1\\n                    arr.sort()\\n                    break\\n                elif arr[j] == 0:\\n                    return sum(arr)\\n                else:\\n                    if k % 2 == 0:\\n                        return sum(arr)\\n                    else:\\n                        arr.sort()\\n                        arr[0] = -arr[0]\\n                        return sum(arr)\\n\\n        return sum(arr)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K > 0:\\n            t = A.index(min(A))\\n            if A[t] < 0:\\n                A[t] = abs(A[t])\\n                K -= 1\\n                continue\\n            if A[t] == 0:\\n                K = 0\\n                continue\\n            if A[t] > 0:\\n                if K % 2 == 0:\\n                    K = 0\\n                    continue\\n                else:\\n                    A[t] = -A[t]\\n                    K = 0\\n                    continue\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for x in range(K):\\n            minNum=A.index(min(A))\\n            A[minNum]=A[minNum]*(-1)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            variable = min(A)\\n            A.remove(variable)\\n            A.append(variable*-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n        \\n            A.remove(val)\\n            A.append(-1 * val)\\n                                \\n        return sum(A)\\n                \\n               \\n\", \"# 5:47 -> 5:50 slowwww\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K:\\n            m = min(A)\\n            m_i = A.index(m)\\n            A[m_i] = -m\\n            K -= 1\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n\\n        for i in range(K):\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        a1 =A\\n        sum1 =0\\n        while K > 0 :\\n            mina = min(a1)\\n            a1.remove(mina)\\n            a1.append(mina * (-1))\\n            K -=1\\n            \\n    \\n        \\n        return sum(a1)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > A[1]:\\n                A.append(A.pop(0))\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        cur = sum(A)\\n        for i in range(K):\\n           # max_ = sum(A)\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        max_sum = -999999\\n        for _ in range(K):\\n            A = sorted(A)\\n            min_index = A.index(min(A))\\n            \\n            A[min_index] = -A[min_index]\\n\\n            max_ = sum(A)\\n            # print(max_)\\n            # max_sum = max(max_sum, max_)\\n        return max_\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        while K > 0:\\n            min_index = A.index(min(A))\\n            A[min_index] = - A[min_index]\\n            K = K - 1\\n        \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n            \\n            #if val <= 0:\\n            A.remove(val)\\n            A.append(-1 * val)\\n                \\n           \\n                \\n        return sum(A)\\n                \\n               \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K != 0:\\n            A[A.index(min(A))] = -min(A)\\n            K -= 1\\n        return(sum(A))\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        c = 0\\n        \\n        while c < K:\\n            c += 1\\n            min_ = min(A)\\n            index = A.index(min_)\\n            A[index] = -1 * min_\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K>0:\\n            K-=1\\n            A[A.index(min(A))]=-min(A)\\n        return sum(A)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        ## find the min element and its index and negate it K times\\n        \\n        ## Complexity = O(kn)\\n        \\n#         def findMin(A):\\n#             minValue = float('inf')\\n#             minIndex = 0\\n\\n#             for i, ele in enumerate(A):\\n#                 if ele < minValue:\\n#                     minValue = ele\\n#                     minIndex = i\\n            \\n#             return minIndex\\n\\n#         for k in range(K):\\n#             minIndex = findMin(A)\\n#             A[minIndex] = -A[minIndex]\\n            \\n        \\n        ### using min heap\\n        \\n        heapq.heapify(A)\\n        for k in range(K):\\n            A[0] = -A[0]\\n            heapq.heapify(A)\\n        \\n        return sum(A)\\n    \\n    ### let's do with heap, no need to pop just negate that element\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K) :\\n            check = min(A)\\n            t = A.index(check)\\n            A[t] = -A[t]\\n        \\n        return sum(A)\\n            \\n        \\n        \\n        \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A[A.index(min(A))] *= -1\\n            \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        if not A: return sum(A)\\n        \\n        if K == 0: return sum(A)\\n        \\n        \\n        for i in range(K):\\n    \\n            A[A.index(min(A))] = - A[A.index(min(A))]\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if not A:\\n            return sum(A)\\n        if K==0:\\n            return sum(A)\\n        for i in range(K):\\n            A[A.index(min(A))] = -A[A.index(min(A))]\\n        return sum(A)\\n\", \"# 5:47 ->\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if K == 0:\\n            return sum(A)\\n        m = min(A)\\n        m_i = A.index(m)\\n        A[m_i] = -m\\n        return self.largestSumAfterKNegations(A, K - 1)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for item in A :\\n            if item < 0 and K > 1:\\n                A[A.index(item)] = item * (-1)\\n                K-= 1\\n            elif item > 0 and K >= 2:\\n                K-=2\\n        else:\\n            A.sort()\\n            while K != 0:\\n                K -=1\\n                A[0] = A[0] * (-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > 0:\\n                A.sort()\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A.sort()\\n            A[0] = -A[0]\\n        \\n        return sum(A)\", \"\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        ans = sum(A)\\n        A.sort()\\n        \\n        for i in range(len(A)):\\n            if not K or A[i]==0:\\n                break\\n            if A[i]<0:\\n                ans -= 2*A[i]\\n                K-=1\\n            else:\\n                if K%2:\\n                    if i and A[i]>abs(A[i-1]):\\n                        ans += 2*A[i-1]\\n                    else:\\n                        ans -= 2*A[i]\\n                break\\n                    \\n        return ans\"]",
        "difficulty": "introductory",
        "input": [
            [
                4,
                2,
                3
            ],
            1
        ],
        "output": 5,
        "halu_type": "Identification Hallucination",
        "fn_name": "largestSumAfterKNegations",
        "starter_code": "\nclass Solution:\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\n        ",
        "url": "https://leetcode.com/problems/maximize-sum-of-array-after-k-negations/"
    },
    {
        "id": 1244,
        "task_id": 4479,
        "test_case_id": 3,
        "question": "Given an array A of integers, we must modify the array in the following way: we choose an i and replace A[i] with -A[i], and we repeat this process K times in total.  (We may choose the same index i multiple times.)\nReturn the largest possible sum of the array after modifying it in this way.\n \nExample 1:\nInput: A = [4,2,3], K = 1\nOutput: 5\nExplanation: Choose indices (1,) and A becomes [4,-2,3].\n\n\nExample 2:\nInput: A = [3,-1,0,2], K = 3\nOutput: 6\nExplanation: Choose indices (1, 2, 2) and A becomes [3,1,0,2].\n\n\nExample 3:\nInput: A = [2,-3,-1,5,-4], K = 2\nOutput: 13\nExplanation: Choose indices (1, 4) and A becomes [2,3,-1,5,4].\n\n\n\n \nNote:\n\n1 <= A.length <= 10000\n1 <= K <= 10000\n-100 <= A[i] <= 100",
        "solutions": "[\"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        \\n        i = 0\\n        while A[i] < 0 and K > 0:\\n            A[i] *= -1\\n            i += 1\\n            K -= 1\\n            \\n        if K % 2 == 1 and 0 not in A:\\n            return sum(A) - 2*min(A)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        i=0\\n        while(i<len(A) and A[i]<0 and i<K):\\n            A[i]=-A[i]\\n            i+=1\\n        \\n        return sum(A)-(K - i) % 2 * min(A) * 2\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        S=0\\n        negs=[]\\n        posmin=100\\n        negmax=-100\\n        for i in range(len(A)):\\n            S+=A[i]\\n            if A[i]<0:\\n                negs.append(A[i])\\n                negmax=max(negmax,A[i])\\n            else:\\n                posmin=min(posmin,A[i])\\n        if K is 0:\\n            return(S)\\n        negs.sort()\\n        if K <= len(negs):\\n            return(S-2*sum(negs[:K]))\\n        elif (K - len(negs))%2 is 0:\\n            return(S-2*sum(negs[:K]))\\n        elif posmin<-negmax:\\n            return(S-2*sum(negs[:K])-2*posmin)\\n        else:\\n            return(S-2*sum(negs[:K])+2*negmax)\", \"class Solution(object):\\n    def largestSumAfterKNegations(self, A, K):\\n        lst = sorted(A)\\n        count = K\\n        i = 0\\n        while count>0:\\n            if lst[i] <= lst[i + 1]:\\n                lst[i] = lst[i] * -1\\n                count -= 1\\n            else:\\n                i += 1\\n        return sum(lst)\\n\", \"import heapq\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for _ in range(K):\\n            heapq.heapreplace(A, -A[0])\\n        return sum(A)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        p = 0\\n        while A[p] < 0:\\n            A[p] = - A[p]\\n            p += 1\\n            K -= 1\\n            if K == 0:\\n                 return sum(A)\\n        if K % 2 == 0:\\n            return sum(A)\\n        else:\\n            return sum(A) - 2 * min(A[p], A[p-1])\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        #heapify is in place algo so take linear time\\n        for x in range(K):\\n            #if x %2 :\\n                #heappreplace(A, -A[0])\\n            #heappush(A, -heapq.heappop(A))\\n            heapreplace(A, -A[0])\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], k: int) -> int:\\n        d = collections.defaultdict(int)\\n        for x in A:\\n            d[x] += 1\\n        \\n        res = 0\\n        for i in range(-100, 0):\\n            if i in d and k > 0:\\n                x = min(k, d[i])\\n                k -= x\\n                d[i] -= x\\n                d[-i] += x\\n            \\n        res = 0\\n        if k % 2 == 1:\\n            j = 0\\n            while j not in d or d[j] <= 0:\\n                j += 1\\n            print(j)\\n            d[j] -= 1\\n            d[-j] += 1\\n        for i in range(-100, 101):\\n            res += d[i] * i\\n        return res\\n        \\n            \\n                \\n                    \\n            \\n\\n                \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        res = 0\\n        A.sort()\\n        i = 0\\n        while i < len(A) and i < K and A[i] < 0:\\n            A[i] = -A[i]\\n            i += 1\\n\\n        return sum(A) - (K - i) % 2 * 2 * min(A[i-1:i] + A[i:i+1])\\n            \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n          A.sort()\\n          A[0] = -A[0]\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        result = 0\\n        count = 0\\n        A.sort()\\n       \\n        if A[0] > 0:\\n            if K%2 == 1:\\n                A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            return(result)\\n        elif A[0] == 0:\\n            for i in A:\\n                result += i\\n            return(result)\\n        else:\\n            for j in range (len(A)):\\n                if A[j] >= 0:\\n                    break\\n                else:\\n                    count += 1\\n            ra = K%count\\n            if K < count:\\n                for m in range(K):\\n                    A[m] = -A[m]\\n            else:\\n                for m in range(count):\\n                    A[m] = -A[m]\\n                K -= count\\n                A.sort()\\n                if K%2 == 1:\\n                    A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            \\n            return(result)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for i in range(K):\\n            m = heapq.heappop(A)\\n            heapq.heappush(A, -m)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        newA = sorted(A)\\n        print(newA)\\n        count = 0\\n        for i in range(K):\\n            newA[0] = -newA[0]\\n            newA = sorted(newA)\\n        print(newA)\\n        return sum(newA)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        B = sorted(A)\\n        for i in range(K):\\n            B[0] = -B[0]\\n            B = sorted(B)\\n        return sum(B)\", \"class Solution:\\n    def largestSumAfterKNegations(self, arr: List[int], k: int) -> int:\\n        ra = k\\n        arr.sort()\\n        for _ in range(ra):\\n            for j in range(len(arr)):\\n                if arr[j] < 0:\\n                    arr[j] = abs(arr[j])\\n                    k -= 1\\n                    arr.sort()\\n                    break\\n                elif arr[j] == 0:\\n                    return sum(arr)\\n                else:\\n                    if k % 2 == 0:\\n                        return sum(arr)\\n                    else:\\n                        arr.sort()\\n                        arr[0] = -arr[0]\\n                        return sum(arr)\\n\\n        return sum(arr)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K > 0:\\n            t = A.index(min(A))\\n            if A[t] < 0:\\n                A[t] = abs(A[t])\\n                K -= 1\\n                continue\\n            if A[t] == 0:\\n                K = 0\\n                continue\\n            if A[t] > 0:\\n                if K % 2 == 0:\\n                    K = 0\\n                    continue\\n                else:\\n                    A[t] = -A[t]\\n                    K = 0\\n                    continue\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for x in range(K):\\n            minNum=A.index(min(A))\\n            A[minNum]=A[minNum]*(-1)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            variable = min(A)\\n            A.remove(variable)\\n            A.append(variable*-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n        \\n            A.remove(val)\\n            A.append(-1 * val)\\n                                \\n        return sum(A)\\n                \\n               \\n\", \"# 5:47 -> 5:50 slowwww\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K:\\n            m = min(A)\\n            m_i = A.index(m)\\n            A[m_i] = -m\\n            K -= 1\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n\\n        for i in range(K):\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        a1 =A\\n        sum1 =0\\n        while K > 0 :\\n            mina = min(a1)\\n            a1.remove(mina)\\n            a1.append(mina * (-1))\\n            K -=1\\n            \\n    \\n        \\n        return sum(a1)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > A[1]:\\n                A.append(A.pop(0))\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        cur = sum(A)\\n        for i in range(K):\\n           # max_ = sum(A)\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        max_sum = -999999\\n        for _ in range(K):\\n            A = sorted(A)\\n            min_index = A.index(min(A))\\n            \\n            A[min_index] = -A[min_index]\\n\\n            max_ = sum(A)\\n            # print(max_)\\n            # max_sum = max(max_sum, max_)\\n        return max_\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        while K > 0:\\n            min_index = A.index(min(A))\\n            A[min_index] = - A[min_index]\\n            K = K - 1\\n        \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n            \\n            #if val <= 0:\\n            A.remove(val)\\n            A.append(-1 * val)\\n                \\n           \\n                \\n        return sum(A)\\n                \\n               \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K != 0:\\n            A[A.index(min(A))] = -min(A)\\n            K -= 1\\n        return(sum(A))\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        c = 0\\n        \\n        while c < K:\\n            c += 1\\n            min_ = min(A)\\n            index = A.index(min_)\\n            A[index] = -1 * min_\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K>0:\\n            K-=1\\n            A[A.index(min(A))]=-min(A)\\n        return sum(A)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        ## find the min element and its index and negate it K times\\n        \\n        ## Complexity = O(kn)\\n        \\n#         def findMin(A):\\n#             minValue = float('inf')\\n#             minIndex = 0\\n\\n#             for i, ele in enumerate(A):\\n#                 if ele < minValue:\\n#                     minValue = ele\\n#                     minIndex = i\\n            \\n#             return minIndex\\n\\n#         for k in range(K):\\n#             minIndex = findMin(A)\\n#             A[minIndex] = -A[minIndex]\\n            \\n        \\n        ### using min heap\\n        \\n        heapq.heapify(A)\\n        for k in range(K):\\n            A[0] = -A[0]\\n            heapq.heapify(A)\\n        \\n        return sum(A)\\n    \\n    ### let's do with heap, no need to pop just negate that element\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K) :\\n            check = min(A)\\n            t = A.index(check)\\n            A[t] = -A[t]\\n        \\n        return sum(A)\\n            \\n        \\n        \\n        \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A[A.index(min(A))] *= -1\\n            \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        if not A: return sum(A)\\n        \\n        if K == 0: return sum(A)\\n        \\n        \\n        for i in range(K):\\n    \\n            A[A.index(min(A))] = - A[A.index(min(A))]\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if not A:\\n            return sum(A)\\n        if K==0:\\n            return sum(A)\\n        for i in range(K):\\n            A[A.index(min(A))] = -A[A.index(min(A))]\\n        return sum(A)\\n\", \"# 5:47 ->\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if K == 0:\\n            return sum(A)\\n        m = min(A)\\n        m_i = A.index(m)\\n        A[m_i] = -m\\n        return self.largestSumAfterKNegations(A, K - 1)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for item in A :\\n            if item < 0 and K > 1:\\n                A[A.index(item)] = item * (-1)\\n                K-= 1\\n            elif item > 0 and K >= 2:\\n                K-=2\\n        else:\\n            A.sort()\\n            while K != 0:\\n                K -=1\\n                A[0] = A[0] * (-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > 0:\\n                A.sort()\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A.sort()\\n            A[0] = -A[0]\\n        \\n        return sum(A)\", \"\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        ans = sum(A)\\n        A.sort()\\n        \\n        for i in range(len(A)):\\n            if not K or A[i]==0:\\n                break\\n            if A[i]<0:\\n                ans -= 2*A[i]\\n                K-=1\\n            else:\\n                if K%2:\\n                    if i and A[i]>abs(A[i-1]):\\n                        ans += 2*A[i-1]\\n                    else:\\n                        ans -= 2*A[i]\\n                break\\n                    \\n        return ans\"]",
        "difficulty": "introductory",
        "input": [
            [
                3,
                -1,
                0,
                2
            ],
            3
        ],
        "output": 6,
        "halu_type": "Identification Hallucination",
        "fn_name": "largestSumAfterKNegations",
        "starter_code": "\nclass Solution:\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\n        ",
        "url": "https://leetcode.com/problems/maximize-sum-of-array-after-k-negations/"
    },
    {
        "id": 1245,
        "task_id": 4479,
        "test_case_id": 4,
        "question": "Given an array A of integers, we must modify the array in the following way: we choose an i and replace A[i] with -A[i], and we repeat this process K times in total.  (We may choose the same index i multiple times.)\nReturn the largest possible sum of the array after modifying it in this way.\n \nExample 1:\nInput: A = [4,2,3], K = 1\nOutput: 5\nExplanation: Choose indices (1,) and A becomes [4,-2,3].\n\n\nExample 2:\nInput: A = [3,-1,0,2], K = 3\nOutput: 6\nExplanation: Choose indices (1, 2, 2) and A becomes [3,1,0,2].\n\n\nExample 3:\nInput: A = [2,-3,-1,5,-4], K = 2\nOutput: 13\nExplanation: Choose indices (1, 4) and A becomes [2,3,-1,5,4].\n\n\n\n \nNote:\n\n1 <= A.length <= 10000\n1 <= K <= 10000\n-100 <= A[i] <= 100",
        "solutions": "[\"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        \\n        i = 0\\n        while A[i] < 0 and K > 0:\\n            A[i] *= -1\\n            i += 1\\n            K -= 1\\n            \\n        if K % 2 == 1 and 0 not in A:\\n            return sum(A) - 2*min(A)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        i=0\\n        while(i<len(A) and A[i]<0 and i<K):\\n            A[i]=-A[i]\\n            i+=1\\n        \\n        return sum(A)-(K - i) % 2 * min(A) * 2\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        S=0\\n        negs=[]\\n        posmin=100\\n        negmax=-100\\n        for i in range(len(A)):\\n            S+=A[i]\\n            if A[i]<0:\\n                negs.append(A[i])\\n                negmax=max(negmax,A[i])\\n            else:\\n                posmin=min(posmin,A[i])\\n        if K is 0:\\n            return(S)\\n        negs.sort()\\n        if K <= len(negs):\\n            return(S-2*sum(negs[:K]))\\n        elif (K - len(negs))%2 is 0:\\n            return(S-2*sum(negs[:K]))\\n        elif posmin<-negmax:\\n            return(S-2*sum(negs[:K])-2*posmin)\\n        else:\\n            return(S-2*sum(negs[:K])+2*negmax)\", \"class Solution(object):\\n    def largestSumAfterKNegations(self, A, K):\\n        lst = sorted(A)\\n        count = K\\n        i = 0\\n        while count>0:\\n            if lst[i] <= lst[i + 1]:\\n                lst[i] = lst[i] * -1\\n                count -= 1\\n            else:\\n                i += 1\\n        return sum(lst)\\n\", \"import heapq\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for _ in range(K):\\n            heapq.heapreplace(A, -A[0])\\n        return sum(A)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        p = 0\\n        while A[p] < 0:\\n            A[p] = - A[p]\\n            p += 1\\n            K -= 1\\n            if K == 0:\\n                 return sum(A)\\n        if K % 2 == 0:\\n            return sum(A)\\n        else:\\n            return sum(A) - 2 * min(A[p], A[p-1])\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        #heapify is in place algo so take linear time\\n        for x in range(K):\\n            #if x %2 :\\n                #heappreplace(A, -A[0])\\n            #heappush(A, -heapq.heappop(A))\\n            heapreplace(A, -A[0])\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], k: int) -> int:\\n        d = collections.defaultdict(int)\\n        for x in A:\\n            d[x] += 1\\n        \\n        res = 0\\n        for i in range(-100, 0):\\n            if i in d and k > 0:\\n                x = min(k, d[i])\\n                k -= x\\n                d[i] -= x\\n                d[-i] += x\\n            \\n        res = 0\\n        if k % 2 == 1:\\n            j = 0\\n            while j not in d or d[j] <= 0:\\n                j += 1\\n            print(j)\\n            d[j] -= 1\\n            d[-j] += 1\\n        for i in range(-100, 101):\\n            res += d[i] * i\\n        return res\\n        \\n            \\n                \\n                    \\n            \\n\\n                \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        res = 0\\n        A.sort()\\n        i = 0\\n        while i < len(A) and i < K and A[i] < 0:\\n            A[i] = -A[i]\\n            i += 1\\n\\n        return sum(A) - (K - i) % 2 * 2 * min(A[i-1:i] + A[i:i+1])\\n            \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n          A.sort()\\n          A[0] = -A[0]\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        result = 0\\n        count = 0\\n        A.sort()\\n       \\n        if A[0] > 0:\\n            if K%2 == 1:\\n                A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            return(result)\\n        elif A[0] == 0:\\n            for i in A:\\n                result += i\\n            return(result)\\n        else:\\n            for j in range (len(A)):\\n                if A[j] >= 0:\\n                    break\\n                else:\\n                    count += 1\\n            ra = K%count\\n            if K < count:\\n                for m in range(K):\\n                    A[m] = -A[m]\\n            else:\\n                for m in range(count):\\n                    A[m] = -A[m]\\n                K -= count\\n                A.sort()\\n                if K%2 == 1:\\n                    A[0] = -A[0]\\n            for i in A:\\n                result += i\\n            \\n            return(result)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        heapq.heapify(A)\\n        for i in range(K):\\n            m = heapq.heappop(A)\\n            heapq.heappush(A, -m)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        newA = sorted(A)\\n        print(newA)\\n        count = 0\\n        for i in range(K):\\n            newA[0] = -newA[0]\\n            newA = sorted(newA)\\n        print(newA)\\n        return sum(newA)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        B = sorted(A)\\n        for i in range(K):\\n            B[0] = -B[0]\\n            B = sorted(B)\\n        return sum(B)\", \"class Solution:\\n    def largestSumAfterKNegations(self, arr: List[int], k: int) -> int:\\n        ra = k\\n        arr.sort()\\n        for _ in range(ra):\\n            for j in range(len(arr)):\\n                if arr[j] < 0:\\n                    arr[j] = abs(arr[j])\\n                    k -= 1\\n                    arr.sort()\\n                    break\\n                elif arr[j] == 0:\\n                    return sum(arr)\\n                else:\\n                    if k % 2 == 0:\\n                        return sum(arr)\\n                    else:\\n                        arr.sort()\\n                        arr[0] = -arr[0]\\n                        return sum(arr)\\n\\n        return sum(arr)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K > 0:\\n            t = A.index(min(A))\\n            if A[t] < 0:\\n                A[t] = abs(A[t])\\n                K -= 1\\n                continue\\n            if A[t] == 0:\\n                K = 0\\n                continue\\n            if A[t] > 0:\\n                if K % 2 == 0:\\n                    K = 0\\n                    continue\\n                else:\\n                    A[t] = -A[t]\\n                    K = 0\\n                    continue\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for x in range(K):\\n            minNum=A.index(min(A))\\n            A[minNum]=A[minNum]*(-1)\\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            variable = min(A)\\n            A.remove(variable)\\n            A.append(variable*-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n        \\n            A.remove(val)\\n            A.append(-1 * val)\\n                                \\n        return sum(A)\\n                \\n               \\n\", \"# 5:47 -> 5:50 slowwww\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K:\\n            m = min(A)\\n            m_i = A.index(m)\\n            A[m_i] = -m\\n            K -= 1\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n\\n        for i in range(K):\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        a1 =A\\n        sum1 =0\\n        while K > 0 :\\n            mina = min(a1)\\n            a1.remove(mina)\\n            a1.append(mina * (-1))\\n            K -=1\\n            \\n    \\n        \\n        return sum(a1)\\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > A[1]:\\n                A.append(A.pop(0))\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        cur = sum(A)\\n        for i in range(K):\\n           # max_ = sum(A)\\n            min_ = min(A)\\n            idx = A.index(min_)\\n            A[idx] = -A[idx]\\n            max_ = sum(A)\\n        return max_\\n            \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        max_sum = -999999\\n        for _ in range(K):\\n            A = sorted(A)\\n            min_index = A.index(min(A))\\n            \\n            A[min_index] = -A[min_index]\\n\\n            max_ = sum(A)\\n            # print(max_)\\n            # max_sum = max(max_sum, max_)\\n        return max_\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        while K > 0:\\n            min_index = A.index(min(A))\\n            A[min_index] = - A[min_index]\\n            K = K - 1\\n        \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K):\\n            \\n            val = min(A)\\n            \\n            #if val <= 0:\\n            A.remove(val)\\n            A.append(-1 * val)\\n                \\n           \\n                \\n        return sum(A)\\n                \\n               \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K != 0:\\n            A[A.index(min(A))] = -min(A)\\n            K -= 1\\n        return(sum(A))\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        c = 0\\n        \\n        while c < K:\\n            c += 1\\n            min_ = min(A)\\n            index = A.index(min_)\\n            A[index] = -1 * min_\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        while K>0:\\n            K-=1\\n            A[A.index(min(A))]=-min(A)\\n        return sum(A)\", \"class Solution:\\n    import heapq\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        ## find the min element and its index and negate it K times\\n        \\n        ## Complexity = O(kn)\\n        \\n#         def findMin(A):\\n#             minValue = float('inf')\\n#             minIndex = 0\\n\\n#             for i, ele in enumerate(A):\\n#                 if ele < minValue:\\n#                     minValue = ele\\n#                     minIndex = i\\n            \\n#             return minIndex\\n\\n#         for k in range(K):\\n#             minIndex = findMin(A)\\n#             A[minIndex] = -A[minIndex]\\n            \\n        \\n        ### using min heap\\n        \\n        heapq.heapify(A)\\n        for k in range(K):\\n            A[0] = -A[0]\\n            heapq.heapify(A)\\n        \\n        return sum(A)\\n    \\n    ### let's do with heap, no need to pop just negate that element\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        for i in range(K) :\\n            check = min(A)\\n            t = A.index(check)\\n            A[t] = -A[t]\\n        \\n        return sum(A)\\n            \\n        \\n        \\n        \\n        \\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A[A.index(min(A))] *= -1\\n            \\n        return sum(A)\\n\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        \\n        if not A: return sum(A)\\n        \\n        if K == 0: return sum(A)\\n        \\n        \\n        for i in range(K):\\n    \\n            A[A.index(min(A))] = - A[A.index(min(A))]\\n        \\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if not A:\\n            return sum(A)\\n        if K==0:\\n            return sum(A)\\n        for i in range(K):\\n            A[A.index(min(A))] = -A[A.index(min(A))]\\n        return sum(A)\\n\", \"# 5:47 ->\\n# Seems greedy: pick smallest number and flip it\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        if K == 0:\\n            return sum(A)\\n        m = min(A)\\n        m_i = A.index(m)\\n        A[m_i] = -m\\n        return self.largestSumAfterKNegations(A, K - 1)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for item in A :\\n            if item < 0 and K > 1:\\n                A[A.index(item)] = item * (-1)\\n                K-= 1\\n            elif item > 0 and K >= 2:\\n                K-=2\\n        else:\\n            A.sort()\\n            while K != 0:\\n                K -=1\\n                A[0] = A[0] * (-1)\\n        return sum(A)\", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        A.sort()\\n        for i in range(K):\\n            A[0] = -1 * A[0]\\n            if A[0] > 0:\\n                A.sort()\\n        return sum(A)        \", \"class Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        for i in range(K):\\n            A.sort()\\n            A[0] = -A[0]\\n        \\n        return sum(A)\", \"\\nclass Solution:\\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\\n        ans = sum(A)\\n        A.sort()\\n        \\n        for i in range(len(A)):\\n            if not K or A[i]==0:\\n                break\\n            if A[i]<0:\\n                ans -= 2*A[i]\\n                K-=1\\n            else:\\n                if K%2:\\n                    if i and A[i]>abs(A[i-1]):\\n                        ans += 2*A[i-1]\\n                    else:\\n                        ans -= 2*A[i]\\n                break\\n                    \\n        return ans\"]",
        "difficulty": "introductory",
        "input": [
            [
                2,
                -3,
                -1,
                5,
                -4
            ],
            2
        ],
        "output": 13,
        "halu_type": "Identification Hallucination",
        "fn_name": "largestSumAfterKNegations",
        "starter_code": "\nclass Solution:\n    def largestSumAfterKNegations(self, A: List[int], K: int) -> int:\n        ",
        "url": "https://leetcode.com/problems/maximize-sum-of-array-after-k-negations/"
    },
    {
        "id": 1246,
        "task_id": 4659,
        "test_case_id": 1,
        "question": "Given a non-negative integer numRows, generate the first numRows of Pascal's triangle.\n\n\nIn Pascal's triangle, each number is the sum of the two numbers directly above it.\n\nExample:\n\n\nInput: 5\nOutput:\n[\n     [1],\n    [1,1],\n   [1,2,1],\n  [1,3,3,1],\n [1,4,6,4,1]\n]",
        "solutions": "[\"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = [[1]]\\n         for i in range(1,numRows):\\n             res.append(list(map(lambda x, y : x + y, res[-1]+[0], [0]+res[-1])))\\n         return res[:numRows]\\n          \\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         if numRows==0:\\n             return []\\n         tem=[0,1]\\n         l=[]\\n         for i in range(numRows):\\n             rowlist=[]\\n             for j in range(len(tem)-1):\\n                 rowlist.append(tem[j]+tem[j+1])\\n             l.append(rowlist)\\n             tem=rowlist[:]\\n             tem.insert(0,0)\\n             tem.append(0)\\n         return  l\\n         '''\\n         ans = [[1] * n for n in range(1, numRows + 1)]\\n         for i in range(1, numRows + 1):\\n             for j in range(0, i - 2):\\n                 ans[i - 1][1 + j] = ans[i - 2][j] + ans[i - 2][j + 1]\\n         return ans'''\\n     \\n             \\n \\n \\n \\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows==0:\\n             return []\\n         res=[[1]]\\n         for i in range(1,numRows):\\n             newRow = [1]\\n             for j in range(1,i):\\n                 newRow.append(res[-1][j-1]+res[-1][j])\\n             newRow.append(1)\\n             res.append(newRow)\\n         return res\\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         '''\\n         if numRows==0:\\n            return []\\n         tem=[0,1]\\n         l=[]\\n         for i in range(numRows):\\n            rowlist=[]\\n             for j in range(len(tem)-1):\\n                rowlist.append(tem[j]+tem[j+1])\\n             l.append(rowlist)\\n             tem=rowlist[:]\\n             tem.insert(0,0)\\n             tem.append(0)\\n         return  l'''\\n         ans = [[1] * n for n in range(1, numRows + 1)]\\n         for i in range(1, numRows + 1):\\n             for j in range(0, i - 2):\\n                 ans[i - 1][1 + j] = ans[i - 2][j] + ans[i - 2][j + 1]\\n         return ans\\n     \\n             \\n \\n \\n \\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         A = []\\n         for i in range(numRows):\\n             A.append([1 for _ in range(i+1)])\\n             for j in range(1, i):\\n                 A[i][j] = A[i-1][j-1] + A[i-1][j]\\n         return A\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows < 1:\\n             return []\\n         base = [1]\\n         triangle = [base]\\n         for i in range(1, numRows):\\n             new_row = [1]\\n             prev_row = triangle[i-1]\\n             for j in range(len(prev_row)-1):\\n                 new_row.append(prev_row[j] + prev_row[j+1])\\n             new_row.append(1)\\n             triangle.append(new_row)\\n         return triangle\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         A=[[1],[1,1]]\\n         if numRows == 0:\\n             return[]\\n         elif numRows<3:\\n             return A[0:numRows]\\n         else:\\n             i=3\\n             while i<=numRows:\\n                 temp=A[i-2][:]\\n                 temp.append(0)\\n                 temp2=temp[::-1]\\n                 temp3=[X+Y for X,Y in zip(temp,temp2)]\\n                 A.append(temp3) \\n                 i+=1          \\n         print(A)\\n         return A\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = []\\n         for i in range(numRows):\\n             result.append([])\\n             for j in range(i + 1):\\n                 if j in (0, i):\\n                     result[i].append(1)\\n                 else:\\n                     result[i].append(result[i - 1][j - 1] + result[i - 1][j])\\n         return result\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n \\n         result = [[]] * numRows\\n         for i in range(numRows):\\n             if i == 0:\\n                 result[0] = [1]\\n                 \\n             if i == 1:\\n                 result[1] = [1, 1]\\n                 \\n             if i > 1:\\n                 temp = []\\n                 for k in range(i+1):                    \\n                     left =  result[i-1][k-1] if k >= 1 else 0\\n                     right = result[i-1][k] if k < i else 0\\n                     temp.append(left + right)\\n                 \\n                 result[i] = temp\\n                 \\n                 \\n         return result\", \"class Solution:\\n     def my_fact(self, n):\\n         if n == 1 or n == 0:\\n             return 1\\n         else:\\n             return n * self.my_fact(n-1)\\n         \\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = [[] for i in range(numRows)]\\n         for n in range(numRows):\\n             for k in range(0, n+1):\\n                 val = self.my_fact(n) / (self.my_fact(n-k) * self.my_fact(k))\\n                 res[n].append(int(val))\\n                 \\n         return res\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows == 0:\\n             return []\\n         result = [[1]]\\n         if numRows == 1:\\n             return result\\n         result.append([1, 1])\\n         if numRows == 2:\\n             return result\\n         for row in range(2, numRows):\\n             l, r = 0, 1\\n             tmp = [1]\\n             while r < len(result[row-1]):\\n                 tmp.append(result[row-1][l]+result[row-1][r])\\n                 l += 1\\n                 r += 1\\n             tmp.append(1)\\n             result.append(tmp)\\n         return result\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows==0:\\n             return []\\n         if numRows==1:\\n             return [[1]]\\n         if numRows==2:\\n             return [[1],[1,1]]\\n         result=[[1],[1,1]]\\n         for i in range(3,numRows+1):\\n             if len(result)<i:\\n                 result.append([])\\n             result[i-1].append(1)\\n             count=0\\n             while len(result[i-1])<i-1:\\n                 result[i-1].append(result[i-2][count]+result[i-2][count+1])\\n                 count+=1\\n             result[i-1].append(1)\\n         return result\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n #         def fac(num):\\n #             factorial = 1\\n #             for i in range(1, num+1):\\n #                 factorial *= i\\n #             return factorial\\n         \\n #         i = 1\\n #         L = []\\n #         while i <= numRows:\\n #             j = 1\\n #             l = []\\n #             while j <= i:\\n #                 l.append(int(fac(i-1)/(fac(j-1)*fac(i-j))))\\n #                 j += 1\\n #             L.append(l)\\n #             i += 1\\n             \\n #         return L\\n \\n             \\n #         if numRows == 0:\\n #             return []\\n         \\n #         i = 1\\n #         L = [[1]]\\n #         while i+1 <= numRows:\\n #             j = 0\\n #             l = []\\n #             while j <= i:\\n #                 if j == 0 or j == i:\\n #                     l.append(1)\\n #                 else:\\n #                     l.append(L[i-1][j-1]+L[i-1][j])\\n #                 j += 1\\n #             L.append(l)\\n #             i += 1\\n             \\n #         return L\\n \\n     \\n         # result = []\\n         # for i in xrange(numRows):\\n         #     result.append([])\\n         #     for j in xrange(i + 1):\\n         #         if j in (0, i):\\n         #             result[i].append(1)\\n         #         else:\\n         #             result[i].append(result[i - 1][j - 1] + result[i - 1][j])\\n         # return result\\n     \\n     \\n         # if not numRows: return []\\n         # res = [[1]]\\n         # for i in range(1, numRows):\\n         #     res += [list(map(lambda x, y: x + y, res[-1] + [0], [0] + res[-1]))]\\n         # return res[:numRows]\\n     \\n         \\n         if numRows == 0: return []\\n         if numRows == 1: return [[1]]\\n         res = [[1], [1, 1]]\\n \\n         def add(nums):\\n             res = nums[:1]\\n             for i, j in enumerate(nums):\\n                 if i < len(nums) - 1:\\n                     res += [nums[i] + nums[i + 1]]\\n             res += nums[:1]\\n             return res\\n \\n         while len(res) < numRows:\\n             # res.extend([add(res[-1])])\\n             res.append(add(res[-1]))\\n         return res\"]",
        "difficulty": "introductory",
        "input": [
            5
        ],
        "output": [
            [
                1
            ],
            [
                1,
                1
            ],
            [
                1,
                2,
                1
            ],
            [
                1,
                3,
                3,
                1
            ],
            [
                1,
                4,
                6,
                4,
                1
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "generate",
        "starter_code": "\nclass Solution:\n    def generate(self, numRows: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/pascals-triangle/"
    },
    {
        "id": 1247,
        "task_id": 4659,
        "test_case_id": 2,
        "question": "Given a non-negative integer numRows, generate the first numRows of Pascal's triangle.\n\n\nIn Pascal's triangle, each number is the sum of the two numbers directly above it.\n\nExample:\n\n\nInput: 5\nOutput:\n[\n     [1],\n    [1,1],\n   [1,2,1],\n  [1,3,3,1],\n [1,4,6,4,1]\n]",
        "solutions": "[\"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = [[1]]\\n         for i in range(1,numRows):\\n             res.append(list(map(lambda x, y : x + y, res[-1]+[0], [0]+res[-1])))\\n         return res[:numRows]\\n          \\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         if numRows==0:\\n             return []\\n         tem=[0,1]\\n         l=[]\\n         for i in range(numRows):\\n             rowlist=[]\\n             for j in range(len(tem)-1):\\n                 rowlist.append(tem[j]+tem[j+1])\\n             l.append(rowlist)\\n             tem=rowlist[:]\\n             tem.insert(0,0)\\n             tem.append(0)\\n         return  l\\n         '''\\n         ans = [[1] * n for n in range(1, numRows + 1)]\\n         for i in range(1, numRows + 1):\\n             for j in range(0, i - 2):\\n                 ans[i - 1][1 + j] = ans[i - 2][j] + ans[i - 2][j + 1]\\n         return ans'''\\n     \\n             \\n \\n \\n \\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows==0:\\n             return []\\n         res=[[1]]\\n         for i in range(1,numRows):\\n             newRow = [1]\\n             for j in range(1,i):\\n                 newRow.append(res[-1][j-1]+res[-1][j])\\n             newRow.append(1)\\n             res.append(newRow)\\n         return res\\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         '''\\n         if numRows==0:\\n            return []\\n         tem=[0,1]\\n         l=[]\\n         for i in range(numRows):\\n            rowlist=[]\\n             for j in range(len(tem)-1):\\n                rowlist.append(tem[j]+tem[j+1])\\n             l.append(rowlist)\\n             tem=rowlist[:]\\n             tem.insert(0,0)\\n             tem.append(0)\\n         return  l'''\\n         ans = [[1] * n for n in range(1, numRows + 1)]\\n         for i in range(1, numRows + 1):\\n             for j in range(0, i - 2):\\n                 ans[i - 1][1 + j] = ans[i - 2][j] + ans[i - 2][j + 1]\\n         return ans\\n     \\n             \\n \\n \\n \\n\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         A = []\\n         for i in range(numRows):\\n             A.append([1 for _ in range(i+1)])\\n             for j in range(1, i):\\n                 A[i][j] = A[i-1][j-1] + A[i-1][j]\\n         return A\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows < 1:\\n             return []\\n         base = [1]\\n         triangle = [base]\\n         for i in range(1, numRows):\\n             new_row = [1]\\n             prev_row = triangle[i-1]\\n             for j in range(len(prev_row)-1):\\n                 new_row.append(prev_row[j] + prev_row[j+1])\\n             new_row.append(1)\\n             triangle.append(new_row)\\n         return triangle\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         A=[[1],[1,1]]\\n         if numRows == 0:\\n             return[]\\n         elif numRows<3:\\n             return A[0:numRows]\\n         else:\\n             i=3\\n             while i<=numRows:\\n                 temp=A[i-2][:]\\n                 temp.append(0)\\n                 temp2=temp[::-1]\\n                 temp3=[X+Y for X,Y in zip(temp,temp2)]\\n                 A.append(temp3) \\n                 i+=1          \\n         print(A)\\n         return A\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = []\\n         for i in range(numRows):\\n             result.append([])\\n             for j in range(i + 1):\\n                 if j in (0, i):\\n                     result[i].append(1)\\n                 else:\\n                     result[i].append(result[i - 1][j - 1] + result[i - 1][j])\\n         return result\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n \\n         result = [[]] * numRows\\n         for i in range(numRows):\\n             if i == 0:\\n                 result[0] = [1]\\n                 \\n             if i == 1:\\n                 result[1] = [1, 1]\\n                 \\n             if i > 1:\\n                 temp = []\\n                 for k in range(i+1):                    \\n                     left =  result[i-1][k-1] if k >= 1 else 0\\n                     right = result[i-1][k] if k < i else 0\\n                     temp.append(left + right)\\n                 \\n                 result[i] = temp\\n                 \\n                 \\n         return result\", \"class Solution:\\n     def my_fact(self, n):\\n         if n == 1 or n == 0:\\n             return 1\\n         else:\\n             return n * self.my_fact(n-1)\\n         \\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = [[] for i in range(numRows)]\\n         for n in range(numRows):\\n             for k in range(0, n+1):\\n                 val = self.my_fact(n) / (self.my_fact(n-k) * self.my_fact(k))\\n                 res[n].append(int(val))\\n                 \\n         return res\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows == 0:\\n             return []\\n         result = [[1]]\\n         if numRows == 1:\\n             return result\\n         result.append([1, 1])\\n         if numRows == 2:\\n             return result\\n         for row in range(2, numRows):\\n             l, r = 0, 1\\n             tmp = [1]\\n             while r < len(result[row-1]):\\n                 tmp.append(result[row-1][l]+result[row-1][r])\\n                 l += 1\\n                 r += 1\\n             tmp.append(1)\\n             result.append(tmp)\\n         return result\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if numRows==0:\\n             return []\\n         if numRows==1:\\n             return [[1]]\\n         if numRows==2:\\n             return [[1],[1,1]]\\n         result=[[1],[1,1]]\\n         for i in range(3,numRows+1):\\n             if len(result)<i:\\n                 result.append([])\\n             result[i-1].append(1)\\n             count=0\\n             while len(result[i-1])<i-1:\\n                 result[i-1].append(result[i-2][count]+result[i-2][count+1])\\n                 count+=1\\n             result[i-1].append(1)\\n         return result\", \"class Solution:\\n     def generate(self, numRows):\\n         \\\"\\\"\\\"\\n         :type numRows: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n #         def fac(num):\\n #             factorial = 1\\n #             for i in range(1, num+1):\\n #                 factorial *= i\\n #             return factorial\\n         \\n #         i = 1\\n #         L = []\\n #         while i <= numRows:\\n #             j = 1\\n #             l = []\\n #             while j <= i:\\n #                 l.append(int(fac(i-1)/(fac(j-1)*fac(i-j))))\\n #                 j += 1\\n #             L.append(l)\\n #             i += 1\\n             \\n #         return L\\n \\n             \\n #         if numRows == 0:\\n #             return []\\n         \\n #         i = 1\\n #         L = [[1]]\\n #         while i+1 <= numRows:\\n #             j = 0\\n #             l = []\\n #             while j <= i:\\n #                 if j == 0 or j == i:\\n #                     l.append(1)\\n #                 else:\\n #                     l.append(L[i-1][j-1]+L[i-1][j])\\n #                 j += 1\\n #             L.append(l)\\n #             i += 1\\n             \\n #         return L\\n \\n     \\n         # result = []\\n         # for i in xrange(numRows):\\n         #     result.append([])\\n         #     for j in xrange(i + 1):\\n         #         if j in (0, i):\\n         #             result[i].append(1)\\n         #         else:\\n         #             result[i].append(result[i - 1][j - 1] + result[i - 1][j])\\n         # return result\\n     \\n     \\n         # if not numRows: return []\\n         # res = [[1]]\\n         # for i in range(1, numRows):\\n         #     res += [list(map(lambda x, y: x + y, res[-1] + [0], [0] + res[-1]))]\\n         # return res[:numRows]\\n     \\n         \\n         if numRows == 0: return []\\n         if numRows == 1: return [[1]]\\n         res = [[1], [1, 1]]\\n \\n         def add(nums):\\n             res = nums[:1]\\n             for i, j in enumerate(nums):\\n                 if i < len(nums) - 1:\\n                     res += [nums[i] + nums[i + 1]]\\n             res += nums[:1]\\n             return res\\n \\n         while len(res) < numRows:\\n             # res.extend([add(res[-1])])\\n             res.append(add(res[-1]))\\n         return res\"]",
        "difficulty": "introductory",
        "input": [
            1
        ],
        "output": [
            [
                1
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "generate",
        "starter_code": "\nclass Solution:\n    def generate(self, numRows: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/pascals-triangle/"
    },
    {
        "id": 1248,
        "task_id": 2628,
        "test_case_id": 1,
        "question": "The gray code is a binary numeral system where two successive values differ in only one bit.\n\nGiven a non-negative integer n representing the total number of bits in the code, print the sequence of gray code. A gray code sequence must begin with 0.\n\nExample 1:\n\n\nInput: 2\nOutput: [0,1,3,2]\nExplanation:\n00 - 0\n01 - 1\n11 - 3\n10 - 2\n\nFor a given n, a gray code sequence may not be uniquely defined.\nFor example, [0,2,3,1] is also a valid gray code sequence.\n\n00 - 0\n10 - 2\n11 - 3\n01 - 1\n\n\nExample 2:\n\n\nInput: 0\nOutput: [0]\nExplanation: We define the gray code sequence to begin with 0.\n             A gray code sequence of n has size = 2n, which for n = 0 the size is 20 = 1.\n             Therefore, for n = 0 the gray code sequence is [0].",
        "solutions": "[\"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = []\\n         for i in range(1<<n):\\n             res.append(i ^ i >>1)\\n         return res\\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n               \\n         result = [0, 1]\\n         if n <= 1:\\n             return result[:n+1]\\n         res_len = 2 ** n\\n         \\n         cnt = 1\\n         while len(result) != res_len:\\n             cnt *= 2\\n             result.extend(result[::-1])\\n             #offset = 0 \\n             start = len(result) // 2\\n             for i in range(start, len(result)):\\n                 #if i > 0 and i % cnt == 0:\\n                 #    offset = cnt;\\n                 #result[i] += offset \\n                 result[i] += cnt\\n             \\n         return result\", \"class Solution:\\n     def grayCode(self, n):\\n         \\n         if n == 0:\\n             return [0]\\n         \\n         concat = ''\\n         \\n         res = [0, 1]\\n         \\n         for i in range(1, n):\\n             newRes = []\\n             \\n             for j in res:\\n                 newRes.append(j)\\n             for j in reversed(res):\\n                 newRes.append(j + (2 ** i))\\n                 \\n             res = newRes\\n             \\n         return res\\n             \\n             \\n         \\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [0]\\n         result = [0, 1]\\n         \\n         # mirror the previous layer\\n         # then padding '1' at the most significant bit position --- for the second half\\n         # 0 1  --> 00 01 | 11 10 (1 + 1; 1 + 0)\\n         \\n         for i in range(2, n + 1):\\n             mask = 1 << (i - 1)\\n             temp = []\\n             for j in range(len(result)):\\n                 temp.append(result[-j - 1] | mask)\\n             result += temp\\n         return result\\n         \\n\", \"class Solution:\\n     def helper(self, n):\\n         if n == 0:\\n             return ['0']\\n         if n == 1:\\n             return ['0', '1']\\n         ret = []\\n         for code in self.helper(n - 1):\\n             ret.append('0' + code)\\n         for code in reversed(self.helper(n - 1)):\\n             ret.append('1' + code)\\n         return ret\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [0]\\n         ret = []\\n         code = self.grayCode(n - 1)\\n         ret += code\\n         for v in reversed(code):\\n             ret.append(2 ** (n - 1) + v)\\n         return ret\\n         \\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = [0]\\n         for i in range(n):\\n             temp = [] \\n             for num in result:\\n                 temp.append(num + 2**i)\\n             result += temp[::-1]\\n             \\n         return result\\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [0]\\n     \\n         q = [0, -1]\\n     \\n         for i in range(n):\\n             tag = 0\\n             while q[0] != -1:\\n                 item = q.pop(0)\\n                 q.append(item * 2 + tag)\\n                 q.append(item * 2 + (1 - tag))\\n                 tag = 1 - tag\\n             q.pop(0)\\n             q.append(-1)\\n     \\n         q.pop(-1)\\n         \\n         return q\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n               \\n         result = [0, 1]\\n         if n <= 1:\\n             return result[:n+1]\\n         res_len = 2 ** n\\n         \\n         cnt = 1\\n         while len(result) != res_len:\\n             cnt *= 2\\n             #orig_len = len(result)\\n             for i in range(cnt - 1, -1, -1):\\n                 result.append(result[i] + cnt)\\n             \\n         return result\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if n < 0:\\n             return []\\n         elif n == 0:\\n             return [0]\\n         elif n == 1:\\n             return [0, 1]\\n         \\n         result = [0, 1]\\n         res_len = 2 ** n\\n         \\n         cnt = 1\\n         while len(result) != res_len:\\n             cnt *= 2\\n             result.extend(result[::-1])\\n             offset = 0 \\n             for i in range(len(result)):\\n                 if i > 0 and i % cnt == 0:\\n                     offset = cnt;\\n                 result[i] += offset\\n             \\n             \\n         return result\\n         \\n         \\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         return [i >> 1 ^ i for i in range(1 << n)]\\n\", \"class Solution:\\n     # @return a list of integers\\n     def grayCode(self, n):\\n         res=[]\\n         size=1<<n\\n         for i in range(size):\\n             res.append((i>>1)^i)\\n         return res\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ans = [0]\\n         mask = 0x1\\n         for l in range(n):\\n             for c in ans[::-1]: \\n                 ans.append(mask | c)\\n             mask <<= 1    \\n         return ans    \"]",
        "difficulty": "interview",
        "input": [
            2
        ],
        "output": [
            0,
            1,
            3,
            2
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "grayCode",
        "starter_code": "\nclass Solution:\n    def grayCode(self, n: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/gray-code/"
    },
    {
        "id": 1249,
        "task_id": 2628,
        "test_case_id": 2,
        "question": "The gray code is a binary numeral system where two successive values differ in only one bit.\n\nGiven a non-negative integer n representing the total number of bits in the code, print the sequence of gray code. A gray code sequence must begin with 0.\n\nExample 1:\n\n\nInput: 2\nOutput: [0,1,3,2]\nExplanation:\n00 - 0\n01 - 1\n11 - 3\n10 - 2\n\nFor a given n, a gray code sequence may not be uniquely defined.\nFor example, [0,2,3,1] is also a valid gray code sequence.\n\n00 - 0\n10 - 2\n11 - 3\n01 - 1\n\n\nExample 2:\n\n\nInput: 0\nOutput: [0]\nExplanation: We define the gray code sequence to begin with 0.\n             A gray code sequence of n has size = 2n, which for n = 0 the size is 20 = 1.\n             Therefore, for n = 0 the gray code sequence is [0].",
        "solutions": "[\"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = []\\n         for i in range(1<<n):\\n             res.append(i ^ i >>1)\\n         return res\\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n               \\n         result = [0, 1]\\n         if n <= 1:\\n             return result[:n+1]\\n         res_len = 2 ** n\\n         \\n         cnt = 1\\n         while len(result) != res_len:\\n             cnt *= 2\\n             result.extend(result[::-1])\\n             #offset = 0 \\n             start = len(result) // 2\\n             for i in range(start, len(result)):\\n                 #if i > 0 and i % cnt == 0:\\n                 #    offset = cnt;\\n                 #result[i] += offset \\n                 result[i] += cnt\\n             \\n         return result\", \"class Solution:\\n     def grayCode(self, n):\\n         \\n         if n == 0:\\n             return [0]\\n         \\n         concat = ''\\n         \\n         res = [0, 1]\\n         \\n         for i in range(1, n):\\n             newRes = []\\n             \\n             for j in res:\\n                 newRes.append(j)\\n             for j in reversed(res):\\n                 newRes.append(j + (2 ** i))\\n                 \\n             res = newRes\\n             \\n         return res\\n             \\n             \\n         \\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [0]\\n         result = [0, 1]\\n         \\n         # mirror the previous layer\\n         # then padding '1' at the most significant bit position --- for the second half\\n         # 0 1  --> 00 01 | 11 10 (1 + 1; 1 + 0)\\n         \\n         for i in range(2, n + 1):\\n             mask = 1 << (i - 1)\\n             temp = []\\n             for j in range(len(result)):\\n                 temp.append(result[-j - 1] | mask)\\n             result += temp\\n         return result\\n         \\n\", \"class Solution:\\n     def helper(self, n):\\n         if n == 0:\\n             return ['0']\\n         if n == 1:\\n             return ['0', '1']\\n         ret = []\\n         for code in self.helper(n - 1):\\n             ret.append('0' + code)\\n         for code in reversed(self.helper(n - 1)):\\n             ret.append('1' + code)\\n         return ret\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [0]\\n         ret = []\\n         code = self.grayCode(n - 1)\\n         ret += code\\n         for v in reversed(code):\\n             ret.append(2 ** (n - 1) + v)\\n         return ret\\n         \\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = [0]\\n         for i in range(n):\\n             temp = [] \\n             for num in result:\\n                 temp.append(num + 2**i)\\n             result += temp[::-1]\\n             \\n         return result\\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [0]\\n     \\n         q = [0, -1]\\n     \\n         for i in range(n):\\n             tag = 0\\n             while q[0] != -1:\\n                 item = q.pop(0)\\n                 q.append(item * 2 + tag)\\n                 q.append(item * 2 + (1 - tag))\\n                 tag = 1 - tag\\n             q.pop(0)\\n             q.append(-1)\\n     \\n         q.pop(-1)\\n         \\n         return q\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n               \\n         result = [0, 1]\\n         if n <= 1:\\n             return result[:n+1]\\n         res_len = 2 ** n\\n         \\n         cnt = 1\\n         while len(result) != res_len:\\n             cnt *= 2\\n             #orig_len = len(result)\\n             for i in range(cnt - 1, -1, -1):\\n                 result.append(result[i] + cnt)\\n             \\n         return result\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if n < 0:\\n             return []\\n         elif n == 0:\\n             return [0]\\n         elif n == 1:\\n             return [0, 1]\\n         \\n         result = [0, 1]\\n         res_len = 2 ** n\\n         \\n         cnt = 1\\n         while len(result) != res_len:\\n             cnt *= 2\\n             result.extend(result[::-1])\\n             offset = 0 \\n             for i in range(len(result)):\\n                 if i > 0 and i % cnt == 0:\\n                     offset = cnt;\\n                 result[i] += offset\\n             \\n             \\n         return result\\n         \\n         \\n\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         return [i >> 1 ^ i for i in range(1 << n)]\\n\", \"class Solution:\\n     # @return a list of integers\\n     def grayCode(self, n):\\n         res=[]\\n         size=1<<n\\n         for i in range(size):\\n             res.append((i>>1)^i)\\n         return res\", \"class Solution:\\n     def grayCode(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ans = [0]\\n         mask = 0x1\\n         for l in range(n):\\n             for c in ans[::-1]: \\n                 ans.append(mask | c)\\n             mask <<= 1    \\n         return ans    \"]",
        "difficulty": "interview",
        "input": [
            1
        ],
        "output": [
            0,
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "grayCode",
        "starter_code": "\nclass Solution:\n    def grayCode(self, n: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/gray-code/"
    },
    {
        "id": 1250,
        "task_id": 2629,
        "test_case_id": 1,
        "question": "Given a positive integer n, generate a square matrix filled with elements from 1 to n2 in spiral order.\n\nExample:\n\n\nInput: 3\nOutput:\n[\n [ 1, 2, 3 ],\n [ 8, 9, 4 ],\n [ 7, 6, 5 ]\n]",
        "solutions": "[\"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n<1:\\n             return []\\n         left=0\\n         right=n-1\\n         up=0\\n         bottom=n-1\\n         self.matrix=[[0 for i in range(n)] for j in range(n)]\\n         count=1\\n         while left<=right:\\n             if left<right:\\n                 count=self.DrawRow(up,left,right,count)\\n                 count=self.DrawCol(right,up,bottom,count)\\n                 count=self.DrawRow(bottom,right,left,count)\\n                 count=self.DrawCol(left,bottom,up,count)\\n             else:\\n                 count=self.DrawCol(left,up,bottom+1,count)\\n                 break\\n             up+=1\\n             bottom+=-1\\n             left+=1\\n             right+=-1\\n         return self.matrix\\n \\n     def DrawRow(self,row,start,end,value):\\n         add=1\\n         if start>end:\\n             add=-1\\n         for i in range(start,end,add):\\n             self.matrix[row][i]=value\\n             value+=1\\n         return value\\n     def DrawCol(self,col,start,end,value):\\n         add=1\\n         if start>end:\\n             add=-1\\n         for i in range(start,end,add):\\n             self.matrix[i][col]=value\\n             value+=1\\n         return value\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         matrix = [[0 for i in range(n)] for i in range(n)]\\n         start, end = 0, n - 1\\n         count = 1\\n         for j in range(int(n/2)):\\n             for i in range(start, end + 1):\\n                 matrix[start][i] = count\\n                 count += 1\\n             for i in range(start + 1, end + 1):\\n                 matrix[i][end] = count\\n                 count += 1\\n             for i in range(end-1, start-1,-1):\\n                 matrix[end][i] = count\\n                 count += 1\\n             for i in range(end-1, start, -1):\\n                 matrix[i][start] = count\\n                 count += 1\\n             start += 1\\n             end -= 1\\n         if n%2!=0: matrix[start][end] = count\\n         return matrix\\n\", \"class Solution:\\n     def generateMatrix(self, n):\\n         res = [[0 for i in range(n)] for j in range(n)]\\n         count, rowNum = 1, n\\n         rowIndex1, rowIndex2, colIndex1, colIndex2 = 0, n - 1, n - 1, 0\\n \\n         while rowNum >= 1:\\n             for i in range(colIndex2, colIndex1 + 1):\\n                 res[rowIndex1][i] = count\\n                 count += 1\\n             rowIndex1 += 1\\n \\n             for i in range(rowIndex1, rowIndex2 + 1):\\n                 res[i][colIndex1] = count\\n                 count += 1\\n             colIndex1 -= 1\\n \\n             for i in range(colIndex1, colIndex2 - 1, -1):\\n                 res[rowIndex2][i] = count\\n                 count += 1\\n             rowIndex2 -= 1\\n \\n             for i in range(rowIndex2, rowIndex1 - 1, -1):\\n                 res[i][colIndex2] = count\\n                 count += 1\\n             colIndex2 += 1\\n             rowNum -= 2\\n         return res\\n\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         elif n == 1:\\n             return [[1]]\\n         \\n         cnt = 1\\n         ret = [[None] * n for _ in range(n)]\\n         def layer(width, margin):\\n             if not width > 0:\\n                 return\\n             nonlocal cnt\\n             for i in range(margin, margin+width):\\n                 ret[margin][i] = cnt\\n                 cnt += 1\\n             for i in range(margin+1, margin+width):\\n                 ret[i][margin+width-1] = cnt\\n                 cnt += 1\\n             for i in range(margin+width-2, margin-1, -1):\\n                 ret[margin+width-1][i] = cnt\\n                 cnt += 1\\n             for i in range(margin+width-2, margin, -1):\\n                 ret[i][margin] = cnt\\n                 cnt += 1\\n             layer(width-2, margin+1)\\n         layer(n, 0)\\n         return ret\", \"class Solution:\\n     def generateMatrix(self, n):\\n         matrix = []\\n         if n == 0:\\n             return matrix\\n \\n         matrix = [[0]*n for i in range(n)]\\n         rowBegin = 0\\n         rowEnd = n - 1\\n         colBegin = 0\\n         colEnd = n - 1\\n         num = 0\\n \\n         while True:\\n             # traverse right\\n             for j in range(colBegin, colEnd+1):\\n                 num += 1\\n                 matrix[rowBegin][j] = num\\n             rowBegin += 1\\n             if rowBegin > rowEnd:\\n                 break\\n                 \\n             # traverse down\\n             for j in range(rowBegin, rowEnd+1):\\n                 num += 1\\n                 matrix[j][colEnd] = num\\n             colEnd -= 1\\n             if colBegin > colEnd:\\n                 break\\n                 \\n             # traverse left\\n             for j in range(colEnd, colBegin-1, -1):\\n                 num += 1\\n                 matrix[rowEnd][j] = num\\n             rowEnd -= 1\\n             if rowBegin > rowEnd:\\n                 break\\n                 \\n             # traverse up\\n             for j in range(rowEnd, rowBegin-1, -1):\\n                 num += 1\\n                 matrix[j][colBegin] = num\\n             colBegin += 1\\n             if colBegin > colEnd:\\n                 break\\n         return matrix\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         t=1\\n         temp=[]\\n         for i in range(n):\\n         \\ttemp.append([None]*n)\\n         rep = int(n/2) if n%2==0 else int(n/2)+1\\n         k=1\\n \\n         for i in range(rep):\\n         \\tfor j in range(i,n-i-1):\\n         \\t\\ttemp[i][j] = k\\n         \\t\\tk+=1\\n         \\tfor j in range(i,n-i-1):\\n         \\t\\ttemp[j][n-i-1] = k\\n         \\t\\tk+=1\\n         \\tfor j in range(n-i-1,i,-1):\\n         \\t\\ttemp[n-i-1][j] = k\\n         \\t\\tk+=1\\n         \\tfor j in range(n-i-1,i,-1):\\n         \\t\\ttemp[j][i] = k\\n         \\t\\tk+=1\\n         if n%2==1:temp[int(n/2)][int(n/2)]=n*n\\n         return temp        \", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         left = 0\\n         top = 0\\n         right = n - 1\\n         bottom = n - 1\\n         \\n         res = [[0 for _ in range(n)] for _ in range(n)]\\n         \\n         num = 1\\n         \\n         while left < right and top < bottom:\\n             for i in range(left, right):\\n                 res[top][i] = num\\n                 num += 1\\n             for i in range(top, bottom):\\n                 res[i][right] = num\\n                 num += 1\\n             for i in range(right, left, -1):\\n                 res[bottom][i] = num\\n                 num += 1\\n             for i in range(bottom, top, -1):\\n                 res[i][left] = num\\n                 num += 1\\n             left += 1\\n             right -= 1\\n             top += 1\\n             bottom -= 1\\n         if left == right and top == bottom:\\n             res[left][top] = num\\n             \\n         return res\\n\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         retv = [[0 for i in range(n)] for j in range(n)]\\n         num = n * n\\n         i,j,di,dj = 0,0,0,1\\n         for k in range(1, num+1):\\n             retv[i][j] = k\\n             if retv[(i+di)%n][(j+dj)%n]:\\n                 di,dj = dj, -di\\n             \\n             i += di\\n             j += dj\\n             \\n         return retv\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         arr = [a for a in range (1, n * n + 1)]\\n         matrix = [[a for a in range (1, n + 1)] for b in range (1, n + 1)]\\n         num = 0\\n         row_s = 0\\n         row_e = n - 1\\n         col_s = 0\\n         col_e = n - 1\\n         while col_e > col_s and row_e > row_s:\\n             for i in range (col_s, col_e):\\n                 matrix[row_s][i] = arr[num]\\n                 num += 1\\n             for i in range (row_s, row_e):\\n                 matrix[i][col_e] = arr[num]\\n                 num += 1\\n             for i in range (col_e, col_s, -1):\\n                 matrix[row_e][i] = arr[num]\\n                 num += 1\\n             for i in range (row_e, row_s, -1):\\n                 matrix[i][col_s] = arr[num]\\n                 num += 1\\n             row_s += 1\\n             col_s += 1\\n             row_e -= 1\\n             col_e -= 1\\n \\n         if col_s == col_e:\\n             for i in range (row_s, row_e + 1):\\n                 matrix[i][col_e] = arr[num]\\n                 num += 1\\n         else:\\n             for i in range (col_s, col_e + 1):\\n                 matrix[row_s][i] = arr[num]\\n                 num += 1\\n         return matrix\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         res = [[0]*n for _ in range(n)]\\n         record = set()\\n         rupper,rdown,cleft,cright,val = 0,n-1,0,n-1,1\\n         while rupper<rdown and cleft<cright:\\n             for col in range(cleft,cright):\\n                 res[rupper][col] = val\\n                 val +=1\\n             for row in range(rupper,rdown):\\n                 res[row][cright] = val\\n                 val +=1\\n             for col in range(cright,cleft,-1):\\n                 res[rdown][col] = val\\n                 val +=1\\n             for row in range(rdown, rupper,-1):\\n                 res[row][cleft] = val\\n                 val +=1\\n             rupper +=1\\n             rdown -=1\\n             cleft +=1\\n             cright -=1\\n         if n%2 !=0:\\n             res[n//2][n//2] = val\\n         return res\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         nums = list(range(1, n*n+1))\\n         matrix = [[nums[-1]]]\\n         start = n*n - 1\\n         \\n         while start > 0:\\n             matrix = matrix[::-1]\\n             matrix = [*zip(*matrix)]\\n             matrix = [nums[(start-len(matrix[0])):start]] + matrix\\n             start = start - len(matrix[0])\\n             \\n         return [*map(list, matrix)]\"]",
        "difficulty": "interview",
        "input": [
            3
        ],
        "output": [
            [
                1,
                2,
                3
            ],
            [
                8,
                9,
                4
            ],
            [
                7,
                6,
                5
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "generateMatrix",
        "starter_code": "\nclass Solution:\n    def generateMatrix(self, n: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/spiral-matrix-ii/"
    },
    {
        "id": 1251,
        "task_id": 2629,
        "test_case_id": 2,
        "question": "Given a positive integer n, generate a square matrix filled with elements from 1 to n2 in spiral order.\n\nExample:\n\n\nInput: 3\nOutput:\n[\n [ 1, 2, 3 ],\n [ 8, 9, 4 ],\n [ 7, 6, 5 ]\n]",
        "solutions": "[\"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n<1:\\n             return []\\n         left=0\\n         right=n-1\\n         up=0\\n         bottom=n-1\\n         self.matrix=[[0 for i in range(n)] for j in range(n)]\\n         count=1\\n         while left<=right:\\n             if left<right:\\n                 count=self.DrawRow(up,left,right,count)\\n                 count=self.DrawCol(right,up,bottom,count)\\n                 count=self.DrawRow(bottom,right,left,count)\\n                 count=self.DrawCol(left,bottom,up,count)\\n             else:\\n                 count=self.DrawCol(left,up,bottom+1,count)\\n                 break\\n             up+=1\\n             bottom+=-1\\n             left+=1\\n             right+=-1\\n         return self.matrix\\n \\n     def DrawRow(self,row,start,end,value):\\n         add=1\\n         if start>end:\\n             add=-1\\n         for i in range(start,end,add):\\n             self.matrix[row][i]=value\\n             value+=1\\n         return value\\n     def DrawCol(self,col,start,end,value):\\n         add=1\\n         if start>end:\\n             add=-1\\n         for i in range(start,end,add):\\n             self.matrix[i][col]=value\\n             value+=1\\n         return value\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         matrix = [[0 for i in range(n)] for i in range(n)]\\n         start, end = 0, n - 1\\n         count = 1\\n         for j in range(int(n/2)):\\n             for i in range(start, end + 1):\\n                 matrix[start][i] = count\\n                 count += 1\\n             for i in range(start + 1, end + 1):\\n                 matrix[i][end] = count\\n                 count += 1\\n             for i in range(end-1, start-1,-1):\\n                 matrix[end][i] = count\\n                 count += 1\\n             for i in range(end-1, start, -1):\\n                 matrix[i][start] = count\\n                 count += 1\\n             start += 1\\n             end -= 1\\n         if n%2!=0: matrix[start][end] = count\\n         return matrix\\n\", \"class Solution:\\n     def generateMatrix(self, n):\\n         res = [[0 for i in range(n)] for j in range(n)]\\n         count, rowNum = 1, n\\n         rowIndex1, rowIndex2, colIndex1, colIndex2 = 0, n - 1, n - 1, 0\\n \\n         while rowNum >= 1:\\n             for i in range(colIndex2, colIndex1 + 1):\\n                 res[rowIndex1][i] = count\\n                 count += 1\\n             rowIndex1 += 1\\n \\n             for i in range(rowIndex1, rowIndex2 + 1):\\n                 res[i][colIndex1] = count\\n                 count += 1\\n             colIndex1 -= 1\\n \\n             for i in range(colIndex1, colIndex2 - 1, -1):\\n                 res[rowIndex2][i] = count\\n                 count += 1\\n             rowIndex2 -= 1\\n \\n             for i in range(rowIndex2, rowIndex1 - 1, -1):\\n                 res[i][colIndex2] = count\\n                 count += 1\\n             colIndex2 += 1\\n             rowNum -= 2\\n         return res\\n\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         elif n == 1:\\n             return [[1]]\\n         \\n         cnt = 1\\n         ret = [[None] * n for _ in range(n)]\\n         def layer(width, margin):\\n             if not width > 0:\\n                 return\\n             nonlocal cnt\\n             for i in range(margin, margin+width):\\n                 ret[margin][i] = cnt\\n                 cnt += 1\\n             for i in range(margin+1, margin+width):\\n                 ret[i][margin+width-1] = cnt\\n                 cnt += 1\\n             for i in range(margin+width-2, margin-1, -1):\\n                 ret[margin+width-1][i] = cnt\\n                 cnt += 1\\n             for i in range(margin+width-2, margin, -1):\\n                 ret[i][margin] = cnt\\n                 cnt += 1\\n             layer(width-2, margin+1)\\n         layer(n, 0)\\n         return ret\", \"class Solution:\\n     def generateMatrix(self, n):\\n         matrix = []\\n         if n == 0:\\n             return matrix\\n \\n         matrix = [[0]*n for i in range(n)]\\n         rowBegin = 0\\n         rowEnd = n - 1\\n         colBegin = 0\\n         colEnd = n - 1\\n         num = 0\\n \\n         while True:\\n             # traverse right\\n             for j in range(colBegin, colEnd+1):\\n                 num += 1\\n                 matrix[rowBegin][j] = num\\n             rowBegin += 1\\n             if rowBegin > rowEnd:\\n                 break\\n                 \\n             # traverse down\\n             for j in range(rowBegin, rowEnd+1):\\n                 num += 1\\n                 matrix[j][colEnd] = num\\n             colEnd -= 1\\n             if colBegin > colEnd:\\n                 break\\n                 \\n             # traverse left\\n             for j in range(colEnd, colBegin-1, -1):\\n                 num += 1\\n                 matrix[rowEnd][j] = num\\n             rowEnd -= 1\\n             if rowBegin > rowEnd:\\n                 break\\n                 \\n             # traverse up\\n             for j in range(rowEnd, rowBegin-1, -1):\\n                 num += 1\\n                 matrix[j][colBegin] = num\\n             colBegin += 1\\n             if colBegin > colEnd:\\n                 break\\n         return matrix\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         t=1\\n         temp=[]\\n         for i in range(n):\\n         \\ttemp.append([None]*n)\\n         rep = int(n/2) if n%2==0 else int(n/2)+1\\n         k=1\\n \\n         for i in range(rep):\\n         \\tfor j in range(i,n-i-1):\\n         \\t\\ttemp[i][j] = k\\n         \\t\\tk+=1\\n         \\tfor j in range(i,n-i-1):\\n         \\t\\ttemp[j][n-i-1] = k\\n         \\t\\tk+=1\\n         \\tfor j in range(n-i-1,i,-1):\\n         \\t\\ttemp[n-i-1][j] = k\\n         \\t\\tk+=1\\n         \\tfor j in range(n-i-1,i,-1):\\n         \\t\\ttemp[j][i] = k\\n         \\t\\tk+=1\\n         if n%2==1:temp[int(n/2)][int(n/2)]=n*n\\n         return temp        \", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         left = 0\\n         top = 0\\n         right = n - 1\\n         bottom = n - 1\\n         \\n         res = [[0 for _ in range(n)] for _ in range(n)]\\n         \\n         num = 1\\n         \\n         while left < right and top < bottom:\\n             for i in range(left, right):\\n                 res[top][i] = num\\n                 num += 1\\n             for i in range(top, bottom):\\n                 res[i][right] = num\\n                 num += 1\\n             for i in range(right, left, -1):\\n                 res[bottom][i] = num\\n                 num += 1\\n             for i in range(bottom, top, -1):\\n                 res[i][left] = num\\n                 num += 1\\n             left += 1\\n             right -= 1\\n             top += 1\\n             bottom -= 1\\n         if left == right and top == bottom:\\n             res[left][top] = num\\n             \\n         return res\\n\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         retv = [[0 for i in range(n)] for j in range(n)]\\n         num = n * n\\n         i,j,di,dj = 0,0,0,1\\n         for k in range(1, num+1):\\n             retv[i][j] = k\\n             if retv[(i+di)%n][(j+dj)%n]:\\n                 di,dj = dj, -di\\n             \\n             i += di\\n             j += dj\\n             \\n         return retv\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         arr = [a for a in range (1, n * n + 1)]\\n         matrix = [[a for a in range (1, n + 1)] for b in range (1, n + 1)]\\n         num = 0\\n         row_s = 0\\n         row_e = n - 1\\n         col_s = 0\\n         col_e = n - 1\\n         while col_e > col_s and row_e > row_s:\\n             for i in range (col_s, col_e):\\n                 matrix[row_s][i] = arr[num]\\n                 num += 1\\n             for i in range (row_s, row_e):\\n                 matrix[i][col_e] = arr[num]\\n                 num += 1\\n             for i in range (col_e, col_s, -1):\\n                 matrix[row_e][i] = arr[num]\\n                 num += 1\\n             for i in range (row_e, row_s, -1):\\n                 matrix[i][col_s] = arr[num]\\n                 num += 1\\n             row_s += 1\\n             col_s += 1\\n             row_e -= 1\\n             col_e -= 1\\n \\n         if col_s == col_e:\\n             for i in range (row_s, row_e + 1):\\n                 matrix[i][col_e] = arr[num]\\n                 num += 1\\n         else:\\n             for i in range (col_s, col_e + 1):\\n                 matrix[row_s][i] = arr[num]\\n                 num += 1\\n         return matrix\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         res = [[0]*n for _ in range(n)]\\n         record = set()\\n         rupper,rdown,cleft,cright,val = 0,n-1,0,n-1,1\\n         while rupper<rdown and cleft<cright:\\n             for col in range(cleft,cright):\\n                 res[rupper][col] = val\\n                 val +=1\\n             for row in range(rupper,rdown):\\n                 res[row][cright] = val\\n                 val +=1\\n             for col in range(cright,cleft,-1):\\n                 res[rdown][col] = val\\n                 val +=1\\n             for row in range(rdown, rupper,-1):\\n                 res[row][cleft] = val\\n                 val +=1\\n             rupper +=1\\n             rdown -=1\\n             cleft +=1\\n             cright -=1\\n         if n%2 !=0:\\n             res[n//2][n//2] = val\\n         return res\", \"class Solution:\\n     def generateMatrix(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         nums = list(range(1, n*n+1))\\n         matrix = [[nums[-1]]]\\n         start = n*n - 1\\n         \\n         while start > 0:\\n             matrix = matrix[::-1]\\n             matrix = [*zip(*matrix)]\\n             matrix = [nums[(start-len(matrix[0])):start]] + matrix\\n             start = start - len(matrix[0])\\n             \\n         return [*map(list, matrix)]\"]",
        "difficulty": "interview",
        "input": [
            1
        ],
        "output": [
            [
                1
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "generateMatrix",
        "starter_code": "\nclass Solution:\n    def generateMatrix(self, n: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/spiral-matrix-ii/"
    },
    {
        "id": 1252,
        "task_id": 2885,
        "test_case_id": 1,
        "question": "Given a set of non-overlapping intervals, insert a new interval into the intervals (merge if necessary).\n\nYou may assume that the intervals were initially sorted according to their start times.\n\nExample 1:\n\n\nInput: intervals = [[1,3],[6,9]], newInterval = [2,5]\nOutput: [[1,5],[6,9]]\n\n\nExample 2:\n\n\nInput: intervals = [[1,2],[3,5],[6,7],[8,10],[12,16]], newInterval = [4,8]\nOutput: [[1,2],[3,10],[12,16]]\nExplanation: Because the new interval [4,8] overlaps with [3,5],[6,7],[8,10].",
        "solutions": "[\"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         # my no6, 56ms, beats 100%\\n         if len(intervals)==0:\\n             return [newInterval]\\n         start = 0; end = 0\\n         foundl = foundr = -1\\n         for i, n in enumerate(intervals):\\n             if foundl==-1 and newInterval.start<=n.end:\\n                 foundl = 1\\n                 start = i\\n                 break\\n             else:\\n                 continue\\n         for i, n in enumerate(intervals[::-1]):\\n             if foundr==-1 and newInterval.end>=n.start:\\n                 foundr = 1\\n                 end = len(intervals)-1-i\\n                 break\\n             else:\\n                 continue\\n         print(start, end,foundl,foundr)\\n         if foundl==1 and foundr==1:\\n             if start<=end:\\n                 s = min(intervals[start].start, newInterval.start)\\n                 e = max(intervals[end].end, newInterval.end)\\n                 return intervals[:start]+[Interval(s,e)]+intervals[end+1:]\\n             else:\\n                 return intervals[:start]+[newInterval]+intervals[end+1:]\\n         elif foundl==1:\\n             return [newInterval]+intervals\\n         else:\\n             return intervals+[newInterval]\\n         \\n         # others'\\n         # s, e = newInterval.start, newInterval.end\\n         # left, right = [], []\\n         # for i in intervals:\\n         #     if i.end < s:\\n         #         left += i,\\n         #     elif i.start > e:\\n         #         right += i,\\n         #     else:\\n         #         s = min(s, i.start)\\n         #         e = max(e, i.end)\\n         # return left + [Interval(s, e)] + right\\n \\n             \\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if intervals == []:\\n             return [newInterval]\\n         \\n         index = 0\\n         while index < len(intervals):\\n             if newInterval.start < intervals[index].start:\\n                 break\\n             index += 1\\n         del_index = 0\\n         insert_index = 0\\n         if index == 0:\\n             s = newInterval.start\\n             e = newInterval.end\\n             insert_index = index \\n         else:\\n             if newInterval.start <= intervals[index - 1].end:\\n                 s = intervals[index - 1].start\\n                 e = max(intervals[index - 1].end, newInterval.end)\\n                 insert_index = index - 1\\n                 del_index += 1\\n             else:\\n                 s = newInterval.start\\n                 e = newInterval.end\\n                 insert_index = index\\n             \\n         while index < len(intervals):\\n             if intervals[index].start <= e:\\n                 e = max(e, intervals[index].end)\\n                 del_index += 1\\n                 index += 1\\n             else:\\n                 break\\n         i = 0\\n         while i < del_index:\\n             intervals.pop(insert_index)\\n             i += 1\\n         intervals.insert(insert_index, Interval(s,e))\\n         return intervals\\n                     \\n                 \\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n import operator\\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         #Leetcode \\u7b2c56\\u9898\\u7684\\u601d\\u8def\\n         returnedList = []\\n         if(len(intervals) == 0):\\n             returnedList.append(newInterval)\\n             return returnedList\\n \\n         p = 1       #\\u5224\\u65ad\\u662f\\u5faa\\u73af\\u7ed3\\u675f\\u8df3\\u51fa\\u5faa\\u73af\\uff0c\\u8fd8\\u662fnewInterval.end < intervals[i].start\\u8df3\\u51fa\\u5faa\\u73af\\uff0c\\u524d\\u8005\\u4e0d\\u8fdb\\u884c\\u62fc\\u63a5\\n         for i in range(len(intervals)):\\n             if(intervals[i].end < newInterval.start):\\n                 returnedList.append(intervals[i])\\n             elif(intervals[i].end == newInterval.start):\\n                 newInterval.start = intervals[i].start\\n             elif(newInterval.start < intervals[i].end and newInterval.end > intervals[i].start): #\\u76f8\\u4ea4\\n                 newInterval.start = min(newInterval.start, intervals[i].start)\\n                 newInterval.end = max(newInterval.end, intervals[i].end)\\n             elif(newInterval.end == intervals[i].start):\\n                 newInterval.end = intervals[i].end\\n             elif(newInterval.end < intervals[i].start):\\n                 p = 0\\n                 break\\n \\n \\n         returnedList.append(newInterval)\\n         if(p == 0):\\n             returnedList.extend(intervals[i:])\\n         return returnedList\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         \\n         ans = []\\n         \\n         # some boundary conditions\\n         if intervals == []:\\n             return [newInterval]\\n         elif newInterval.end < intervals[0].start:\\n             return [newInterval] + intervals\\n         elif newInterval.start > intervals[-1].end:\\n             return intervals + [newInterval]\\n         \\n         new_start = newInterval.start\\n         new_end = newInterval.end\\n         \\n         for i in range(0, len(intervals)):\\n             if intervals[i].end < new_start: # intertval[i].start <= interval[i].end < new_start\\n                 ans.append(intervals[i])\\n             elif i and (intervals[i-1].end < new_start) and (new_end < intervals[i].start): # i >= 1\\n                 ans.append(newInterval)\\n                 for j in range(i, len(intervals)):\\n                     ans.append(intervals[j])\\n                 return ans\\n             else:\\n                 merge_start = min(intervals[i].start, new_start)\\n                 merge_end = max(intervals[i].end, new_end)\\n                 break\\n                 \\n         #print(\\\"i:\\\", i)\\n         #print(merge_start, merge_end)\\n         \\n         notStop = 1\\n         for ii in range(i+1, len(intervals)):\\n             if intervals[ii].start > merge_end: # merge_start <= merge_end < intervals[ii].start\\n                 notStop = 0\\n                 break\\n         \\n         if notStop: # after traversing remaining intervals, merge_intvl overlaps the last (i.e. entire remaining) intervals\\n             merge_end = max(intervals[-1].end, merge_end) # or index ii\\n             ans.append(Interval(merge_start, merge_end))\\n             return ans\\n \\n         merge_end = max(intervals[ii-1].end, merge_end)\\n         #print(\\\"ii:\\\", ii)\\n         #print(merge_start, merge_end)\\n         ans.append(Interval(merge_start, merge_end))\\n         \\n         \\n         for iii in range(ii, len(intervals)):\\n             ans.append(intervals[iii])\\n         \\n         return ans\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         start, end = newInterval.start, newInterval.end\\n         left = [i for i in intervals if start > i.end]\\n         right = [i for i in intervals if end < i.start]\\n         if left + right != intervals:\\n             start = min(start, intervals[len(left)].start)\\n             end = max(end, intervals[~len(right)].end)\\n         return left + [Interval(start, end)] + right\\n             \\n                 \\n             \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         s, e = newInterval.start, newInterval.end\\n         left, right = [], []\\n         for i in intervals:\\n             if i.end < s:\\n                 left += i,\\n             elif i.start > e:\\n                 right += i,\\n             else:\\n                 s = min(s, i.start)\\n                 e = max(e, i.end)\\n         return left + [Interval(s, e)] + right\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         start, end = newInterval.start, newInterval.end\\n         left, right = [], []\\n         for i in intervals:\\n             if i.end < start:\\n                 left.append(i)\\n             elif i.start > end:\\n                 right.append(i)\\n             else:\\n                 start = min(start, i.start)\\n                 end = max(end,i.end)\\n         return left + [Interval(start, end)] + right\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         left = 0\\n         right = len(intervals) - 1\\n         tar = newInterval.start\\n         while True:\\n             if left > right:\\n                 break\\n             mid = (left + right) // 2\\n             value = intervals[mid].start\\n             if value == tar:\\n                 left = mid\\n                 break\\n             elif value > tar:\\n                 right = mid - 1\\n             else:\\n                 left = mid + 1\\n         intervals.insert(left, newInterval)\\n         ret = []\\n         tmp = intervals[0]\\n         ret.append(tmp)\\n         it = iter(intervals)\\n         next(it)\\n         for i in it:\\n             if i.start <= ret[-1].end:\\n                 prev = ret.pop()\\n                 ret.append(Interval(prev.start, max(i.end, prev.end)))\\n             else:\\n                 ret.append(i)\\n         return ret\\n             \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if not intervals:\\n             return [newInterval]\\n         v = []\\n         a =newInterval.start\\n         b =newInterval.end\\n         flag =0\\n         s=a\\n         e=b\\n         for i in intervals:\\n             if i.start<=a<=i.end:\\n                 s = i.start\\n             if i.start<=b<=i.end:\\n                 e = i.end\\n                 v.append(Interval(s,e))\\n                 flag =1\\n             if not flag and i.start >b:\\n                 v.append(Interval(s,e))\\n                 flag =1\\n             if i.end<a or i.start>b:\\n                 v.append(i)\\n         if not flag:\\n             v.append(Interval(s,e))\\n         return v\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         res = []\\n         for interval in intervals:\\n             if self.merge(interval, newInterval):\\n                 newInterval = Interval(min(interval.start, newInterval.start), max(interval.end, newInterval.end))\\n             else:\\n                 if newInterval != None and newInterval.end < interval.start:\\n                     res.append(newInterval)\\n                     newInterval = None\\n                 res.append(interval)\\n         if newInterval != None:\\n             res.append(newInterval)\\n         return res\\n     \\n     def merge(self, a, b):\\n         if b is None or a.start > b.end or a.end < b.start:\\n             return False\\n         return True\", \"class Solution:\\n     def insert(self, intervals, newInterval):\\n         start = len(intervals)\\n         for i in range(len(intervals)):\\n             if newInterval.start<intervals[i].start:\\n                 start = min(start, i)\\n             if not (newInterval.start > intervals[i].end or newInterval.end < intervals[i].start):\\n                 newInterval = Interval(min(newInterval.start, intervals[i].start), max(newInterval.end, intervals[i].end))\\n                 start, intervals[i] = min(i,start), []\\n         intervals = [ele for ele in intervals if ele!=[]]\\n         intervals.insert(start,newInterval)\\n         return intervals\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         intervals.append(newInterval)\\n         if len(intervals) <= 1:\\n           return intervals\\n         \\n         intervals.sort(key=lambda x:x.start)\\n         \\n         i = 0\\n         while i < len(intervals) - 1:\\n           left = intervals[i]\\n           right = intervals[i+1]\\n           if left.end >= right.start:\\n             intervals[i] = Interval(s=left.start, e=max(left.end, right.end))\\n             del intervals[i+1]\\n           else:\\n             i += 1\\n         \\n         return intervals\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         result = []\\n         for interval in intervals:\\n             if newInterval and newInterval.end >= interval.start and newInterval.start <= interval.end:\\n                 newInterval.start = min(interval.start, newInterval.start)\\n                 newInterval.end = max(interval.end, newInterval.end)\\n             else:\\n                 if newInterval and newInterval.end < interval.start:\\n                     result.append(newInterval)\\n                     result.append(interval)\\n                     newInterval = None\\n                 else:\\n                     result.append(interval)\\n         if newInterval:\\n             result.append(newInterval)\\n         return result\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    3
                ],
                [
                    6,
                    9
                ]
            ],
            [
                2,
                5
            ]
        ],
        "output": [
            [
                1,
                5
            ],
            [
                6,
                9
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "insert",
        "starter_code": "\nclass Solution:\n    def insert(self, intervals: List[List[int]], newInterval: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/insert-interval/"
    },
    {
        "id": 1253,
        "task_id": 2885,
        "test_case_id": 2,
        "question": "Given a set of non-overlapping intervals, insert a new interval into the intervals (merge if necessary).\n\nYou may assume that the intervals were initially sorted according to their start times.\n\nExample 1:\n\n\nInput: intervals = [[1,3],[6,9]], newInterval = [2,5]\nOutput: [[1,5],[6,9]]\n\n\nExample 2:\n\n\nInput: intervals = [[1,2],[3,5],[6,7],[8,10],[12,16]], newInterval = [4,8]\nOutput: [[1,2],[3,10],[12,16]]\nExplanation: Because the new interval [4,8] overlaps with [3,5],[6,7],[8,10].",
        "solutions": "[\"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         # my no6, 56ms, beats 100%\\n         if len(intervals)==0:\\n             return [newInterval]\\n         start = 0; end = 0\\n         foundl = foundr = -1\\n         for i, n in enumerate(intervals):\\n             if foundl==-1 and newInterval.start<=n.end:\\n                 foundl = 1\\n                 start = i\\n                 break\\n             else:\\n                 continue\\n         for i, n in enumerate(intervals[::-1]):\\n             if foundr==-1 and newInterval.end>=n.start:\\n                 foundr = 1\\n                 end = len(intervals)-1-i\\n                 break\\n             else:\\n                 continue\\n         print(start, end,foundl,foundr)\\n         if foundl==1 and foundr==1:\\n             if start<=end:\\n                 s = min(intervals[start].start, newInterval.start)\\n                 e = max(intervals[end].end, newInterval.end)\\n                 return intervals[:start]+[Interval(s,e)]+intervals[end+1:]\\n             else:\\n                 return intervals[:start]+[newInterval]+intervals[end+1:]\\n         elif foundl==1:\\n             return [newInterval]+intervals\\n         else:\\n             return intervals+[newInterval]\\n         \\n         # others'\\n         # s, e = newInterval.start, newInterval.end\\n         # left, right = [], []\\n         # for i in intervals:\\n         #     if i.end < s:\\n         #         left += i,\\n         #     elif i.start > e:\\n         #         right += i,\\n         #     else:\\n         #         s = min(s, i.start)\\n         #         e = max(e, i.end)\\n         # return left + [Interval(s, e)] + right\\n \\n             \\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if intervals == []:\\n             return [newInterval]\\n         \\n         index = 0\\n         while index < len(intervals):\\n             if newInterval.start < intervals[index].start:\\n                 break\\n             index += 1\\n         del_index = 0\\n         insert_index = 0\\n         if index == 0:\\n             s = newInterval.start\\n             e = newInterval.end\\n             insert_index = index \\n         else:\\n             if newInterval.start <= intervals[index - 1].end:\\n                 s = intervals[index - 1].start\\n                 e = max(intervals[index - 1].end, newInterval.end)\\n                 insert_index = index - 1\\n                 del_index += 1\\n             else:\\n                 s = newInterval.start\\n                 e = newInterval.end\\n                 insert_index = index\\n             \\n         while index < len(intervals):\\n             if intervals[index].start <= e:\\n                 e = max(e, intervals[index].end)\\n                 del_index += 1\\n                 index += 1\\n             else:\\n                 break\\n         i = 0\\n         while i < del_index:\\n             intervals.pop(insert_index)\\n             i += 1\\n         intervals.insert(insert_index, Interval(s,e))\\n         return intervals\\n                     \\n                 \\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n import operator\\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         #Leetcode \\u7b2c56\\u9898\\u7684\\u601d\\u8def\\n         returnedList = []\\n         if(len(intervals) == 0):\\n             returnedList.append(newInterval)\\n             return returnedList\\n \\n         p = 1       #\\u5224\\u65ad\\u662f\\u5faa\\u73af\\u7ed3\\u675f\\u8df3\\u51fa\\u5faa\\u73af\\uff0c\\u8fd8\\u662fnewInterval.end < intervals[i].start\\u8df3\\u51fa\\u5faa\\u73af\\uff0c\\u524d\\u8005\\u4e0d\\u8fdb\\u884c\\u62fc\\u63a5\\n         for i in range(len(intervals)):\\n             if(intervals[i].end < newInterval.start):\\n                 returnedList.append(intervals[i])\\n             elif(intervals[i].end == newInterval.start):\\n                 newInterval.start = intervals[i].start\\n             elif(newInterval.start < intervals[i].end and newInterval.end > intervals[i].start): #\\u76f8\\u4ea4\\n                 newInterval.start = min(newInterval.start, intervals[i].start)\\n                 newInterval.end = max(newInterval.end, intervals[i].end)\\n             elif(newInterval.end == intervals[i].start):\\n                 newInterval.end = intervals[i].end\\n             elif(newInterval.end < intervals[i].start):\\n                 p = 0\\n                 break\\n \\n \\n         returnedList.append(newInterval)\\n         if(p == 0):\\n             returnedList.extend(intervals[i:])\\n         return returnedList\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         \\n         ans = []\\n         \\n         # some boundary conditions\\n         if intervals == []:\\n             return [newInterval]\\n         elif newInterval.end < intervals[0].start:\\n             return [newInterval] + intervals\\n         elif newInterval.start > intervals[-1].end:\\n             return intervals + [newInterval]\\n         \\n         new_start = newInterval.start\\n         new_end = newInterval.end\\n         \\n         for i in range(0, len(intervals)):\\n             if intervals[i].end < new_start: # intertval[i].start <= interval[i].end < new_start\\n                 ans.append(intervals[i])\\n             elif i and (intervals[i-1].end < new_start) and (new_end < intervals[i].start): # i >= 1\\n                 ans.append(newInterval)\\n                 for j in range(i, len(intervals)):\\n                     ans.append(intervals[j])\\n                 return ans\\n             else:\\n                 merge_start = min(intervals[i].start, new_start)\\n                 merge_end = max(intervals[i].end, new_end)\\n                 break\\n                 \\n         #print(\\\"i:\\\", i)\\n         #print(merge_start, merge_end)\\n         \\n         notStop = 1\\n         for ii in range(i+1, len(intervals)):\\n             if intervals[ii].start > merge_end: # merge_start <= merge_end < intervals[ii].start\\n                 notStop = 0\\n                 break\\n         \\n         if notStop: # after traversing remaining intervals, merge_intvl overlaps the last (i.e. entire remaining) intervals\\n             merge_end = max(intervals[-1].end, merge_end) # or index ii\\n             ans.append(Interval(merge_start, merge_end))\\n             return ans\\n \\n         merge_end = max(intervals[ii-1].end, merge_end)\\n         #print(\\\"ii:\\\", ii)\\n         #print(merge_start, merge_end)\\n         ans.append(Interval(merge_start, merge_end))\\n         \\n         \\n         for iii in range(ii, len(intervals)):\\n             ans.append(intervals[iii])\\n         \\n         return ans\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         start, end = newInterval.start, newInterval.end\\n         left = [i for i in intervals if start > i.end]\\n         right = [i for i in intervals if end < i.start]\\n         if left + right != intervals:\\n             start = min(start, intervals[len(left)].start)\\n             end = max(end, intervals[~len(right)].end)\\n         return left + [Interval(start, end)] + right\\n             \\n                 \\n             \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         s, e = newInterval.start, newInterval.end\\n         left, right = [], []\\n         for i in intervals:\\n             if i.end < s:\\n                 left += i,\\n             elif i.start > e:\\n                 right += i,\\n             else:\\n                 s = min(s, i.start)\\n                 e = max(e, i.end)\\n         return left + [Interval(s, e)] + right\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         start, end = newInterval.start, newInterval.end\\n         left, right = [], []\\n         for i in intervals:\\n             if i.end < start:\\n                 left.append(i)\\n             elif i.start > end:\\n                 right.append(i)\\n             else:\\n                 start = min(start, i.start)\\n                 end = max(end,i.end)\\n         return left + [Interval(start, end)] + right\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         left = 0\\n         right = len(intervals) - 1\\n         tar = newInterval.start\\n         while True:\\n             if left > right:\\n                 break\\n             mid = (left + right) // 2\\n             value = intervals[mid].start\\n             if value == tar:\\n                 left = mid\\n                 break\\n             elif value > tar:\\n                 right = mid - 1\\n             else:\\n                 left = mid + 1\\n         intervals.insert(left, newInterval)\\n         ret = []\\n         tmp = intervals[0]\\n         ret.append(tmp)\\n         it = iter(intervals)\\n         next(it)\\n         for i in it:\\n             if i.start <= ret[-1].end:\\n                 prev = ret.pop()\\n                 ret.append(Interval(prev.start, max(i.end, prev.end)))\\n             else:\\n                 ret.append(i)\\n         return ret\\n             \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if not intervals:\\n             return [newInterval]\\n         v = []\\n         a =newInterval.start\\n         b =newInterval.end\\n         flag =0\\n         s=a\\n         e=b\\n         for i in intervals:\\n             if i.start<=a<=i.end:\\n                 s = i.start\\n             if i.start<=b<=i.end:\\n                 e = i.end\\n                 v.append(Interval(s,e))\\n                 flag =1\\n             if not flag and i.start >b:\\n                 v.append(Interval(s,e))\\n                 flag =1\\n             if i.end<a or i.start>b:\\n                 v.append(i)\\n         if not flag:\\n             v.append(Interval(s,e))\\n         return v\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         res = []\\n         for interval in intervals:\\n             if self.merge(interval, newInterval):\\n                 newInterval = Interval(min(interval.start, newInterval.start), max(interval.end, newInterval.end))\\n             else:\\n                 if newInterval != None and newInterval.end < interval.start:\\n                     res.append(newInterval)\\n                     newInterval = None\\n                 res.append(interval)\\n         if newInterval != None:\\n             res.append(newInterval)\\n         return res\\n     \\n     def merge(self, a, b):\\n         if b is None or a.start > b.end or a.end < b.start:\\n             return False\\n         return True\", \"class Solution:\\n     def insert(self, intervals, newInterval):\\n         start = len(intervals)\\n         for i in range(len(intervals)):\\n             if newInterval.start<intervals[i].start:\\n                 start = min(start, i)\\n             if not (newInterval.start > intervals[i].end or newInterval.end < intervals[i].start):\\n                 newInterval = Interval(min(newInterval.start, intervals[i].start), max(newInterval.end, intervals[i].end))\\n                 start, intervals[i] = min(i,start), []\\n         intervals = [ele for ele in intervals if ele!=[]]\\n         intervals.insert(start,newInterval)\\n         return intervals\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         intervals.append(newInterval)\\n         if len(intervals) <= 1:\\n           return intervals\\n         \\n         intervals.sort(key=lambda x:x.start)\\n         \\n         i = 0\\n         while i < len(intervals) - 1:\\n           left = intervals[i]\\n           right = intervals[i+1]\\n           if left.end >= right.start:\\n             intervals[i] = Interval(s=left.start, e=max(left.end, right.end))\\n             del intervals[i+1]\\n           else:\\n             i += 1\\n         \\n         return intervals\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def insert(self, intervals, newInterval):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :type newInterval: Interval\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         result = []\\n         for interval in intervals:\\n             if newInterval and newInterval.end >= interval.start and newInterval.start <= interval.end:\\n                 newInterval.start = min(interval.start, newInterval.start)\\n                 newInterval.end = max(interval.end, newInterval.end)\\n             else:\\n                 if newInterval and newInterval.end < interval.start:\\n                     result.append(newInterval)\\n                     result.append(interval)\\n                     newInterval = None\\n                 else:\\n                     result.append(interval)\\n         if newInterval:\\n             result.append(newInterval)\\n         return result\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    2
                ],
                [
                    3,
                    5
                ],
                [
                    6,
                    7
                ],
                [
                    8,
                    10
                ],
                [
                    12,
                    16
                ]
            ],
            [
                4,
                8
            ]
        ],
        "output": [
            [
                1,
                2
            ],
            [
                3,
                10
            ],
            [
                12,
                16
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "insert",
        "starter_code": "\nclass Solution:\n    def insert(self, intervals: List[List[int]], newInterval: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/insert-interval/"
    },
    {
        "id": 1254,
        "task_id": 4751,
        "test_case_id": 1,
        "question": "Given head, the head of a linked list, determine if the linked list has a cycle in it.\n\nThere is a cycle in a linked list if there is some node in the list that can be reached again by continuously following the next pointer. Internally, pos is used to denote the index of the node that tail's next pointer is connected to. Note that pos is not passed as a parameter.\n\nReturn true if there is a cycle in the linked list. Otherwise, return false.\n\nInput: head = [3,2,0,-4], pos = 1\nOutput: true\nExplanation: There is a cycle in the linked list, where the tail connects to the 1st node (0-indexed).\n\n\nInput: head = [1,2], pos = 0\nOutput: true\nExplanation: There is a cycle in the linked list, where the tail connects to the 0th node.\n\n\nInput: head = [1], pos = -1\nOutput: false\nExplanation: There is no cycle in the linked list.\n\n\nConstraints:\n\nThe number of the nodes in the list is in the range [0, 104].\n-105 <= Node.val <= 105\npos is -1 or a valid index in the linked-list.",
        "solutions": "[\"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        if head == None:\\n            return False\\n        \\n        slow = head\\n        fast = head.next\\n        \\n        while slow != fast:\\n            if fast is None or fast.next is None:\\n                return False\\n            slow = slow.next\\n            fast = fast.next.next\\n        \\n        return True\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        slow = head\\n        fast = head\\n        while fast and fast.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        \\n        slow = head\\n        fast = head\\n        \\n        \\n        while fast and fast.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        nodes_seen = set()\\n        while head is not None:\\n            if head in nodes_seen:\\n                return True\\n            nodes_seen.add(head)\\n            head = head.next\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        cur = head\\n        check = set()\\n        while cur:\\n            if cur in check:\\n                return True\\n            check.add(cur)\\n            cur = cur.next\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        try:\\n            slow = head\\n            fast = head.next \\n            while slow is not fast:\\n                slow = slow.next\\n                fast = fast.next.next\\n            return True\\n        except:\\n            return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        if head is None:\\n            return False\\n\\n        slow, fast = head, head\\n        while fast.next and fast.next.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        return False\"]",
        "difficulty": "introductory",
        "input": [
            [
                3,
                2,
                0,
                -4
            ],
            1
        ],
        "output": [
            true
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "hasCycle",
        "starter_code": "\n# Definition for singly-linked list.\n# class ListNode:\n#     def __init__(self, x):\n#         self.val = x\n#         self.next = None\nclass Solution:\n    def hasCycle(self, head: ListNode) -> bool:\n        ",
        "url": "https://leetcode.com/problems/linked-list-cycle/"
    },
    {
        "id": 1255,
        "task_id": 4751,
        "test_case_id": 2,
        "question": "Given head, the head of a linked list, determine if the linked list has a cycle in it.\n\nThere is a cycle in a linked list if there is some node in the list that can be reached again by continuously following the next pointer. Internally, pos is used to denote the index of the node that tail's next pointer is connected to. Note that pos is not passed as a parameter.\n\nReturn true if there is a cycle in the linked list. Otherwise, return false.\n\nInput: head = [3,2,0,-4], pos = 1\nOutput: true\nExplanation: There is a cycle in the linked list, where the tail connects to the 1st node (0-indexed).\n\n\nInput: head = [1,2], pos = 0\nOutput: true\nExplanation: There is a cycle in the linked list, where the tail connects to the 0th node.\n\n\nInput: head = [1], pos = -1\nOutput: false\nExplanation: There is no cycle in the linked list.\n\n\nConstraints:\n\nThe number of the nodes in the list is in the range [0, 104].\n-105 <= Node.val <= 105\npos is -1 or a valid index in the linked-list.",
        "solutions": "[\"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        if head == None:\\n            return False\\n        \\n        slow = head\\n        fast = head.next\\n        \\n        while slow != fast:\\n            if fast is None or fast.next is None:\\n                return False\\n            slow = slow.next\\n            fast = fast.next.next\\n        \\n        return True\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        slow = head\\n        fast = head\\n        while fast and fast.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        \\n        slow = head\\n        fast = head\\n        \\n        \\n        while fast and fast.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        nodes_seen = set()\\n        while head is not None:\\n            if head in nodes_seen:\\n                return True\\n            nodes_seen.add(head)\\n            head = head.next\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        cur = head\\n        check = set()\\n        while cur:\\n            if cur in check:\\n                return True\\n            check.add(cur)\\n            cur = cur.next\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        try:\\n            slow = head\\n            fast = head.next \\n            while slow is not fast:\\n                slow = slow.next\\n                fast = fast.next.next\\n            return True\\n        except:\\n            return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        if head is None:\\n            return False\\n\\n        slow, fast = head, head\\n        while fast.next and fast.next.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        return False\"]",
        "difficulty": "introductory",
        "input": [
            [
                1,
                2
            ],
            0
        ],
        "output": [
            true
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "hasCycle",
        "starter_code": "\n# Definition for singly-linked list.\n# class ListNode:\n#     def __init__(self, x):\n#         self.val = x\n#         self.next = None\nclass Solution:\n    def hasCycle(self, head: ListNode) -> bool:\n        ",
        "url": "https://leetcode.com/problems/linked-list-cycle/"
    },
    {
        "id": 1256,
        "task_id": 4751,
        "test_case_id": 3,
        "question": "Given head, the head of a linked list, determine if the linked list has a cycle in it.\n\nThere is a cycle in a linked list if there is some node in the list that can be reached again by continuously following the next pointer. Internally, pos is used to denote the index of the node that tail's next pointer is connected to. Note that pos is not passed as a parameter.\n\nReturn true if there is a cycle in the linked list. Otherwise, return false.\n\nInput: head = [3,2,0,-4], pos = 1\nOutput: true\nExplanation: There is a cycle in the linked list, where the tail connects to the 1st node (0-indexed).\n\n\nInput: head = [1,2], pos = 0\nOutput: true\nExplanation: There is a cycle in the linked list, where the tail connects to the 0th node.\n\n\nInput: head = [1], pos = -1\nOutput: false\nExplanation: There is no cycle in the linked list.\n\n\nConstraints:\n\nThe number of the nodes in the list is in the range [0, 104].\n-105 <= Node.val <= 105\npos is -1 or a valid index in the linked-list.",
        "solutions": "[\"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        if head == None:\\n            return False\\n        \\n        slow = head\\n        fast = head.next\\n        \\n        while slow != fast:\\n            if fast is None or fast.next is None:\\n                return False\\n            slow = slow.next\\n            fast = fast.next.next\\n        \\n        return True\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        slow = head\\n        fast = head\\n        while fast and fast.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        \\n        slow = head\\n        fast = head\\n        \\n        \\n        while fast and fast.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        nodes_seen = set()\\n        while head is not None:\\n            if head in nodes_seen:\\n                return True\\n            nodes_seen.add(head)\\n            head = head.next\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        cur = head\\n        check = set()\\n        while cur:\\n            if cur in check:\\n                return True\\n            check.add(cur)\\n            cur = cur.next\\n        return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        try:\\n            slow = head\\n            fast = head.next \\n            while slow is not fast:\\n                slow = slow.next\\n                fast = fast.next.next\\n            return True\\n        except:\\n            return False\", \"class Solution:\\n    def hasCycle(self, head: ListNode) -> bool:\\n        if head is None:\\n            return False\\n\\n        slow, fast = head, head\\n        while fast.next and fast.next.next:\\n            slow = slow.next\\n            fast = fast.next.next\\n            if slow == fast:\\n                return True\\n        return False\"]",
        "difficulty": "introductory",
        "input": [
            [
                1
            ],
            -1
        ],
        "output": [
            false
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "hasCycle",
        "starter_code": "\n# Definition for singly-linked list.\n# class ListNode:\n#     def __init__(self, x):\n#         self.val = x\n#         self.next = None\nclass Solution:\n    def hasCycle(self, head: ListNode) -> bool:\n        ",
        "url": "https://leetcode.com/problems/linked-list-cycle/"
    },
    {
        "id": 1257,
        "task_id": 2470,
        "test_case_id": 1,
        "question": "Given two integer arrays arr1 and arr2, return the minimum number of operations (possibly zero) needed to make arr1 strictly increasing.\nIn one operation, you can choose two indices 0 <= i < arr1.length and 0 <= j < arr2.length and do the assignment arr1[i] = arr2[j].\nIf there is no way to make arr1 strictly increasing, return -1.\n \nExample 1:\nInput: arr1 = [1,5,3,6,7], arr2 = [1,3,2,4]\nOutput: 1\nExplanation: Replace 5 with 2, then arr1 = [1, 2, 3, 6, 7].\n\nExample 2:\nInput: arr1 = [1,5,3,6,7], arr2 = [4,3,1]\nOutput: 2\nExplanation: Replace 5 with 3 and then replace 3 with 4. arr1 = [1, 3, 4, 6, 7].\n\nExample 3:\nInput: arr1 = [1,5,3,6,7], arr2 = [1,6,3,3]\nOutput: -1\nExplanation: You can't make arr1 strictly increasing.\n \nConstraints:\n\n1 <= arr1.length, arr2.length <= 2000\n0 <= arr1[i], arr2[i] <= 10^9",
        "solutions": "[\"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            prev_with_op = min_last_value_given_operations(n - 1, ops - 1)\\n            b = find_larger_value_in_B(prev_with_op)\\n            # dp(n - 1, ops) <= dp(n - 1, ops - 1)\\n            # if dp(n - 1, ops - 1) < A[n - 1] => dp(n - 1, ops) < A[n - 1]\\n            if prev_with_op < A[n - 1]:\\n                return min(A[n - 1], b)\\n            elif b <= A[n - 1]:\\n                return b\\n            elif min_last_value_given_operations(n - 1, ops) < A[n - 1]:\\n                return A[n - 1]\\n            return b\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        from bisect import bisect_right as br\\n        arr2.sort()\\n        \\n        dp = {0:-math.inf}\\n        # min_cnt = 0\\n        for n1 in arr1:\\n            # print(n1)\\n            new_dp = {}\\n            # cnt = min_cnt\\n            for cnt in dp:\\n                if n1 > dp[cnt]:\\n                    new_dp[cnt] = min(new_dp.get(cnt, math.inf), n1)\\n                i2 = br(arr2, dp[cnt])\\n                if i2 < len(arr2):\\n                    new_dp[cnt+1] = min(new_dp.get(cnt+1, math.inf), arr2[i2])\\n                cnt += 1\\n            if len(new_dp) == 0:\\n                return -1\\n            # while min_cnt not in new_dp:\\n            #     min_cnt += 1\\n            dp = new_dp\\n            # print(dp)\\n        return min(dp.keys())\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = list(set(arr2))\\n        arr2.sort()\\n        m, n = len(arr1), len(arr2)\\n        keep = [float('inf')]*m\\n        swap = [[float('inf') for _ in range(n)] for _ in range(m)]\\n        keep[0] = 0\\n        for i in range(n):\\n            swap[0][i] = 1\\n        for i in range(1, m):\\n            min_keep, min_swap = float('inf'), float('inf')\\n            for j in range(n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[i-1][j-1]+1)\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j])\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i-1]+1\\n                keep[i] = min(keep[i], min_keep)\\n                swap[i][j] = min(swap[i][j], min_swap)\\n        s = float('inf')\\n        for i in range(n):\\n            s = min(s, swap[m-1][i])\\n        res = min(s, keep[m-1])\\n        return -1 if res >= float('inf') else res\\n\\n# \\u89c1\\u82b1\\u82b1\\u9171\\u89c6\\u9891\\uff1ahttps://www.bilibili.com/video/av67133426/?spm_id_from=333.788.b_636f6d6d656e74.7\\n                    \\n        \\n\", \"import numpy as np\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        m = len(arr1)\\n        arr2 = sorted(np.unique(arr2))\\n        n = len(arr2)\\n        \\n        keep = [float('inf')] * m\\n        keep[0] = 0\\n        swap = [1] * n\\n        \\n        for i in range(1, m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            temp = [float('inf')] * n\\n            for j in range(n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[j - 1] + 1)\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[j])\\n                if arr1[i] > arr1[i - 1]:\\n                    keep[i] = keep[i - 1]\\n                if arr2[j] > arr1[i - 1]:\\n                    temp[j] = keep[i - 1] + 1\\n                temp[j] = min(temp[j], min_swap)\\n                keep[i] = min(keep[i], min_keep)\\n            for j in range(n):\\n                temp[j], swap[j] = swap[j], temp[j]\\n        \\n        s = min(swap)\\n        k = keep[-1]\\n        ans = min(s, k)\\n        return ans if ans < float('inf') else -1\\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        m = len(arr1)\\n        arr2 = list(set(arr2))\\n        arr2.sort()\\n        n = len(arr2)\\n        keep = [float('inf')] * m\\n        keep[0] = 0\\n        swap = [[float('inf') for j in range(n)] for i in range(m)]\\n        for j in range(n):\\n            swap[0][j] = 1\\n        for i in range(1,m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            for j in range(0, n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[i-1][j-1] + 1 )\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j]  )\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i -1]\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i -1] + 1\\n                swap[i][j] = min(swap[i][j], min_swap)\\n                keep[i]  = min(keep[i], min_keep )\\n        \\n        k = keep[-1]\\n        min_swap = min(swap[-1])\\n        ans = min (min_swap, k)\\n        return ans if ans < float('inf') else -1\", \"from bisect import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        d = {float('-inf'): 0}\\n        arr2 = sorted(set(arr2))\\n        for i in arr1:\\n            d2 = {}\\n            for k, v in list(d.items()):\\n                if i > k:\\n                    if i in d2:\\n                        d2[i] = min(d2[i], v)\\n                    else:\\n                        d2[i] = v\\n\\n                if k < arr2[-1]:\\n                    j = arr2[bisect(arr2, k)]\\n                    if j in d2:\\n                        d2[j] = min(d2[j], v + 1)\\n                    else:\\n                        d2[j] = v + 1\\n            \\n            d = d2\\n        \\n        if d:\\n            return min(d.values())\\n        return -1\\n                        \\n        \\n        \\n        \\n        \\n\", \"from collections import defaultdict\\nfrom math import inf\\nfrom bisect import bisect_right\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        n = len(arr2)\\n        dp = {-1: 0}\\n        for i in arr1:\\n            next_dp = defaultdict(lambda: inf)\\n            for key in dp:\\n                if i > key:\\n                    next_dp[i] = min(next_dp[i], dp[key])\\n                loc = bisect_right(arr2, key)\\n                if loc < n:\\n                    next_dp[arr2[loc]] = min(next_dp[arr2[loc]], dp[key] + 1)\\n            dp = next_dp\\n\\n        return min(dp.values()) if dp else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        from bisect import bisect_right as br\\n        arr2.sort()\\n        \\n        dp = {0:-math.inf}\\n        min_cnt = 0\\n        for n1 in arr1:\\n            # print(n1)\\n            new_dp = {}\\n            cnt = min_cnt\\n            while cnt in dp:\\n                if n1 > dp[cnt]:\\n                    new_dp[cnt] = min(new_dp.get(cnt, math.inf), n1)\\n                    # candidate = min(new_dp.get(cnt, math.inf), n1)\\n                    # if candidate < new_dp.get(cnt-1, math.inf):\\n                    #     new_dp[cnt] = candidate\\n                i2 = br(arr2, dp[cnt])\\n                # if i2 < len(arr2) and arr2[i2] < new_dp.get(cnt, math.inf):\\n                if i2 < len(arr2):\\n                    new_dp[cnt+1] = arr2[i2]\\n                # print(new_dp)\\n                # if new_dp.get(cnt+1, -math.inf) >= new_dp.get(cnt, math.inf):\\n                #     new_dp.pop(cnt+1)\\n                # if new_dp.get(cnt, -math.inf) >= new_dp.get(cnt-1, math.inf):\\n                #     new_dp.pop(cnt)\\n                cnt += 1\\n            if len(new_dp) == 0:\\n                return -1\\n            while min_cnt not in new_dp:\\n                min_cnt += 1\\n            dp = new_dp\\n            print(dp)\\n        return min(dp.keys())\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        import bisect\\n        # dp\\u5b58\\u50a8\\u6240\\u6709\\u6f5c\\u5728\\u7684\\u5f53\\u524d\\u72b6\\u6001\\uff08\\u6bcf\\u6b21dp\\u90fd\\u662f\\u904d\\u5386arr1\\u65f6\\u524d\\u4e00\\u4e2a\\u4f4d\\u7f6e\\u7684\\u72b6\\u6001\\uff09\\n        # \\u8fd9\\u91cc\\u7684\\u72b6\\u6001\\u662f\\u4e00\\u4e2a\\u952e\\u503c\\u5bf9\\uff0ckey\\u4ee3\\u8868\\u5f53\\u524d\\u4f4d\\u7f6e\\u7684\\u6570\\u5b57\\uff0c\\n        # \\u53ef\\u4ee5\\u662farr1\\u91cc\\u9762\\u7684\\uff0c\\u4e5f\\u53ef\\u4ee5\\u662farr2\\u91cc\\u9762\\u7528\\u6765\\u66ff\\u6362\\u7684\\n        # value\\u5c31\\u662f\\u6211\\u4eec\\u9700\\u8981\\u64cd\\u4f5c\\u7684\\u6b21\\u6570\\n        dp = {-1: 0}\\n        arr2.sort()\\n        for i in arr1:\\n            tmp = collections.defaultdict(lambda: float('inf'))\\n            for key in dp:\\n                if i > key:\\n                    tmp[i] = min(tmp[i], dp[key])\\n                loc = bisect.bisect_right(arr2, key)\\n                if loc < len(arr2):\\n                    tmp[arr2[loc]] = min(tmp[arr2[loc]], dp[key] + 1)\\n            dp = tmp\\n        return min(dp.values()) if dp else -1\\n\", \"from bisect import bisect_right as br\\nimport math\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i == len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if prev < arr1[i] else math.inf\\n            return min(swap, noswap)  \\n        ret = dfs(0, -math.inf)\\n        return ret if ret != math.inf else -1\\n        \\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"from bisect import bisect_right as br\\nimport math\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))   \\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i >= len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if prev < arr1[i] else math.inf\\n            return min(swap, noswap)\\n        ans = dfs(0, -1)\\n        return ans if ans != math.inf else -1\\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        import bisect\\n        dp = {-1: 0}\\n        arr2.sort()\\n        for i in arr1:\\n            tmp = collections.defaultdict(lambda: float('inf'))\\n            for key in dp:\\n                if i > key:\\n                    tmp[i] = min(tmp[i], dp[key])\\n                loc = bisect.bisect_right(arr2, key)\\n                if loc < len(arr2):\\n                    tmp[arr2[loc]] = min(tmp[arr2[loc]], dp[key] + 1)\\n            dp = tmp\\n        return min(dp.values()) if dp else -1\\n\", \"import bisect\\nimport functools\\n    \\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        \\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"from bisect import bisect_right as br\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i >= len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n            return min(swap, noswap)\\n        \\n        \\n        ans = dfs(0, -math.inf)\\n        return ans if ans!=math.inf else -1\\n        \\n        \\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"import bisect\\nimport functools\\n    \\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # arr1i\\n        # arr2j\\n        arr2 = sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(prev, i):\\n            if i == len(arr1):\\n                return 0\\n            j = bisect.bisect_right(arr2, prev)\\n            swap = 1 + dfs(arr2[j], i+1) if j < len(arr2) else float('inf')\\n            noswap = dfs(arr1[i], i+1) if prev < arr1[i] else float('inf')\\n            return min(swap, noswap)\\n        changes = dfs(float('-inf'), 0)\\n        return changes if changes != float('inf') else -1\\n\", \"class Solution:\\n  def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n    # TC: O(min(M, N)*N+MlogM), SC: O(min(M, N))\\n    m, n = len(arr1), len(arr2)\\n    # sort O(MlogM)\\n    arr2.sort()\\n    # maintain a list of (num-of-swaps, last-value, next-to-swap-index-arr2),\\n    # where the last value should be decrease as the num of swaps is increase\\n    ss = [(0, arr1[0], 0)]\\n    if arr2[0] < arr1[0]:\\n      ss.append((1, arr2[0], 1))\\n    # O(N) iteration\\n    for i in range(1, m):\\n      st = []\\n      # O(min(M, N))\\n      for s, x, j in ss:\\n        if x < arr1[i]:\\n          if st:\\n            if s == st[-1][0]:\\n              if arr1[i] < st[-1][1]:\\n                st[-1] = (s, arr1[i], j)\\n            elif s > st[-1][0]:\\n              if arr1[i] < st[-1][1]:\\n                st.append((s, arr1[i], j))\\n          else:\\n            st.append((s, arr1[i], j))\\n        # amortized O(1)\\n        while j < n and arr2[j] <= x:\\n          j += 1\\n        if j < n:\\n          st.append((s + 1, arr2[j], j))\\n      # since each swap takes at most 1 entry, and at most O(min(M, N)) swap, so SC: O(min(M, N))\\n      ss = st\\n    return ss[0][0] if ss else -1\", \"import bisect\\nfrom typing import List\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        array = arr1\\n        replace = list(sorted(set(arr2)))\\n\\n        @lru_cache(None)\\n        def dfs(array_pos: int, prev_number: int) -> int:\\n            min_replacements = len(replace) * 2\\n\\n            if array_pos == len(array):\\n                return 0\\n\\n            next_replace_pos = 0\\n\\n            if array_pos > 0:\\n                next_replace_pos = bisect.bisect(replace, prev_number)\\n\\n            if array_pos == 0 or (\\n                next_replace_pos < len(replace)\\n            ):\\n                tmp = array[array_pos]\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, replace[next_replace_pos]) + 1)\\n\\n            if array_pos == 0 or prev_number < array[array_pos]:\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]))\\n\\n            return min_replacements\\n\\n        result = dfs(0, -100)\\n\\n        if result > len(replace):\\n            return -1\\n\\n        return result\\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        b2idx = {b: i for i, b in enumerate(B)}\\n        \\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            if val in b2idx:\\n                return B[b2idx[val] + 1]\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j = bisect.bisect_right(arr2,prev)\\n            swap = 1 + dfs(i+1,arr2[j]) if j < len(arr2)  else math.inf\\n            noswap = dfs(i+1,arr1[i])   if arr1[i] > prev else math.inf\\n            return min(swap,noswap)\\n        changes = dfs(0,-math.inf)\\n        return changes if changes != math.inf else -1\\n\", \"import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        N = len(arr1)\\n        arr1 = [0] + arr1\\n        # \\u8868\\u793a\\u524d i \\u4e2a\\u5143\\u7d20\\uff0c\\u6267\\u884c\\u4e86 k \\u6b21\\u64cd\\u4f5c\\u540e\\uff0c\\u662f\\u6709\\u5e8f\\u7684\\n        dp = [[float('inf')] * (N + 1) for _ in range(N + 1)]\\n        dp[0][0] = -float('inf')\\n        \\n        arr2.sort()\\n        for i in range(1, N + 1):\\n            for k in range(i + 1):\\n                # \\u524di-1\\u4e2a\\u5143\\u7d20\\uff0c\\u5df2\\u7ecf\\u5b8c\\u6210\\u4e86k\\u6b21\\u4ea4\\u6362\\n                if arr1[i] > dp[i-1][k]:\\n                    dp[i][k] = min(dp[i][k], arr1[i])\\n                \\n                # \\u524d i-1 \\u4e2a\\u5143\\u7d20\\uff0c\\u5df2\\u7ecf\\u5b8c\\u6210\\u4e86 k-1\\u6b21\\u4ea4\\u6362\\uff0c\\u6240\\u4ee5\\u8fd9\\u4e00\\u6b21\\u4e00\\u5b9a\\u8981\\u4ea4\\u6362\\n                if k >= 1:\\n                    idx_2 = bisect.bisect_right(arr2, dp[i-1][k-1])\\n                    if idx_2 != len(arr2):                    \\n                        dp[i][k] = min(dp[i][k], arr2[idx_2])\\n        \\n        res = float('inf')\\n        for i in range(1, N+1):\\n            if dp[N][i] != float('inf'):\\n                res = min(res, i)\\n        return res if res != float('inf') else -1\", \"import bisect\\nfrom typing import List\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        array = arr1\\n        replace = list(sorted(set(arr2)))\\n\\n        @lru_cache(None)\\n        def dfs(array_pos: int, prev_number: int) -> int:\\n            min_replacements = len(replace) * 2\\n\\n            if array_pos == len(array):\\n                return 0\\n\\n            next_replace_pos = 0\\n\\n            if array_pos > 0:\\n                next_replace_pos = bisect.bisect(replace, array[array_pos - 1])\\n\\n            if array_pos == 0 or (\\n                next_replace_pos < len(replace)\\n            ):\\n                tmp = array[array_pos]\\n                array[array_pos] = replace[next_replace_pos]\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]) + 1)\\n                array[array_pos] = tmp\\n\\n            if array_pos == 0 or array[array_pos - 1] < array[array_pos]:\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]))\\n\\n            return min_replacements\\n\\n        result = dfs(0, -100)\\n\\n        if result > len(replace):\\n            return -1\\n\\n        return result\\n\", \"import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1, arr2) -> int:\\n        N = len(arr1)\\n        arr1 = [0] + arr1\\n        arr2.sort()\\n        dp = [[float('inf')] * (N + 1) for _ in range(N + 1)]\\n        dp[0][0] = -float('inf')\\n        \\n        for i in range(1, N + 1):\\n            for j in range(0, i + 1):\\n                if dp[i-1][j] < arr1[i]:\\n                    dp[i][j] = min(dp[i][j], arr1[i])\\n                if j >= 1:\\n                    # \\u8981\\u5728 arr2 \\u4e2d\\u627e\\u5230\\u4e00\\u4e2a\\u6bd4 arr1[i] \\u7a0d\\u5fae\\u5927\\u4e00\\u70b9\\u7684\\u6570\\n                    idx_2 = bisect.bisect_right(arr2, dp[i-1][j-1])\\n                    if idx_2 != len(arr2):\\n                        dp[i][j] = min(dp[i][j], arr2[idx_2])\\n        \\n        # \\u6ee1\\u8db3\\u6761\\u4ef6\\uff0c\\u5e76\\u4e14\\u80fd\\u591f\\u6267\\u884c\\u6700\\u5c11\\u7684\\u6b21\\u6570\\u7684 K \\u7684\\u503c\\n        res = float('inf')\\n        for i in range(1, N + 1):\\n            if dp[N][i] != float('inf'):\\n                res = min(res, i)\\n        return -1 if res == float('inf') else res\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        \\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        \\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        def bs(arr, l, r, target):\\n            while l<=r:\\n                m = l+(r-l)//2\\n                if arr[m]>target:\\n                    r = m-1\\n                else:\\n                    l = m+1\\n            return l\\n        \\n        arr2.sort()\\n        N = len(arr2)\\n        dp = {-1:0}\\n        \\n        for a in arr1:\\n            dp2 = {}\\n            for prev in dp:\\n                if a>prev:\\n                    dp2[a] = min(dp2.get(a, float('inf')), dp[prev])\\n                \\n                idx = bs(arr2, 0, N-1, prev)\\n                if idx<N:\\n                    dp2[arr2[idx]] = min(dp2.get(arr2[idx], float('inf')), dp[prev]+1)\\n                    \\n            dp = dp2\\n            if not dp:\\n                return -1\\n            \\n        return min(dp.values())\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        \\n        def find_lower_val_in_B(val):\\n            larger_equal_idx = bisect.bisect_left(B, val)\\n            if larger_equal_idx > 0:\\n                return B[larger_equal_idx - 1]\\n            return None  # no lower value in B\\n\\n        @lru_cache(None)\\n        def make_prefix_increasing(n, upper=float('inf')):\\n            if n == 0:\\n                return 0\\n\\n            swap_b = find_lower_val_in_B(upper)\\n            ret = float('inf')\\n            if A[n - 1] < upper:\\n                ret = min(make_prefix_increasing(n - 1, upper=A[n - 1]), ret)\\n            if swap_b is not None:\\n                ret = min(1 + make_prefix_increasing(n - 1, upper=swap_b), ret)\\n\\n            return ret\\n\\n        ret = make_prefix_increasing(len(A))\\n        return ret if ret < float('inf') else -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        \\n        dp = {-1:0}\\n        B = sorted(B)\\n        \\n        for cur in A:\\n            temp = collections.defaultdict(lambda: float('inf'))\\n            for prev in dp:\\n                if prev<cur:\\n                    temp[cur] = min(temp[cur], dp[prev])\\n                idx = self.upper_bound(B, prev)\\n                if idx<len(B):\\n                    temp[B[idx]] = min(temp[B[idx]], dp[prev]+1) \\n            dp = temp\\n        \\n        if dp:\\n            return min(dp.values())\\n        return -1\\n    \\n    def upper_bound(self, B, target):\\n        l=0\\n        r=len(B)\\n        while l<r:\\n            mid=l+(r-l)//2\\n            if B[mid]<=target:\\n                l=mid+1\\n            else:\\n                r=mid\\n        return l\\n                \\n                \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n            return min(\\n                A[n - 1] if min_last_value_given_operations(n - 1, ops) < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        if not arr1: return 0\\n        if not arr2: return arr1 == sorted(arr1)\\n        \\n        arr2 = sorted(list(set(arr2)), reverse=True)\\n        \\n        n, m, inf = len(arr1), len(arr2), float('inf')\\n        f = [[inf]*(n+1) for _ in range(n)]\\n        for i in range(n+1):\\n            f[0][i] = min(arr1[0],arr2[-1])\\n        f[0][0] = arr1[0]\\n        for i in range(1,n):\\n            found_k = 0\\n            for j in range(n+1):\\n                if f[i-1][j] < arr1[i]: f[i][j] = arr1[i]\\n                if not j: continue\\n                va = f[i-1][j-1]\\n                if not found_k:\\n                    if arr2[0] > va:\\n                        l,r = 0,m-1\\n                        while l < r:\\n                            mid = l+r+1 >> 1\\n                            if arr2[mid] > va:\\n                                l = mid\\n                            else: \\n                                r = mid - 1\\n                        k = l\\n                        found_k = 1\\n                        f[i][j] = min(f[i][j],arr2[k])\\n                else:\\n                    while k+1 < m and arr2[k+1] > va: k += 1\\n                    f[i][j] = min(f[i][j],arr2[k])\\n        for i,v in enumerate(f[-1]): \\n            if v < inf: return i\\n        return -1\\n                \\n                \\n                \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            prev_with_op = min_last_value_given_operations(n - 1, ops - 1)\\n            b = find_larger_value_in_B(prev_with_op)\\n            # dp(n - 1, ops) <= dp(n - 1, ops - 1)\\n            # if dp(n - 1, ops - 1) < A[n - 1] => dp(n - 1, ops) < A[n - 1]\\n            if prev_with_op < A[n - 1]:\\n                return min(A[n - 1], b)\\n            elif b <= A[n - 1]:\\n                return b\\n            elif min_last_value_given_operations(n - 1, ops) < A[n - 1]:\\n                return A[n - 1]\\n            return b\\n\\n        for ops in range(min(len(A), len(B)) + 1):\\n            if min_last_value_given_operations(len(A), ops) < float('inf'):\\n                return ops\\n\\n        return -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"def binsearch(arr,x):\\n    if(arr[0]>x):\\n        return 0\\n    l=0\\n    h=len(arr)-1\\n    ret=-1\\n    while(l<=h):\\n        mid=(l+h)//2\\n        if(arr[mid]<=x):\\n            l=mid+1\\n        elif(arr[mid]>x):\\n            ret=mid\\n            h=mid-1\\n    return ret\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m=len(arr2)\\n        n=len(arr1)\\n        dp={}\\n        # print(binsearch(arr2,0))\\n        def dfs(arr1,arr2,left,curr,dp):\\n            if(curr>=len(arr1)):\\n                return 0\\n            if((curr,left) in dp):\\n                return dp[(curr,left)]\\n            res1=sys.maxsize\\n            res2=0\\n            if(arr1[curr]>left):\\n                res1=dfs(arr1,arr2,arr1[curr],curr+1,dp)\\n            mid=binsearch(arr2,left)\\n            if(mid==-1):\\n                res2=sys.maxsize-1\\n            else:\\n                res2=dfs(arr1,arr2,arr2[mid],curr+1,dp)\\n            dp[(curr,left)]=min(res1,1+res2)\\n            return dp[(curr,left)]\\n        x=dfs(arr1,arr2,-sys.maxsize,0,dp)\\n        if(x>=(sys.maxsize-1)):\\n            return -1\\n        return x\\n            \\n\", \"def binsearch(arr,x):\\n    if(arr[0]>x):\\n        return 0\\n    l=0\\n    h=len(arr)-1\\n    ret=-1\\n    while(l<=h):\\n        mid=(l+h)//2\\n        if(arr[mid]<=x):\\n            l=mid+1\\n        elif(arr[mid]>x):\\n            ret=mid\\n            h=mid-1\\n    return ret\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m=len(arr2)\\n        n=len(arr1)\\n        dp={}\\n        def dfs(arr1,arr2,left,curr,dp):\\n            if(curr>=len(arr1)):\\n                return 0\\n            if((curr,left) in dp):\\n                return dp[(curr,left)]\\n            res1=sys.maxsize\\n            res2=0\\n            if(arr1[curr]>left):\\n                res1=dfs(arr1,arr2,arr1[curr],curr+1,dp)\\n            mid=binsearch(arr2,left)\\n            if(mid==-1):\\n                res2=sys.maxsize-1\\n            else:\\n                res2=dfs(arr1,arr2,arr2[mid],curr+1,dp)\\n            dp[(curr,left)]=min(res1,1+res2)\\n            return dp[(curr,left)]\\n        x=dfs(arr1,arr2,-sys.maxsize,0,dp)\\n        if(x==(sys.maxsize)):\\n            return -1\\n        return x\\n            \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        b2idx = {b: i for i, b in enumerate(B)}\\n        \\n        # @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            if val in b2idx:\\n                return B[b2idx[val] + 1]\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        arr2.sort()\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        \\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=n:\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,m-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        arr2.sort()\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans\\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                \\n                    \\n#                 for k in range(j,len(arr2)):\\n#                     # can we use binary search here\\n#                     # we got to find out the minumum value in arr2 which is greater than prev\\n                    \\n#                     if arr2[k]>prev:\\n#                         # we can probably use binary search here\\n#                         # get the first index which is strictly greater than prev\\n#                         ans=min(ans,1+helper(i+1,k+1,arr2[k]))\\n#                         break\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"import bisect\\nclass Solution:\\n    def recurse(self,arr1,arr2,m,n,idx,prev,dp):\\n        if idx>=n:\\n            return 0\\n        k=bisect.bisect_right(arr2,prev)\\n        if dp[idx][k]!=-1:\\n            return dp[idx][k]\\n        c1=self.recurse(arr1,arr2,m,n,idx+1,arr1[idx],dp) if arr1[idx]>prev else 2000\\n        c2=1+self.recurse(arr1,arr2,m,n,idx+1,arr2[k],dp) if k<m else 2000\\n        dp[idx][k]=min(c1,c2)\\n        return dp[idx][k]\\n     \\n\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        dp=[[-1 for i in range(2001)] for j in range(2001)]\\n        arr2.sort()\\n        self.recurse(arr1,arr2,len(arr2),len(arr1),0,-10**9,dp)\\n        if dp[0][0]>=2000:\\n            return -1\\n        else:\\n            return dp[0][0]\\n      \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # Maintain two dp array: 1 for keep arr1[i], 1 for swap arr1[i] with arr2[j]\\n        # Convert arr2 to sorted unique set\\n        a = arr1\\n        b = sorted(list(set(arr2)))\\n        n, m = len(a), len(b)\\n        keep = [0, math.inf]  # keep[i] = keep a[i]\\n        swap = [[1] * m, [math.inf] * m]  # swap[i][j] = a[i] replaced with b[j]\\n\\n        prev, curr = 1, 0\\n        for i in range(1, n):\\n            prev, curr = curr, prev\\n            # must init every time\\n            swap[curr] = [math.inf] * m\\n            keep[curr] = math.inf\\n            # keep[i] case 1: a[i] is bigger than previous keep[i-1]\\n            if a[i] > a[i - 1]:\\n                keep[curr] = keep[prev]\\n            for j in range(m):\\n                # keep[i] case 2: a[i] is bigger then previous swap value.\\n                if a[i] > b[j]:\\n                    keep[curr] = min(keep[curr], swap[prev][j])\\n                # swap case 1: when a[i-1] keeps\\n                if b[j] > a[i - 1]:\\n                    swap[curr][j] = min(swap[curr][j], 1 + keep[prev])\\n                # Swap case 2: when a[i-1] swapped\\n                if j > 0:\\n                    # only need to check last swap[i-1][j-1] since swap[][j] decreases when j increases.\\n                    swap[curr][j] = min(swap[curr][j], 1 + swap[prev][j - 1])\\n\\n        res = min(keep[curr], swap[curr][m - 1])\\n        return res if res < math.inf else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=n:\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                # pick first index from arr2 starting j which is greater than prev\\n                idx=bsearch(j,m-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n\\n    \\n                \\n                \\n        \\n        \\n\", \"import bisect\\nclass Solution:\\n    def recurse(self,arr1,arr2,m,n,idx,prev,dp):\\n        if idx>=n:\\n            return 0\\n        k=bisect.bisect_right(arr2,prev)\\n        if dp[idx][k]!=-1:\\n            return dp[idx][k]\\n        c1=self.recurse(arr1,arr2,m,n,idx+1,arr1[idx],dp) if arr1[idx]>prev else 2000\\n        c2=1+self.recurse(arr1,arr2,m,n,idx+1,arr2[k],dp) if k<m else 2000\\n        dp[idx][k]=min(c1,c2)\\n        return dp[idx][k]\\n     \\n\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        dp=[[-1 for i in range(2001)] for j in range(2001)]\\n        arr2.sort()\\n        ans=self.recurse(arr1,arr2,len(arr2),len(arr1),0,-10**9,dp)\\n        if ans>=2000:\\n            return -1\\n        else:\\n            return ans\\n      \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # Maintain two dp array: 1 for keep arr1[i], 1 for swap arr1[i] with arr2[j]\\n        # Convert arr2 to sorted unique set\\n        a = arr1\\n        b = sorted(list(set(arr2)))\\n        n, m = len(a), len(b)\\n        keep = [math.inf] * n  # keep[i] = keep a[i]\\n        swap = [[math.inf] * m for _ in range(n)]  # swap[i][j] = a[i] replaced with b[j]\\n        # init\\n        keep[0] = 0  # keep a[0], no swap\\n        swap[0] = [1] * m  # can be swapped with any a[j]\\n\\n        for i in range(1, n):\\n            # keep[i] case 1: a[i] is bigger than previous keep[i-1]\\n            if a[i] > a[i - 1]:\\n                keep[i] = keep[i - 1]\\n            for j in range(m):\\n                # keep[i] case 2: a[i] is bigger then previous swap value.\\n                if a[i] > b[j]:\\n                    keep[i] = min(keep[i], swap[i - 1][j])\\n                # swap case 1: when a[i-1] keeps\\n                if b[j] > a[i - 1]:\\n                    swap[i][j] = min(swap[i][j], 1 + keep[i - 1])\\n                # Swap case 2: when a[i-1] swapped\\n                if j > 0:\\n                    # only need to check last swap[i-1][j-1] since swap[][j] decreases when j increases.\\n                    swap[i][j] = min(swap[i][j], 1 + swap[i - 1][j - 1])\\n\\n        res = min(keep[n - 1], swap[n - 1][m - 1])\\n        return res if res < math.inf else -1        \", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m = len(arr1)\\n        n = len(arr2)\\n        newarr2 = [arr2[0]]\\n        for i in range(1, n):\\n            if arr2[i] != newarr2[-1]:\\n                newarr2.append(arr2[i])\\n                \\n        arr2 = newarr2\\n        n = len(arr2)\\n\\n        IL = 10 ** 9 + 7\\n        \\n        dp = [[IL for j in range(n + 1)] for i in range(m)]\\n        \\n        dp[0][n] = 0\\n        for i in range(n):\\n            if arr2[i] < arr1[0]:\\n                dp[0][i] = 1\\n            else:\\n                break\\n        \\n        for i in range(1, m):\\n            idx = 0\\n            if arr2[0] > arr1[i - 1]:\\n                dp[i][0] = dp[i - 1][n] + 1\\n            \\n            for k in range(1, n):\\n                a = dp[i - 1][n] + 1 if dp[i - 1][n] != IL and arr1[i - 1] < arr2[k] else IL\\n                b = dp[i - 1][idx] + 1 if dp[i - 1][idx] != IL else IL\\n                dp[i][k] = min(a, b)\\n                if dp[i - 1][k] < dp[i - 1][idx]:\\n                    idx = k\\n                    \\n            if dp[i - 1][n] != IL and arr1[i] > arr1[i -1]:\\n                dp[i][n] = dp[i - 1][n]\\n            \\n            for k in range(n):\\n                if dp[i - 1][k] != IL and arr2[k] < arr1[i]:\\n                    dp[i][n] = min(dp[i][n], dp[i - 1][k])\\n                    \\n        m = min(dp[m - 1])\\n        \\n        return m if m != IL else -1\\n        \\n                \\n\", \"class Solution:    \\n    def binary_search_right(self, arr, l, r, t):\\n        while l < r:\\n            m = l + (r - l) // 2\\n            if arr[m] <= t:\\n                l = m + 1\\n            else:\\n                r = m\\n        return l\\n    \\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        \\n        dp = {}\\n        def help(i1, i2):\\n            if i1 == len(arr1):\\n                return 0\\n            \\n            if i1 != 0:\\n                key = (i1, i2, arr1[i1 - 1])\\n            else:\\n                key = (i1, i2, None)\\n            \\n            if key in dp:\\n                return dp[key]\\n            \\n            # print(key)\\n            \\n            # all possible results\\n            pos = []\\n            \\n            # don't make changes here\\n            if i1 == 0 or arr1[i1 - 1] < arr1[i1]:\\n                res = help(i1 + 1, i2)\\n                if res != -1:\\n                    pos.append(res)\\n                    \\n            if i1 != 0:\\n                # make change\\n                i2 = self.binary_search_right(arr2, i2, len(arr2), arr1[i1 - 1])\\n                if i2 != len(arr2):\\n                    tmp = arr1[i1]\\n                    arr1[i1] = arr2[i2]\\n                    res = help(i1 + 1, i2 + 1)\\n                    if res != -1:\\n                        pos.append(res + 1)\\n                    arr1[i1] = tmp\\n            else:\\n                # make change\\n                if i2 < len(arr2) and arr2[i2] < arr1[i1]:\\n                    tmp = arr1[i1]\\n                    arr1[i1] = arr2[i2]\\n                    res = help(i1 + 1, i2 + 1)\\n                    if res != -1:\\n                        pos.append(res + 1)\\n                    arr1[i1] = tmp\\n            \\n            if len(pos) == 0:\\n                dp[key] = -1\\n            else:\\n                dp[key] = min(pos)\\n            return dp[key]\\n        \\n        return help(0, 0)\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        \\n        # dp[i][k] means the minimum number we can have at ith position with k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                if dp[i - 1][k] < arr1[i]:\\n                    # not assign\\n                    dp[i][k] = arr1[i]\\n                \\n                num = self.helper(arr2, dp[i - 1][k - 1])\\n                if num > dp[i - 1][k - 1]:\\n                    dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n\\n    def helper(self, arr, val):\\n        # find in arr the smallest number that is larger than val\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            else:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        size1, size2 = len(arr1), len(arr2)\\n        Inf = float('inf')\\n        keep = [Inf] * size1\\n        keep[0] = 0\\n        swap = [[Inf] * size2 for _ in range(size1)]\\n        swap[0] = [1] * size2\\n        for i in range(1, size1):\\n            minKeep = minSwap = Inf\\n            for j in range(size2):\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                if arr1[i] > arr2[j]:\\n                    minKeep = min(minKeep, swap[i-1][j])\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i-1] + 1\\n                if j > 0: # arr2[j] > arr2[j-1] is always True\\n                    minSwap = min(minSwap, swap[i-1][j-1] + 1)\\n                keep[i] = min(keep[i], minKeep)\\n                swap[i][j] = min(swap[i][j], minSwap)\\n        res = min(min(swap[-1]), keep[-1])\\n        return -1 if res == Inf else res\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        \\n        # dp[i][k] means the minimum number we can have at ith position with k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                if dp[i - 1][k] < arr1[i]:\\n                    # not assign\\n                    dp[i][k] = arr1[i]\\n                \\n                if k >= 1:\\n                    num = self.helper(arr2, dp[i - 1][k - 1])\\n                    if num > dp[i - 1][k - 1]:\\n                        dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n\\n    def helper(self, arr, val):\\n        # find in arr the smallest number that is larger than val\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            else:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        # dp[i][k] means the minimum number we can get at arr1[i] using k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                # not changing for arr1[i]\\n                if dp[i - 1][k] < arr1[i]:\\n                    dp[i][k] = arr1[i]\\n                \\n                # changing for arr1[i]\\n                # find the smallested number in arr2 that is larger than dp[i - 1][k - 1]\\n                if k >= 1:\\n                    num = self.helper(arr2, dp[i - 1][k - 1])\\n                    if num > dp[i - 1][k - 1]:\\n                        dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(0, n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n    \\n    def helper(self, arr, val):\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            elif arr[mid] > val:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        set2 = set(arr2)\\n        arr2 = sorted(set2)\\n        N, M = len(arr1), len(arr2)\\n        to_index = {}\\n        left = 0\\n        for x in sorted(arr1):\\n            while left < M and arr2[left] <= x:\\n                left += 1\\n            to_index[x] = left - 1 if left < M else M\\n        \\n        print(to_index, arr2)\\n        dp = [[[None, None] for _ in range(M + 1)] for _ in range(N)]\\n        \\n        def solve(i, j, k):\\n            if i == N: return 0\\n            if j > M or (j == M and k == 1): return N+1\\n            if dp[i][j][k] is None:\\n                if i == 0: \\n                    dp[i][j][k] = min(solve(i+1, to_index[arr1[0]], 0), 1 + solve(i+1, 0, 1))\\n                else:\\n                    result = N + 1\\n                    if j < M - 1: \\n                        result = 1 + solve(i+1, j+1, 1)\\n                    if k == 0 and arr1[i] > arr1[i-1]:\\n                        result = min(result, solve(i+1, to_index[arr1[i]], 0))\\n                    if k == 1 and arr1[i] > arr2[j]:\\n                        result = min(result, solve(i+1, to_index[arr1[i]], 0))\\n                    dp[i][j][k] = result\\n            return dp[i][j][k]\\n        result = solve(0, 0, 0)\\n        # for i in range(N):\\n        #     for j in range(M):\\n        #         for k in [0, 1]:\\n        #             print(dp[i][j][k], i, j, k)\\n        return result if result <= N else -1\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            if n == 0:\\n                return float('-inf')\\n                # return A[n - 1] if first_n_of_A_is_sorted(n) else float('inf')\\n\\n            ops = min(n, ops)\\n            return min(\\n                A[n - 1] if min_last_value_given_operations(n - 1, ops) < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        for ops in range(min(len(A), len(B)) + 1):\\n            if min_last_value_given_operations(len(A), ops) < float('inf'):\\n                return ops\\n\\n        return -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        ## https://www.youtube.com/watch?v=8ttxdMCU2GE\\n        m = len(arr1)\\n        ## remove dulpicates and sort arr2\\n        arr2 = sorted(list(set(arr2)))\\n        # print(arr2)\\n        n = len(arr2)\\n        swap = [[float('inf') for j in range(n)] for i in range(m)]\\n        keep = [float('inf') for i in range(m)]\\n        \\n        ## initialization\\n        keep[0] = 0\\n        for j in range(n):\\n            swap[0][j] = 1\\n            \\n        for i in range(1, m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            for j in range(n):\\n                ## two variables to help compute case 3 & 4\\n                ## case 4: the last two elements of current valid array are both from arr2\\n                if j>0:\\n                    min_swap = min(min_swap, swap[i-1][j-1]+1)\\n\\n                ## case 3: the second to last element is replaced by the previous element arr2[j-1] or earlier elements\\n                if arr1[i]>arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j])\\n                    \\n                ## case 1: no need to swap; keep arr1[i]\\n                if arr1[i]>arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                \\n                ## case 2: ## replace arr1[i] by arr2[j]\\n                if arr2[j]>arr1[i-1]:\\n                    swap[i][j] = keep[i-1] + 1\\n                \\n                ## update\\n                swap[i][j] = min(swap[i][j], min_swap)\\n                keep[i] = min(keep[i], min_keep)\\n                \\n        # for i in range(m):\\n        #     print(keep[i], swap[i])\\n                \\n        res = min(min(swap[m-1]), keep[m-1])\\n        if res == float('inf'):\\n            return -1\\n        else:\\n            return res\\n\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                5,
                3,
                6,
                7
            ],
            [
                1,
                3,
                2,
                4
            ]
        ],
        "output": 1,
        "halu_type": "Identification Hallucination",
        "fn_name": "makeArrayIncreasing",
        "starter_code": "\nclass Solution:\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\n        ",
        "url": "https://leetcode.com/problems/make-array-strictly-increasing/"
    },
    {
        "id": 1258,
        "task_id": 2470,
        "test_case_id": 2,
        "question": "Given two integer arrays arr1 and arr2, return the minimum number of operations (possibly zero) needed to make arr1 strictly increasing.\nIn one operation, you can choose two indices 0 <= i < arr1.length and 0 <= j < arr2.length and do the assignment arr1[i] = arr2[j].\nIf there is no way to make arr1 strictly increasing, return -1.\n \nExample 1:\nInput: arr1 = [1,5,3,6,7], arr2 = [1,3,2,4]\nOutput: 1\nExplanation: Replace 5 with 2, then arr1 = [1, 2, 3, 6, 7].\n\nExample 2:\nInput: arr1 = [1,5,3,6,7], arr2 = [4,3,1]\nOutput: 2\nExplanation: Replace 5 with 3 and then replace 3 with 4. arr1 = [1, 3, 4, 6, 7].\n\nExample 3:\nInput: arr1 = [1,5,3,6,7], arr2 = [1,6,3,3]\nOutput: -1\nExplanation: You can't make arr1 strictly increasing.\n \nConstraints:\n\n1 <= arr1.length, arr2.length <= 2000\n0 <= arr1[i], arr2[i] <= 10^9",
        "solutions": "[\"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            prev_with_op = min_last_value_given_operations(n - 1, ops - 1)\\n            b = find_larger_value_in_B(prev_with_op)\\n            # dp(n - 1, ops) <= dp(n - 1, ops - 1)\\n            # if dp(n - 1, ops - 1) < A[n - 1] => dp(n - 1, ops) < A[n - 1]\\n            if prev_with_op < A[n - 1]:\\n                return min(A[n - 1], b)\\n            elif b <= A[n - 1]:\\n                return b\\n            elif min_last_value_given_operations(n - 1, ops) < A[n - 1]:\\n                return A[n - 1]\\n            return b\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        from bisect import bisect_right as br\\n        arr2.sort()\\n        \\n        dp = {0:-math.inf}\\n        # min_cnt = 0\\n        for n1 in arr1:\\n            # print(n1)\\n            new_dp = {}\\n            # cnt = min_cnt\\n            for cnt in dp:\\n                if n1 > dp[cnt]:\\n                    new_dp[cnt] = min(new_dp.get(cnt, math.inf), n1)\\n                i2 = br(arr2, dp[cnt])\\n                if i2 < len(arr2):\\n                    new_dp[cnt+1] = min(new_dp.get(cnt+1, math.inf), arr2[i2])\\n                cnt += 1\\n            if len(new_dp) == 0:\\n                return -1\\n            # while min_cnt not in new_dp:\\n            #     min_cnt += 1\\n            dp = new_dp\\n            # print(dp)\\n        return min(dp.keys())\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = list(set(arr2))\\n        arr2.sort()\\n        m, n = len(arr1), len(arr2)\\n        keep = [float('inf')]*m\\n        swap = [[float('inf') for _ in range(n)] for _ in range(m)]\\n        keep[0] = 0\\n        for i in range(n):\\n            swap[0][i] = 1\\n        for i in range(1, m):\\n            min_keep, min_swap = float('inf'), float('inf')\\n            for j in range(n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[i-1][j-1]+1)\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j])\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i-1]+1\\n                keep[i] = min(keep[i], min_keep)\\n                swap[i][j] = min(swap[i][j], min_swap)\\n        s = float('inf')\\n        for i in range(n):\\n            s = min(s, swap[m-1][i])\\n        res = min(s, keep[m-1])\\n        return -1 if res >= float('inf') else res\\n\\n# \\u89c1\\u82b1\\u82b1\\u9171\\u89c6\\u9891\\uff1ahttps://www.bilibili.com/video/av67133426/?spm_id_from=333.788.b_636f6d6d656e74.7\\n                    \\n        \\n\", \"import numpy as np\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        m = len(arr1)\\n        arr2 = sorted(np.unique(arr2))\\n        n = len(arr2)\\n        \\n        keep = [float('inf')] * m\\n        keep[0] = 0\\n        swap = [1] * n\\n        \\n        for i in range(1, m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            temp = [float('inf')] * n\\n            for j in range(n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[j - 1] + 1)\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[j])\\n                if arr1[i] > arr1[i - 1]:\\n                    keep[i] = keep[i - 1]\\n                if arr2[j] > arr1[i - 1]:\\n                    temp[j] = keep[i - 1] + 1\\n                temp[j] = min(temp[j], min_swap)\\n                keep[i] = min(keep[i], min_keep)\\n            for j in range(n):\\n                temp[j], swap[j] = swap[j], temp[j]\\n        \\n        s = min(swap)\\n        k = keep[-1]\\n        ans = min(s, k)\\n        return ans if ans < float('inf') else -1\\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        m = len(arr1)\\n        arr2 = list(set(arr2))\\n        arr2.sort()\\n        n = len(arr2)\\n        keep = [float('inf')] * m\\n        keep[0] = 0\\n        swap = [[float('inf') for j in range(n)] for i in range(m)]\\n        for j in range(n):\\n            swap[0][j] = 1\\n        for i in range(1,m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            for j in range(0, n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[i-1][j-1] + 1 )\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j]  )\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i -1]\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i -1] + 1\\n                swap[i][j] = min(swap[i][j], min_swap)\\n                keep[i]  = min(keep[i], min_keep )\\n        \\n        k = keep[-1]\\n        min_swap = min(swap[-1])\\n        ans = min (min_swap, k)\\n        return ans if ans < float('inf') else -1\", \"from bisect import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        d = {float('-inf'): 0}\\n        arr2 = sorted(set(arr2))\\n        for i in arr1:\\n            d2 = {}\\n            for k, v in list(d.items()):\\n                if i > k:\\n                    if i in d2:\\n                        d2[i] = min(d2[i], v)\\n                    else:\\n                        d2[i] = v\\n\\n                if k < arr2[-1]:\\n                    j = arr2[bisect(arr2, k)]\\n                    if j in d2:\\n                        d2[j] = min(d2[j], v + 1)\\n                    else:\\n                        d2[j] = v + 1\\n            \\n            d = d2\\n        \\n        if d:\\n            return min(d.values())\\n        return -1\\n                        \\n        \\n        \\n        \\n        \\n\", \"from collections import defaultdict\\nfrom math import inf\\nfrom bisect import bisect_right\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        n = len(arr2)\\n        dp = {-1: 0}\\n        for i in arr1:\\n            next_dp = defaultdict(lambda: inf)\\n            for key in dp:\\n                if i > key:\\n                    next_dp[i] = min(next_dp[i], dp[key])\\n                loc = bisect_right(arr2, key)\\n                if loc < n:\\n                    next_dp[arr2[loc]] = min(next_dp[arr2[loc]], dp[key] + 1)\\n            dp = next_dp\\n\\n        return min(dp.values()) if dp else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        from bisect import bisect_right as br\\n        arr2.sort()\\n        \\n        dp = {0:-math.inf}\\n        min_cnt = 0\\n        for n1 in arr1:\\n            # print(n1)\\n            new_dp = {}\\n            cnt = min_cnt\\n            while cnt in dp:\\n                if n1 > dp[cnt]:\\n                    new_dp[cnt] = min(new_dp.get(cnt, math.inf), n1)\\n                    # candidate = min(new_dp.get(cnt, math.inf), n1)\\n                    # if candidate < new_dp.get(cnt-1, math.inf):\\n                    #     new_dp[cnt] = candidate\\n                i2 = br(arr2, dp[cnt])\\n                # if i2 < len(arr2) and arr2[i2] < new_dp.get(cnt, math.inf):\\n                if i2 < len(arr2):\\n                    new_dp[cnt+1] = arr2[i2]\\n                # print(new_dp)\\n                # if new_dp.get(cnt+1, -math.inf) >= new_dp.get(cnt, math.inf):\\n                #     new_dp.pop(cnt+1)\\n                # if new_dp.get(cnt, -math.inf) >= new_dp.get(cnt-1, math.inf):\\n                #     new_dp.pop(cnt)\\n                cnt += 1\\n            if len(new_dp) == 0:\\n                return -1\\n            while min_cnt not in new_dp:\\n                min_cnt += 1\\n            dp = new_dp\\n            print(dp)\\n        return min(dp.keys())\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        import bisect\\n        # dp\\u5b58\\u50a8\\u6240\\u6709\\u6f5c\\u5728\\u7684\\u5f53\\u524d\\u72b6\\u6001\\uff08\\u6bcf\\u6b21dp\\u90fd\\u662f\\u904d\\u5386arr1\\u65f6\\u524d\\u4e00\\u4e2a\\u4f4d\\u7f6e\\u7684\\u72b6\\u6001\\uff09\\n        # \\u8fd9\\u91cc\\u7684\\u72b6\\u6001\\u662f\\u4e00\\u4e2a\\u952e\\u503c\\u5bf9\\uff0ckey\\u4ee3\\u8868\\u5f53\\u524d\\u4f4d\\u7f6e\\u7684\\u6570\\u5b57\\uff0c\\n        # \\u53ef\\u4ee5\\u662farr1\\u91cc\\u9762\\u7684\\uff0c\\u4e5f\\u53ef\\u4ee5\\u662farr2\\u91cc\\u9762\\u7528\\u6765\\u66ff\\u6362\\u7684\\n        # value\\u5c31\\u662f\\u6211\\u4eec\\u9700\\u8981\\u64cd\\u4f5c\\u7684\\u6b21\\u6570\\n        dp = {-1: 0}\\n        arr2.sort()\\n        for i in arr1:\\n            tmp = collections.defaultdict(lambda: float('inf'))\\n            for key in dp:\\n                if i > key:\\n                    tmp[i] = min(tmp[i], dp[key])\\n                loc = bisect.bisect_right(arr2, key)\\n                if loc < len(arr2):\\n                    tmp[arr2[loc]] = min(tmp[arr2[loc]], dp[key] + 1)\\n            dp = tmp\\n        return min(dp.values()) if dp else -1\\n\", \"from bisect import bisect_right as br\\nimport math\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i == len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if prev < arr1[i] else math.inf\\n            return min(swap, noswap)  \\n        ret = dfs(0, -math.inf)\\n        return ret if ret != math.inf else -1\\n        \\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"from bisect import bisect_right as br\\nimport math\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))   \\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i >= len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if prev < arr1[i] else math.inf\\n            return min(swap, noswap)\\n        ans = dfs(0, -1)\\n        return ans if ans != math.inf else -1\\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        import bisect\\n        dp = {-1: 0}\\n        arr2.sort()\\n        for i in arr1:\\n            tmp = collections.defaultdict(lambda: float('inf'))\\n            for key in dp:\\n                if i > key:\\n                    tmp[i] = min(tmp[i], dp[key])\\n                loc = bisect.bisect_right(arr2, key)\\n                if loc < len(arr2):\\n                    tmp[arr2[loc]] = min(tmp[arr2[loc]], dp[key] + 1)\\n            dp = tmp\\n        return min(dp.values()) if dp else -1\\n\", \"import bisect\\nimport functools\\n    \\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        \\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"from bisect import bisect_right as br\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i >= len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n            return min(swap, noswap)\\n        \\n        \\n        ans = dfs(0, -math.inf)\\n        return ans if ans!=math.inf else -1\\n        \\n        \\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"import bisect\\nimport functools\\n    \\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # arr1i\\n        # arr2j\\n        arr2 = sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(prev, i):\\n            if i == len(arr1):\\n                return 0\\n            j = bisect.bisect_right(arr2, prev)\\n            swap = 1 + dfs(arr2[j], i+1) if j < len(arr2) else float('inf')\\n            noswap = dfs(arr1[i], i+1) if prev < arr1[i] else float('inf')\\n            return min(swap, noswap)\\n        changes = dfs(float('-inf'), 0)\\n        return changes if changes != float('inf') else -1\\n\", \"class Solution:\\n  def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n    # TC: O(min(M, N)*N+MlogM), SC: O(min(M, N))\\n    m, n = len(arr1), len(arr2)\\n    # sort O(MlogM)\\n    arr2.sort()\\n    # maintain a list of (num-of-swaps, last-value, next-to-swap-index-arr2),\\n    # where the last value should be decrease as the num of swaps is increase\\n    ss = [(0, arr1[0], 0)]\\n    if arr2[0] < arr1[0]:\\n      ss.append((1, arr2[0], 1))\\n    # O(N) iteration\\n    for i in range(1, m):\\n      st = []\\n      # O(min(M, N))\\n      for s, x, j in ss:\\n        if x < arr1[i]:\\n          if st:\\n            if s == st[-1][0]:\\n              if arr1[i] < st[-1][1]:\\n                st[-1] = (s, arr1[i], j)\\n            elif s > st[-1][0]:\\n              if arr1[i] < st[-1][1]:\\n                st.append((s, arr1[i], j))\\n          else:\\n            st.append((s, arr1[i], j))\\n        # amortized O(1)\\n        while j < n and arr2[j] <= x:\\n          j += 1\\n        if j < n:\\n          st.append((s + 1, arr2[j], j))\\n      # since each swap takes at most 1 entry, and at most O(min(M, N)) swap, so SC: O(min(M, N))\\n      ss = st\\n    return ss[0][0] if ss else -1\", \"import bisect\\nfrom typing import List\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        array = arr1\\n        replace = list(sorted(set(arr2)))\\n\\n        @lru_cache(None)\\n        def dfs(array_pos: int, prev_number: int) -> int:\\n            min_replacements = len(replace) * 2\\n\\n            if array_pos == len(array):\\n                return 0\\n\\n            next_replace_pos = 0\\n\\n            if array_pos > 0:\\n                next_replace_pos = bisect.bisect(replace, prev_number)\\n\\n            if array_pos == 0 or (\\n                next_replace_pos < len(replace)\\n            ):\\n                tmp = array[array_pos]\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, replace[next_replace_pos]) + 1)\\n\\n            if array_pos == 0 or prev_number < array[array_pos]:\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]))\\n\\n            return min_replacements\\n\\n        result = dfs(0, -100)\\n\\n        if result > len(replace):\\n            return -1\\n\\n        return result\\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        b2idx = {b: i for i, b in enumerate(B)}\\n        \\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            if val in b2idx:\\n                return B[b2idx[val] + 1]\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j = bisect.bisect_right(arr2,prev)\\n            swap = 1 + dfs(i+1,arr2[j]) if j < len(arr2)  else math.inf\\n            noswap = dfs(i+1,arr1[i])   if arr1[i] > prev else math.inf\\n            return min(swap,noswap)\\n        changes = dfs(0,-math.inf)\\n        return changes if changes != math.inf else -1\\n\", \"import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        N = len(arr1)\\n        arr1 = [0] + arr1\\n        # \\u8868\\u793a\\u524d i \\u4e2a\\u5143\\u7d20\\uff0c\\u6267\\u884c\\u4e86 k \\u6b21\\u64cd\\u4f5c\\u540e\\uff0c\\u662f\\u6709\\u5e8f\\u7684\\n        dp = [[float('inf')] * (N + 1) for _ in range(N + 1)]\\n        dp[0][0] = -float('inf')\\n        \\n        arr2.sort()\\n        for i in range(1, N + 1):\\n            for k in range(i + 1):\\n                # \\u524di-1\\u4e2a\\u5143\\u7d20\\uff0c\\u5df2\\u7ecf\\u5b8c\\u6210\\u4e86k\\u6b21\\u4ea4\\u6362\\n                if arr1[i] > dp[i-1][k]:\\n                    dp[i][k] = min(dp[i][k], arr1[i])\\n                \\n                # \\u524d i-1 \\u4e2a\\u5143\\u7d20\\uff0c\\u5df2\\u7ecf\\u5b8c\\u6210\\u4e86 k-1\\u6b21\\u4ea4\\u6362\\uff0c\\u6240\\u4ee5\\u8fd9\\u4e00\\u6b21\\u4e00\\u5b9a\\u8981\\u4ea4\\u6362\\n                if k >= 1:\\n                    idx_2 = bisect.bisect_right(arr2, dp[i-1][k-1])\\n                    if idx_2 != len(arr2):                    \\n                        dp[i][k] = min(dp[i][k], arr2[idx_2])\\n        \\n        res = float('inf')\\n        for i in range(1, N+1):\\n            if dp[N][i] != float('inf'):\\n                res = min(res, i)\\n        return res if res != float('inf') else -1\", \"import bisect\\nfrom typing import List\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        array = arr1\\n        replace = list(sorted(set(arr2)))\\n\\n        @lru_cache(None)\\n        def dfs(array_pos: int, prev_number: int) -> int:\\n            min_replacements = len(replace) * 2\\n\\n            if array_pos == len(array):\\n                return 0\\n\\n            next_replace_pos = 0\\n\\n            if array_pos > 0:\\n                next_replace_pos = bisect.bisect(replace, array[array_pos - 1])\\n\\n            if array_pos == 0 or (\\n                next_replace_pos < len(replace)\\n            ):\\n                tmp = array[array_pos]\\n                array[array_pos] = replace[next_replace_pos]\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]) + 1)\\n                array[array_pos] = tmp\\n\\n            if array_pos == 0 or array[array_pos - 1] < array[array_pos]:\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]))\\n\\n            return min_replacements\\n\\n        result = dfs(0, -100)\\n\\n        if result > len(replace):\\n            return -1\\n\\n        return result\\n\", \"import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1, arr2) -> int:\\n        N = len(arr1)\\n        arr1 = [0] + arr1\\n        arr2.sort()\\n        dp = [[float('inf')] * (N + 1) for _ in range(N + 1)]\\n        dp[0][0] = -float('inf')\\n        \\n        for i in range(1, N + 1):\\n            for j in range(0, i + 1):\\n                if dp[i-1][j] < arr1[i]:\\n                    dp[i][j] = min(dp[i][j], arr1[i])\\n                if j >= 1:\\n                    # \\u8981\\u5728 arr2 \\u4e2d\\u627e\\u5230\\u4e00\\u4e2a\\u6bd4 arr1[i] \\u7a0d\\u5fae\\u5927\\u4e00\\u70b9\\u7684\\u6570\\n                    idx_2 = bisect.bisect_right(arr2, dp[i-1][j-1])\\n                    if idx_2 != len(arr2):\\n                        dp[i][j] = min(dp[i][j], arr2[idx_2])\\n        \\n        # \\u6ee1\\u8db3\\u6761\\u4ef6\\uff0c\\u5e76\\u4e14\\u80fd\\u591f\\u6267\\u884c\\u6700\\u5c11\\u7684\\u6b21\\u6570\\u7684 K \\u7684\\u503c\\n        res = float('inf')\\n        for i in range(1, N + 1):\\n            if dp[N][i] != float('inf'):\\n                res = min(res, i)\\n        return -1 if res == float('inf') else res\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        \\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        \\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        def bs(arr, l, r, target):\\n            while l<=r:\\n                m = l+(r-l)//2\\n                if arr[m]>target:\\n                    r = m-1\\n                else:\\n                    l = m+1\\n            return l\\n        \\n        arr2.sort()\\n        N = len(arr2)\\n        dp = {-1:0}\\n        \\n        for a in arr1:\\n            dp2 = {}\\n            for prev in dp:\\n                if a>prev:\\n                    dp2[a] = min(dp2.get(a, float('inf')), dp[prev])\\n                \\n                idx = bs(arr2, 0, N-1, prev)\\n                if idx<N:\\n                    dp2[arr2[idx]] = min(dp2.get(arr2[idx], float('inf')), dp[prev]+1)\\n                    \\n            dp = dp2\\n            if not dp:\\n                return -1\\n            \\n        return min(dp.values())\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        \\n        def find_lower_val_in_B(val):\\n            larger_equal_idx = bisect.bisect_left(B, val)\\n            if larger_equal_idx > 0:\\n                return B[larger_equal_idx - 1]\\n            return None  # no lower value in B\\n\\n        @lru_cache(None)\\n        def make_prefix_increasing(n, upper=float('inf')):\\n            if n == 0:\\n                return 0\\n\\n            swap_b = find_lower_val_in_B(upper)\\n            ret = float('inf')\\n            if A[n - 1] < upper:\\n                ret = min(make_prefix_increasing(n - 1, upper=A[n - 1]), ret)\\n            if swap_b is not None:\\n                ret = min(1 + make_prefix_increasing(n - 1, upper=swap_b), ret)\\n\\n            return ret\\n\\n        ret = make_prefix_increasing(len(A))\\n        return ret if ret < float('inf') else -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        \\n        dp = {-1:0}\\n        B = sorted(B)\\n        \\n        for cur in A:\\n            temp = collections.defaultdict(lambda: float('inf'))\\n            for prev in dp:\\n                if prev<cur:\\n                    temp[cur] = min(temp[cur], dp[prev])\\n                idx = self.upper_bound(B, prev)\\n                if idx<len(B):\\n                    temp[B[idx]] = min(temp[B[idx]], dp[prev]+1) \\n            dp = temp\\n        \\n        if dp:\\n            return min(dp.values())\\n        return -1\\n    \\n    def upper_bound(self, B, target):\\n        l=0\\n        r=len(B)\\n        while l<r:\\n            mid=l+(r-l)//2\\n            if B[mid]<=target:\\n                l=mid+1\\n            else:\\n                r=mid\\n        return l\\n                \\n                \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n            return min(\\n                A[n - 1] if min_last_value_given_operations(n - 1, ops) < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        if not arr1: return 0\\n        if not arr2: return arr1 == sorted(arr1)\\n        \\n        arr2 = sorted(list(set(arr2)), reverse=True)\\n        \\n        n, m, inf = len(arr1), len(arr2), float('inf')\\n        f = [[inf]*(n+1) for _ in range(n)]\\n        for i in range(n+1):\\n            f[0][i] = min(arr1[0],arr2[-1])\\n        f[0][0] = arr1[0]\\n        for i in range(1,n):\\n            found_k = 0\\n            for j in range(n+1):\\n                if f[i-1][j] < arr1[i]: f[i][j] = arr1[i]\\n                if not j: continue\\n                va = f[i-1][j-1]\\n                if not found_k:\\n                    if arr2[0] > va:\\n                        l,r = 0,m-1\\n                        while l < r:\\n                            mid = l+r+1 >> 1\\n                            if arr2[mid] > va:\\n                                l = mid\\n                            else: \\n                                r = mid - 1\\n                        k = l\\n                        found_k = 1\\n                        f[i][j] = min(f[i][j],arr2[k])\\n                else:\\n                    while k+1 < m and arr2[k+1] > va: k += 1\\n                    f[i][j] = min(f[i][j],arr2[k])\\n        for i,v in enumerate(f[-1]): \\n            if v < inf: return i\\n        return -1\\n                \\n                \\n                \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            prev_with_op = min_last_value_given_operations(n - 1, ops - 1)\\n            b = find_larger_value_in_B(prev_with_op)\\n            # dp(n - 1, ops) <= dp(n - 1, ops - 1)\\n            # if dp(n - 1, ops - 1) < A[n - 1] => dp(n - 1, ops) < A[n - 1]\\n            if prev_with_op < A[n - 1]:\\n                return min(A[n - 1], b)\\n            elif b <= A[n - 1]:\\n                return b\\n            elif min_last_value_given_operations(n - 1, ops) < A[n - 1]:\\n                return A[n - 1]\\n            return b\\n\\n        for ops in range(min(len(A), len(B)) + 1):\\n            if min_last_value_given_operations(len(A), ops) < float('inf'):\\n                return ops\\n\\n        return -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"def binsearch(arr,x):\\n    if(arr[0]>x):\\n        return 0\\n    l=0\\n    h=len(arr)-1\\n    ret=-1\\n    while(l<=h):\\n        mid=(l+h)//2\\n        if(arr[mid]<=x):\\n            l=mid+1\\n        elif(arr[mid]>x):\\n            ret=mid\\n            h=mid-1\\n    return ret\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m=len(arr2)\\n        n=len(arr1)\\n        dp={}\\n        # print(binsearch(arr2,0))\\n        def dfs(arr1,arr2,left,curr,dp):\\n            if(curr>=len(arr1)):\\n                return 0\\n            if((curr,left) in dp):\\n                return dp[(curr,left)]\\n            res1=sys.maxsize\\n            res2=0\\n            if(arr1[curr]>left):\\n                res1=dfs(arr1,arr2,arr1[curr],curr+1,dp)\\n            mid=binsearch(arr2,left)\\n            if(mid==-1):\\n                res2=sys.maxsize-1\\n            else:\\n                res2=dfs(arr1,arr2,arr2[mid],curr+1,dp)\\n            dp[(curr,left)]=min(res1,1+res2)\\n            return dp[(curr,left)]\\n        x=dfs(arr1,arr2,-sys.maxsize,0,dp)\\n        if(x>=(sys.maxsize-1)):\\n            return -1\\n        return x\\n            \\n\", \"def binsearch(arr,x):\\n    if(arr[0]>x):\\n        return 0\\n    l=0\\n    h=len(arr)-1\\n    ret=-1\\n    while(l<=h):\\n        mid=(l+h)//2\\n        if(arr[mid]<=x):\\n            l=mid+1\\n        elif(arr[mid]>x):\\n            ret=mid\\n            h=mid-1\\n    return ret\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m=len(arr2)\\n        n=len(arr1)\\n        dp={}\\n        def dfs(arr1,arr2,left,curr,dp):\\n            if(curr>=len(arr1)):\\n                return 0\\n            if((curr,left) in dp):\\n                return dp[(curr,left)]\\n            res1=sys.maxsize\\n            res2=0\\n            if(arr1[curr]>left):\\n                res1=dfs(arr1,arr2,arr1[curr],curr+1,dp)\\n            mid=binsearch(arr2,left)\\n            if(mid==-1):\\n                res2=sys.maxsize-1\\n            else:\\n                res2=dfs(arr1,arr2,arr2[mid],curr+1,dp)\\n            dp[(curr,left)]=min(res1,1+res2)\\n            return dp[(curr,left)]\\n        x=dfs(arr1,arr2,-sys.maxsize,0,dp)\\n        if(x==(sys.maxsize)):\\n            return -1\\n        return x\\n            \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        b2idx = {b: i for i, b in enumerate(B)}\\n        \\n        # @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            if val in b2idx:\\n                return B[b2idx[val] + 1]\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        arr2.sort()\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        \\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=n:\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,m-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        arr2.sort()\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans\\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                \\n                    \\n#                 for k in range(j,len(arr2)):\\n#                     # can we use binary search here\\n#                     # we got to find out the minumum value in arr2 which is greater than prev\\n                    \\n#                     if arr2[k]>prev:\\n#                         # we can probably use binary search here\\n#                         # get the first index which is strictly greater than prev\\n#                         ans=min(ans,1+helper(i+1,k+1,arr2[k]))\\n#                         break\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"import bisect\\nclass Solution:\\n    def recurse(self,arr1,arr2,m,n,idx,prev,dp):\\n        if idx>=n:\\n            return 0\\n        k=bisect.bisect_right(arr2,prev)\\n        if dp[idx][k]!=-1:\\n            return dp[idx][k]\\n        c1=self.recurse(arr1,arr2,m,n,idx+1,arr1[idx],dp) if arr1[idx]>prev else 2000\\n        c2=1+self.recurse(arr1,arr2,m,n,idx+1,arr2[k],dp) if k<m else 2000\\n        dp[idx][k]=min(c1,c2)\\n        return dp[idx][k]\\n     \\n\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        dp=[[-1 for i in range(2001)] for j in range(2001)]\\n        arr2.sort()\\n        self.recurse(arr1,arr2,len(arr2),len(arr1),0,-10**9,dp)\\n        if dp[0][0]>=2000:\\n            return -1\\n        else:\\n            return dp[0][0]\\n      \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # Maintain two dp array: 1 for keep arr1[i], 1 for swap arr1[i] with arr2[j]\\n        # Convert arr2 to sorted unique set\\n        a = arr1\\n        b = sorted(list(set(arr2)))\\n        n, m = len(a), len(b)\\n        keep = [0, math.inf]  # keep[i] = keep a[i]\\n        swap = [[1] * m, [math.inf] * m]  # swap[i][j] = a[i] replaced with b[j]\\n\\n        prev, curr = 1, 0\\n        for i in range(1, n):\\n            prev, curr = curr, prev\\n            # must init every time\\n            swap[curr] = [math.inf] * m\\n            keep[curr] = math.inf\\n            # keep[i] case 1: a[i] is bigger than previous keep[i-1]\\n            if a[i] > a[i - 1]:\\n                keep[curr] = keep[prev]\\n            for j in range(m):\\n                # keep[i] case 2: a[i] is bigger then previous swap value.\\n                if a[i] > b[j]:\\n                    keep[curr] = min(keep[curr], swap[prev][j])\\n                # swap case 1: when a[i-1] keeps\\n                if b[j] > a[i - 1]:\\n                    swap[curr][j] = min(swap[curr][j], 1 + keep[prev])\\n                # Swap case 2: when a[i-1] swapped\\n                if j > 0:\\n                    # only need to check last swap[i-1][j-1] since swap[][j] decreases when j increases.\\n                    swap[curr][j] = min(swap[curr][j], 1 + swap[prev][j - 1])\\n\\n        res = min(keep[curr], swap[curr][m - 1])\\n        return res if res < math.inf else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=n:\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                # pick first index from arr2 starting j which is greater than prev\\n                idx=bsearch(j,m-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n\\n    \\n                \\n                \\n        \\n        \\n\", \"import bisect\\nclass Solution:\\n    def recurse(self,arr1,arr2,m,n,idx,prev,dp):\\n        if idx>=n:\\n            return 0\\n        k=bisect.bisect_right(arr2,prev)\\n        if dp[idx][k]!=-1:\\n            return dp[idx][k]\\n        c1=self.recurse(arr1,arr2,m,n,idx+1,arr1[idx],dp) if arr1[idx]>prev else 2000\\n        c2=1+self.recurse(arr1,arr2,m,n,idx+1,arr2[k],dp) if k<m else 2000\\n        dp[idx][k]=min(c1,c2)\\n        return dp[idx][k]\\n     \\n\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        dp=[[-1 for i in range(2001)] for j in range(2001)]\\n        arr2.sort()\\n        ans=self.recurse(arr1,arr2,len(arr2),len(arr1),0,-10**9,dp)\\n        if ans>=2000:\\n            return -1\\n        else:\\n            return ans\\n      \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # Maintain two dp array: 1 for keep arr1[i], 1 for swap arr1[i] with arr2[j]\\n        # Convert arr2 to sorted unique set\\n        a = arr1\\n        b = sorted(list(set(arr2)))\\n        n, m = len(a), len(b)\\n        keep = [math.inf] * n  # keep[i] = keep a[i]\\n        swap = [[math.inf] * m for _ in range(n)]  # swap[i][j] = a[i] replaced with b[j]\\n        # init\\n        keep[0] = 0  # keep a[0], no swap\\n        swap[0] = [1] * m  # can be swapped with any a[j]\\n\\n        for i in range(1, n):\\n            # keep[i] case 1: a[i] is bigger than previous keep[i-1]\\n            if a[i] > a[i - 1]:\\n                keep[i] = keep[i - 1]\\n            for j in range(m):\\n                # keep[i] case 2: a[i] is bigger then previous swap value.\\n                if a[i] > b[j]:\\n                    keep[i] = min(keep[i], swap[i - 1][j])\\n                # swap case 1: when a[i-1] keeps\\n                if b[j] > a[i - 1]:\\n                    swap[i][j] = min(swap[i][j], 1 + keep[i - 1])\\n                # Swap case 2: when a[i-1] swapped\\n                if j > 0:\\n                    # only need to check last swap[i-1][j-1] since swap[][j] decreases when j increases.\\n                    swap[i][j] = min(swap[i][j], 1 + swap[i - 1][j - 1])\\n\\n        res = min(keep[n - 1], swap[n - 1][m - 1])\\n        return res if res < math.inf else -1        \", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m = len(arr1)\\n        n = len(arr2)\\n        newarr2 = [arr2[0]]\\n        for i in range(1, n):\\n            if arr2[i] != newarr2[-1]:\\n                newarr2.append(arr2[i])\\n                \\n        arr2 = newarr2\\n        n = len(arr2)\\n\\n        IL = 10 ** 9 + 7\\n        \\n        dp = [[IL for j in range(n + 1)] for i in range(m)]\\n        \\n        dp[0][n] = 0\\n        for i in range(n):\\n            if arr2[i] < arr1[0]:\\n                dp[0][i] = 1\\n            else:\\n                break\\n        \\n        for i in range(1, m):\\n            idx = 0\\n            if arr2[0] > arr1[i - 1]:\\n                dp[i][0] = dp[i - 1][n] + 1\\n            \\n            for k in range(1, n):\\n                a = dp[i - 1][n] + 1 if dp[i - 1][n] != IL and arr1[i - 1] < arr2[k] else IL\\n                b = dp[i - 1][idx] + 1 if dp[i - 1][idx] != IL else IL\\n                dp[i][k] = min(a, b)\\n                if dp[i - 1][k] < dp[i - 1][idx]:\\n                    idx = k\\n                    \\n            if dp[i - 1][n] != IL and arr1[i] > arr1[i -1]:\\n                dp[i][n] = dp[i - 1][n]\\n            \\n            for k in range(n):\\n                if dp[i - 1][k] != IL and arr2[k] < arr1[i]:\\n                    dp[i][n] = min(dp[i][n], dp[i - 1][k])\\n                    \\n        m = min(dp[m - 1])\\n        \\n        return m if m != IL else -1\\n        \\n                \\n\", \"class Solution:    \\n    def binary_search_right(self, arr, l, r, t):\\n        while l < r:\\n            m = l + (r - l) // 2\\n            if arr[m] <= t:\\n                l = m + 1\\n            else:\\n                r = m\\n        return l\\n    \\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        \\n        dp = {}\\n        def help(i1, i2):\\n            if i1 == len(arr1):\\n                return 0\\n            \\n            if i1 != 0:\\n                key = (i1, i2, arr1[i1 - 1])\\n            else:\\n                key = (i1, i2, None)\\n            \\n            if key in dp:\\n                return dp[key]\\n            \\n            # print(key)\\n            \\n            # all possible results\\n            pos = []\\n            \\n            # don't make changes here\\n            if i1 == 0 or arr1[i1 - 1] < arr1[i1]:\\n                res = help(i1 + 1, i2)\\n                if res != -1:\\n                    pos.append(res)\\n                    \\n            if i1 != 0:\\n                # make change\\n                i2 = self.binary_search_right(arr2, i2, len(arr2), arr1[i1 - 1])\\n                if i2 != len(arr2):\\n                    tmp = arr1[i1]\\n                    arr1[i1] = arr2[i2]\\n                    res = help(i1 + 1, i2 + 1)\\n                    if res != -1:\\n                        pos.append(res + 1)\\n                    arr1[i1] = tmp\\n            else:\\n                # make change\\n                if i2 < len(arr2) and arr2[i2] < arr1[i1]:\\n                    tmp = arr1[i1]\\n                    arr1[i1] = arr2[i2]\\n                    res = help(i1 + 1, i2 + 1)\\n                    if res != -1:\\n                        pos.append(res + 1)\\n                    arr1[i1] = tmp\\n            \\n            if len(pos) == 0:\\n                dp[key] = -1\\n            else:\\n                dp[key] = min(pos)\\n            return dp[key]\\n        \\n        return help(0, 0)\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        \\n        # dp[i][k] means the minimum number we can have at ith position with k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                if dp[i - 1][k] < arr1[i]:\\n                    # not assign\\n                    dp[i][k] = arr1[i]\\n                \\n                num = self.helper(arr2, dp[i - 1][k - 1])\\n                if num > dp[i - 1][k - 1]:\\n                    dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n\\n    def helper(self, arr, val):\\n        # find in arr the smallest number that is larger than val\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            else:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        size1, size2 = len(arr1), len(arr2)\\n        Inf = float('inf')\\n        keep = [Inf] * size1\\n        keep[0] = 0\\n        swap = [[Inf] * size2 for _ in range(size1)]\\n        swap[0] = [1] * size2\\n        for i in range(1, size1):\\n            minKeep = minSwap = Inf\\n            for j in range(size2):\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                if arr1[i] > arr2[j]:\\n                    minKeep = min(minKeep, swap[i-1][j])\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i-1] + 1\\n                if j > 0: # arr2[j] > arr2[j-1] is always True\\n                    minSwap = min(minSwap, swap[i-1][j-1] + 1)\\n                keep[i] = min(keep[i], minKeep)\\n                swap[i][j] = min(swap[i][j], minSwap)\\n        res = min(min(swap[-1]), keep[-1])\\n        return -1 if res == Inf else res\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        \\n        # dp[i][k] means the minimum number we can have at ith position with k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                if dp[i - 1][k] < arr1[i]:\\n                    # not assign\\n                    dp[i][k] = arr1[i]\\n                \\n                if k >= 1:\\n                    num = self.helper(arr2, dp[i - 1][k - 1])\\n                    if num > dp[i - 1][k - 1]:\\n                        dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n\\n    def helper(self, arr, val):\\n        # find in arr the smallest number that is larger than val\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            else:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        # dp[i][k] means the minimum number we can get at arr1[i] using k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                # not changing for arr1[i]\\n                if dp[i - 1][k] < arr1[i]:\\n                    dp[i][k] = arr1[i]\\n                \\n                # changing for arr1[i]\\n                # find the smallested number in arr2 that is larger than dp[i - 1][k - 1]\\n                if k >= 1:\\n                    num = self.helper(arr2, dp[i - 1][k - 1])\\n                    if num > dp[i - 1][k - 1]:\\n                        dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(0, n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n    \\n    def helper(self, arr, val):\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            elif arr[mid] > val:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        set2 = set(arr2)\\n        arr2 = sorted(set2)\\n        N, M = len(arr1), len(arr2)\\n        to_index = {}\\n        left = 0\\n        for x in sorted(arr1):\\n            while left < M and arr2[left] <= x:\\n                left += 1\\n            to_index[x] = left - 1 if left < M else M\\n        \\n        print(to_index, arr2)\\n        dp = [[[None, None] for _ in range(M + 1)] for _ in range(N)]\\n        \\n        def solve(i, j, k):\\n            if i == N: return 0\\n            if j > M or (j == M and k == 1): return N+1\\n            if dp[i][j][k] is None:\\n                if i == 0: \\n                    dp[i][j][k] = min(solve(i+1, to_index[arr1[0]], 0), 1 + solve(i+1, 0, 1))\\n                else:\\n                    result = N + 1\\n                    if j < M - 1: \\n                        result = 1 + solve(i+1, j+1, 1)\\n                    if k == 0 and arr1[i] > arr1[i-1]:\\n                        result = min(result, solve(i+1, to_index[arr1[i]], 0))\\n                    if k == 1 and arr1[i] > arr2[j]:\\n                        result = min(result, solve(i+1, to_index[arr1[i]], 0))\\n                    dp[i][j][k] = result\\n            return dp[i][j][k]\\n        result = solve(0, 0, 0)\\n        # for i in range(N):\\n        #     for j in range(M):\\n        #         for k in [0, 1]:\\n        #             print(dp[i][j][k], i, j, k)\\n        return result if result <= N else -1\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            if n == 0:\\n                return float('-inf')\\n                # return A[n - 1] if first_n_of_A_is_sorted(n) else float('inf')\\n\\n            ops = min(n, ops)\\n            return min(\\n                A[n - 1] if min_last_value_given_operations(n - 1, ops) < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        for ops in range(min(len(A), len(B)) + 1):\\n            if min_last_value_given_operations(len(A), ops) < float('inf'):\\n                return ops\\n\\n        return -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        ## https://www.youtube.com/watch?v=8ttxdMCU2GE\\n        m = len(arr1)\\n        ## remove dulpicates and sort arr2\\n        arr2 = sorted(list(set(arr2)))\\n        # print(arr2)\\n        n = len(arr2)\\n        swap = [[float('inf') for j in range(n)] for i in range(m)]\\n        keep = [float('inf') for i in range(m)]\\n        \\n        ## initialization\\n        keep[0] = 0\\n        for j in range(n):\\n            swap[0][j] = 1\\n            \\n        for i in range(1, m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            for j in range(n):\\n                ## two variables to help compute case 3 & 4\\n                ## case 4: the last two elements of current valid array are both from arr2\\n                if j>0:\\n                    min_swap = min(min_swap, swap[i-1][j-1]+1)\\n\\n                ## case 3: the second to last element is replaced by the previous element arr2[j-1] or earlier elements\\n                if arr1[i]>arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j])\\n                    \\n                ## case 1: no need to swap; keep arr1[i]\\n                if arr1[i]>arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                \\n                ## case 2: ## replace arr1[i] by arr2[j]\\n                if arr2[j]>arr1[i-1]:\\n                    swap[i][j] = keep[i-1] + 1\\n                \\n                ## update\\n                swap[i][j] = min(swap[i][j], min_swap)\\n                keep[i] = min(keep[i], min_keep)\\n                \\n        # for i in range(m):\\n        #     print(keep[i], swap[i])\\n                \\n        res = min(min(swap[m-1]), keep[m-1])\\n        if res == float('inf'):\\n            return -1\\n        else:\\n            return res\\n\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                5,
                3,
                6,
                7
            ],
            [
                4,
                3,
                1
            ]
        ],
        "output": 2,
        "halu_type": "Identification Hallucination",
        "fn_name": "makeArrayIncreasing",
        "starter_code": "\nclass Solution:\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\n        ",
        "url": "https://leetcode.com/problems/make-array-strictly-increasing/"
    },
    {
        "id": 1259,
        "task_id": 2470,
        "test_case_id": 3,
        "question": "Given two integer arrays arr1 and arr2, return the minimum number of operations (possibly zero) needed to make arr1 strictly increasing.\nIn one operation, you can choose two indices 0 <= i < arr1.length and 0 <= j < arr2.length and do the assignment arr1[i] = arr2[j].\nIf there is no way to make arr1 strictly increasing, return -1.\n \nExample 1:\nInput: arr1 = [1,5,3,6,7], arr2 = [1,3,2,4]\nOutput: 1\nExplanation: Replace 5 with 2, then arr1 = [1, 2, 3, 6, 7].\n\nExample 2:\nInput: arr1 = [1,5,3,6,7], arr2 = [4,3,1]\nOutput: 2\nExplanation: Replace 5 with 3 and then replace 3 with 4. arr1 = [1, 3, 4, 6, 7].\n\nExample 3:\nInput: arr1 = [1,5,3,6,7], arr2 = [1,6,3,3]\nOutput: -1\nExplanation: You can't make arr1 strictly increasing.\n \nConstraints:\n\n1 <= arr1.length, arr2.length <= 2000\n0 <= arr1[i], arr2[i] <= 10^9",
        "solutions": "[\"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            prev_with_op = min_last_value_given_operations(n - 1, ops - 1)\\n            b = find_larger_value_in_B(prev_with_op)\\n            # dp(n - 1, ops) <= dp(n - 1, ops - 1)\\n            # if dp(n - 1, ops - 1) < A[n - 1] => dp(n - 1, ops) < A[n - 1]\\n            if prev_with_op < A[n - 1]:\\n                return min(A[n - 1], b)\\n            elif b <= A[n - 1]:\\n                return b\\n            elif min_last_value_given_operations(n - 1, ops) < A[n - 1]:\\n                return A[n - 1]\\n            return b\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        from bisect import bisect_right as br\\n        arr2.sort()\\n        \\n        dp = {0:-math.inf}\\n        # min_cnt = 0\\n        for n1 in arr1:\\n            # print(n1)\\n            new_dp = {}\\n            # cnt = min_cnt\\n            for cnt in dp:\\n                if n1 > dp[cnt]:\\n                    new_dp[cnt] = min(new_dp.get(cnt, math.inf), n1)\\n                i2 = br(arr2, dp[cnt])\\n                if i2 < len(arr2):\\n                    new_dp[cnt+1] = min(new_dp.get(cnt+1, math.inf), arr2[i2])\\n                cnt += 1\\n            if len(new_dp) == 0:\\n                return -1\\n            # while min_cnt not in new_dp:\\n            #     min_cnt += 1\\n            dp = new_dp\\n            # print(dp)\\n        return min(dp.keys())\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = list(set(arr2))\\n        arr2.sort()\\n        m, n = len(arr1), len(arr2)\\n        keep = [float('inf')]*m\\n        swap = [[float('inf') for _ in range(n)] for _ in range(m)]\\n        keep[0] = 0\\n        for i in range(n):\\n            swap[0][i] = 1\\n        for i in range(1, m):\\n            min_keep, min_swap = float('inf'), float('inf')\\n            for j in range(n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[i-1][j-1]+1)\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j])\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i-1]+1\\n                keep[i] = min(keep[i], min_keep)\\n                swap[i][j] = min(swap[i][j], min_swap)\\n        s = float('inf')\\n        for i in range(n):\\n            s = min(s, swap[m-1][i])\\n        res = min(s, keep[m-1])\\n        return -1 if res >= float('inf') else res\\n\\n# \\u89c1\\u82b1\\u82b1\\u9171\\u89c6\\u9891\\uff1ahttps://www.bilibili.com/video/av67133426/?spm_id_from=333.788.b_636f6d6d656e74.7\\n                    \\n        \\n\", \"import numpy as np\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        m = len(arr1)\\n        arr2 = sorted(np.unique(arr2))\\n        n = len(arr2)\\n        \\n        keep = [float('inf')] * m\\n        keep[0] = 0\\n        swap = [1] * n\\n        \\n        for i in range(1, m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            temp = [float('inf')] * n\\n            for j in range(n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[j - 1] + 1)\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[j])\\n                if arr1[i] > arr1[i - 1]:\\n                    keep[i] = keep[i - 1]\\n                if arr2[j] > arr1[i - 1]:\\n                    temp[j] = keep[i - 1] + 1\\n                temp[j] = min(temp[j], min_swap)\\n                keep[i] = min(keep[i], min_keep)\\n            for j in range(n):\\n                temp[j], swap[j] = swap[j], temp[j]\\n        \\n        s = min(swap)\\n        k = keep[-1]\\n        ans = min(s, k)\\n        return ans if ans < float('inf') else -1\\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        m = len(arr1)\\n        arr2 = list(set(arr2))\\n        arr2.sort()\\n        n = len(arr2)\\n        keep = [float('inf')] * m\\n        keep[0] = 0\\n        swap = [[float('inf') for j in range(n)] for i in range(m)]\\n        for j in range(n):\\n            swap[0][j] = 1\\n        for i in range(1,m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            for j in range(0, n):\\n                if j > 0:\\n                    min_swap = min(min_swap, swap[i-1][j-1] + 1 )\\n                if arr1[i] > arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j]  )\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i -1]\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i -1] + 1\\n                swap[i][j] = min(swap[i][j], min_swap)\\n                keep[i]  = min(keep[i], min_keep )\\n        \\n        k = keep[-1]\\n        min_swap = min(swap[-1])\\n        ans = min (min_swap, k)\\n        return ans if ans < float('inf') else -1\", \"from bisect import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        d = {float('-inf'): 0}\\n        arr2 = sorted(set(arr2))\\n        for i in arr1:\\n            d2 = {}\\n            for k, v in list(d.items()):\\n                if i > k:\\n                    if i in d2:\\n                        d2[i] = min(d2[i], v)\\n                    else:\\n                        d2[i] = v\\n\\n                if k < arr2[-1]:\\n                    j = arr2[bisect(arr2, k)]\\n                    if j in d2:\\n                        d2[j] = min(d2[j], v + 1)\\n                    else:\\n                        d2[j] = v + 1\\n            \\n            d = d2\\n        \\n        if d:\\n            return min(d.values())\\n        return -1\\n                        \\n        \\n        \\n        \\n        \\n\", \"from collections import defaultdict\\nfrom math import inf\\nfrom bisect import bisect_right\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        n = len(arr2)\\n        dp = {-1: 0}\\n        for i in arr1:\\n            next_dp = defaultdict(lambda: inf)\\n            for key in dp:\\n                if i > key:\\n                    next_dp[i] = min(next_dp[i], dp[key])\\n                loc = bisect_right(arr2, key)\\n                if loc < n:\\n                    next_dp[arr2[loc]] = min(next_dp[arr2[loc]], dp[key] + 1)\\n            dp = next_dp\\n\\n        return min(dp.values()) if dp else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        from bisect import bisect_right as br\\n        arr2.sort()\\n        \\n        dp = {0:-math.inf}\\n        min_cnt = 0\\n        for n1 in arr1:\\n            # print(n1)\\n            new_dp = {}\\n            cnt = min_cnt\\n            while cnt in dp:\\n                if n1 > dp[cnt]:\\n                    new_dp[cnt] = min(new_dp.get(cnt, math.inf), n1)\\n                    # candidate = min(new_dp.get(cnt, math.inf), n1)\\n                    # if candidate < new_dp.get(cnt-1, math.inf):\\n                    #     new_dp[cnt] = candidate\\n                i2 = br(arr2, dp[cnt])\\n                # if i2 < len(arr2) and arr2[i2] < new_dp.get(cnt, math.inf):\\n                if i2 < len(arr2):\\n                    new_dp[cnt+1] = arr2[i2]\\n                # print(new_dp)\\n                # if new_dp.get(cnt+1, -math.inf) >= new_dp.get(cnt, math.inf):\\n                #     new_dp.pop(cnt+1)\\n                # if new_dp.get(cnt, -math.inf) >= new_dp.get(cnt-1, math.inf):\\n                #     new_dp.pop(cnt)\\n                cnt += 1\\n            if len(new_dp) == 0:\\n                return -1\\n            while min_cnt not in new_dp:\\n                min_cnt += 1\\n            dp = new_dp\\n            print(dp)\\n        return min(dp.keys())\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        import bisect\\n        # dp\\u5b58\\u50a8\\u6240\\u6709\\u6f5c\\u5728\\u7684\\u5f53\\u524d\\u72b6\\u6001\\uff08\\u6bcf\\u6b21dp\\u90fd\\u662f\\u904d\\u5386arr1\\u65f6\\u524d\\u4e00\\u4e2a\\u4f4d\\u7f6e\\u7684\\u72b6\\u6001\\uff09\\n        # \\u8fd9\\u91cc\\u7684\\u72b6\\u6001\\u662f\\u4e00\\u4e2a\\u952e\\u503c\\u5bf9\\uff0ckey\\u4ee3\\u8868\\u5f53\\u524d\\u4f4d\\u7f6e\\u7684\\u6570\\u5b57\\uff0c\\n        # \\u53ef\\u4ee5\\u662farr1\\u91cc\\u9762\\u7684\\uff0c\\u4e5f\\u53ef\\u4ee5\\u662farr2\\u91cc\\u9762\\u7528\\u6765\\u66ff\\u6362\\u7684\\n        # value\\u5c31\\u662f\\u6211\\u4eec\\u9700\\u8981\\u64cd\\u4f5c\\u7684\\u6b21\\u6570\\n        dp = {-1: 0}\\n        arr2.sort()\\n        for i in arr1:\\n            tmp = collections.defaultdict(lambda: float('inf'))\\n            for key in dp:\\n                if i > key:\\n                    tmp[i] = min(tmp[i], dp[key])\\n                loc = bisect.bisect_right(arr2, key)\\n                if loc < len(arr2):\\n                    tmp[arr2[loc]] = min(tmp[arr2[loc]], dp[key] + 1)\\n            dp = tmp\\n        return min(dp.values()) if dp else -1\\n\", \"from bisect import bisect_right as br\\nimport math\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i == len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if prev < arr1[i] else math.inf\\n            return min(swap, noswap)  \\n        ret = dfs(0, -math.inf)\\n        return ret if ret != math.inf else -1\\n        \\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"from bisect import bisect_right as br\\nimport math\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))   \\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i >= len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if prev < arr1[i] else math.inf\\n            return min(swap, noswap)\\n        ans = dfs(0, -1)\\n        return ans if ans != math.inf else -1\\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        import bisect\\n        dp = {-1: 0}\\n        arr2.sort()\\n        for i in arr1:\\n            tmp = collections.defaultdict(lambda: float('inf'))\\n            for key in dp:\\n                if i > key:\\n                    tmp[i] = min(tmp[i], dp[key])\\n                loc = bisect.bisect_right(arr2, key)\\n                if loc < len(arr2):\\n                    tmp[arr2[loc]] = min(tmp[arr2[loc]], dp[key] + 1)\\n            dp = tmp\\n        return min(dp.values()) if dp else -1\\n\", \"import bisect\\nimport functools\\n    \\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        \\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"from bisect import bisect_right as br\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        @lru_cache(None)\\n        def dfs(i, prev):\\n            if i >= len(arr1): return 0\\n            j = br(arr2, prev)\\n            swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n            noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n            return min(swap, noswap)\\n        \\n        \\n        ans = dfs(0, -math.inf)\\n        return ans if ans!=math.inf else -1\\n        \\n        \\n        \\n        \\n        \\n        \\n        # arr2=sorted(set(arr2))\\n        # @functools.lru_cache(None)\\n        # def dfs(i,prev):\\n        #     if i >= len(arr1):\\n        #         return 0\\n        #     j = br(arr2,prev)\\n        #     swap = 1 + dfs(i+1, arr2[j]) if j < len(arr2) else math.inf\\n        #     noswap = dfs(i+1, arr1[i]) if arr1[i] > prev else math.inf\\n        #     return min(swap,noswap)\\n        # changes=dfs(0, -math.inf)\\n        # return changes if changes!=math.inf else -1\\n\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"import bisect\\nimport functools\\n    \\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # arr1i\\n        # arr2j\\n        arr2 = sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(prev, i):\\n            if i == len(arr1):\\n                return 0\\n            j = bisect.bisect_right(arr2, prev)\\n            swap = 1 + dfs(arr2[j], i+1) if j < len(arr2) else float('inf')\\n            noswap = dfs(arr1[i], i+1) if prev < arr1[i] else float('inf')\\n            return min(swap, noswap)\\n        changes = dfs(float('-inf'), 0)\\n        return changes if changes != float('inf') else -1\\n\", \"class Solution:\\n  def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n    # TC: O(min(M, N)*N+MlogM), SC: O(min(M, N))\\n    m, n = len(arr1), len(arr2)\\n    # sort O(MlogM)\\n    arr2.sort()\\n    # maintain a list of (num-of-swaps, last-value, next-to-swap-index-arr2),\\n    # where the last value should be decrease as the num of swaps is increase\\n    ss = [(0, arr1[0], 0)]\\n    if arr2[0] < arr1[0]:\\n      ss.append((1, arr2[0], 1))\\n    # O(N) iteration\\n    for i in range(1, m):\\n      st = []\\n      # O(min(M, N))\\n      for s, x, j in ss:\\n        if x < arr1[i]:\\n          if st:\\n            if s == st[-1][0]:\\n              if arr1[i] < st[-1][1]:\\n                st[-1] = (s, arr1[i], j)\\n            elif s > st[-1][0]:\\n              if arr1[i] < st[-1][1]:\\n                st.append((s, arr1[i], j))\\n          else:\\n            st.append((s, arr1[i], j))\\n        # amortized O(1)\\n        while j < n and arr2[j] <= x:\\n          j += 1\\n        if j < n:\\n          st.append((s + 1, arr2[j], j))\\n      # since each swap takes at most 1 entry, and at most O(min(M, N)) swap, so SC: O(min(M, N))\\n      ss = st\\n    return ss[0][0] if ss else -1\", \"import bisect\\nfrom typing import List\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        array = arr1\\n        replace = list(sorted(set(arr2)))\\n\\n        @lru_cache(None)\\n        def dfs(array_pos: int, prev_number: int) -> int:\\n            min_replacements = len(replace) * 2\\n\\n            if array_pos == len(array):\\n                return 0\\n\\n            next_replace_pos = 0\\n\\n            if array_pos > 0:\\n                next_replace_pos = bisect.bisect(replace, prev_number)\\n\\n            if array_pos == 0 or (\\n                next_replace_pos < len(replace)\\n            ):\\n                tmp = array[array_pos]\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, replace[next_replace_pos]) + 1)\\n\\n            if array_pos == 0 or prev_number < array[array_pos]:\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]))\\n\\n            return min_replacements\\n\\n        result = dfs(0, -100)\\n\\n        if result > len(replace):\\n            return -1\\n\\n        return result\\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        b2idx = {b: i for i, b in enumerate(B)}\\n        \\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            if val in b2idx:\\n                return B[b2idx[val] + 1]\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j = bisect.bisect_right(arr2,prev)\\n            swap = 1 + dfs(i+1,arr2[j]) if j < len(arr2)  else math.inf\\n            noswap = dfs(i+1,arr1[i])   if arr1[i] > prev else math.inf\\n            return min(swap,noswap)\\n        changes = dfs(0,-math.inf)\\n        return changes if changes != math.inf else -1\\n\", \"import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        N = len(arr1)\\n        arr1 = [0] + arr1\\n        # \\u8868\\u793a\\u524d i \\u4e2a\\u5143\\u7d20\\uff0c\\u6267\\u884c\\u4e86 k \\u6b21\\u64cd\\u4f5c\\u540e\\uff0c\\u662f\\u6709\\u5e8f\\u7684\\n        dp = [[float('inf')] * (N + 1) for _ in range(N + 1)]\\n        dp[0][0] = -float('inf')\\n        \\n        arr2.sort()\\n        for i in range(1, N + 1):\\n            for k in range(i + 1):\\n                # \\u524di-1\\u4e2a\\u5143\\u7d20\\uff0c\\u5df2\\u7ecf\\u5b8c\\u6210\\u4e86k\\u6b21\\u4ea4\\u6362\\n                if arr1[i] > dp[i-1][k]:\\n                    dp[i][k] = min(dp[i][k], arr1[i])\\n                \\n                # \\u524d i-1 \\u4e2a\\u5143\\u7d20\\uff0c\\u5df2\\u7ecf\\u5b8c\\u6210\\u4e86 k-1\\u6b21\\u4ea4\\u6362\\uff0c\\u6240\\u4ee5\\u8fd9\\u4e00\\u6b21\\u4e00\\u5b9a\\u8981\\u4ea4\\u6362\\n                if k >= 1:\\n                    idx_2 = bisect.bisect_right(arr2, dp[i-1][k-1])\\n                    if idx_2 != len(arr2):                    \\n                        dp[i][k] = min(dp[i][k], arr2[idx_2])\\n        \\n        res = float('inf')\\n        for i in range(1, N+1):\\n            if dp[N][i] != float('inf'):\\n                res = min(res, i)\\n        return res if res != float('inf') else -1\", \"import bisect\\nfrom typing import List\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        array = arr1\\n        replace = list(sorted(set(arr2)))\\n\\n        @lru_cache(None)\\n        def dfs(array_pos: int, prev_number: int) -> int:\\n            min_replacements = len(replace) * 2\\n\\n            if array_pos == len(array):\\n                return 0\\n\\n            next_replace_pos = 0\\n\\n            if array_pos > 0:\\n                next_replace_pos = bisect.bisect(replace, array[array_pos - 1])\\n\\n            if array_pos == 0 or (\\n                next_replace_pos < len(replace)\\n            ):\\n                tmp = array[array_pos]\\n                array[array_pos] = replace[next_replace_pos]\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]) + 1)\\n                array[array_pos] = tmp\\n\\n            if array_pos == 0 or array[array_pos - 1] < array[array_pos]:\\n                min_replacements = min(min_replacements, dfs(array_pos + 1, array[array_pos]))\\n\\n            return min_replacements\\n\\n        result = dfs(0, -100)\\n\\n        if result > len(replace):\\n            return -1\\n\\n        return result\\n\", \"import bisect\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1, arr2) -> int:\\n        N = len(arr1)\\n        arr1 = [0] + arr1\\n        arr2.sort()\\n        dp = [[float('inf')] * (N + 1) for _ in range(N + 1)]\\n        dp[0][0] = -float('inf')\\n        \\n        for i in range(1, N + 1):\\n            for j in range(0, i + 1):\\n                if dp[i-1][j] < arr1[i]:\\n                    dp[i][j] = min(dp[i][j], arr1[i])\\n                if j >= 1:\\n                    # \\u8981\\u5728 arr2 \\u4e2d\\u627e\\u5230\\u4e00\\u4e2a\\u6bd4 arr1[i] \\u7a0d\\u5fae\\u5927\\u4e00\\u70b9\\u7684\\u6570\\n                    idx_2 = bisect.bisect_right(arr2, dp[i-1][j-1])\\n                    if idx_2 != len(arr2):\\n                        dp[i][j] = min(dp[i][j], arr2[idx_2])\\n        \\n        # \\u6ee1\\u8db3\\u6761\\u4ef6\\uff0c\\u5e76\\u4e14\\u80fd\\u591f\\u6267\\u884c\\u6700\\u5c11\\u7684\\u6b21\\u6570\\u7684 K \\u7684\\u503c\\n        res = float('inf')\\n        for i in range(1, N + 1):\\n            if dp[N][i] != float('inf'):\\n                res = min(res, i)\\n        return -1 if res == float('inf') else res\", \"import bisect\\nimport functools\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2=sorted(set(arr2))\\n        \\n        @functools.lru_cache(None)\\n        def dfs(i,prev):\\n            if i>=len(arr1):\\n                return 0\\n            j=bisect.bisect_right(arr2,prev)\\n            swap=1+dfs(i+1,arr2[j]) if j<len(arr2) else math.inf\\n            noswap=dfs(i+1,arr1[i]) if arr1[i]>prev else math.inf\\n            return min(swap,noswap)\\n        \\n        changes=dfs(0,-math.inf)\\n        return changes if changes!=math.inf else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        def bs(arr, l, r, target):\\n            while l<=r:\\n                m = l+(r-l)//2\\n                if arr[m]>target:\\n                    r = m-1\\n                else:\\n                    l = m+1\\n            return l\\n        \\n        arr2.sort()\\n        N = len(arr2)\\n        dp = {-1:0}\\n        \\n        for a in arr1:\\n            dp2 = {}\\n            for prev in dp:\\n                if a>prev:\\n                    dp2[a] = min(dp2.get(a, float('inf')), dp[prev])\\n                \\n                idx = bs(arr2, 0, N-1, prev)\\n                if idx<N:\\n                    dp2[arr2[idx]] = min(dp2.get(arr2[idx], float('inf')), dp[prev]+1)\\n                    \\n            dp = dp2\\n            if not dp:\\n                return -1\\n            \\n        return min(dp.values())\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        \\n        def find_lower_val_in_B(val):\\n            larger_equal_idx = bisect.bisect_left(B, val)\\n            if larger_equal_idx > 0:\\n                return B[larger_equal_idx - 1]\\n            return None  # no lower value in B\\n\\n        @lru_cache(None)\\n        def make_prefix_increasing(n, upper=float('inf')):\\n            if n == 0:\\n                return 0\\n\\n            swap_b = find_lower_val_in_B(upper)\\n            ret = float('inf')\\n            if A[n - 1] < upper:\\n                ret = min(make_prefix_increasing(n - 1, upper=A[n - 1]), ret)\\n            if swap_b is not None:\\n                ret = min(1 + make_prefix_increasing(n - 1, upper=swap_b), ret)\\n\\n            return ret\\n\\n        ret = make_prefix_increasing(len(A))\\n        return ret if ret < float('inf') else -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        \\n        dp = {-1:0}\\n        B = sorted(B)\\n        \\n        for cur in A:\\n            temp = collections.defaultdict(lambda: float('inf'))\\n            for prev in dp:\\n                if prev<cur:\\n                    temp[cur] = min(temp[cur], dp[prev])\\n                idx = self.upper_bound(B, prev)\\n                if idx<len(B):\\n                    temp[B[idx]] = min(temp[B[idx]], dp[prev]+1) \\n            dp = temp\\n        \\n        if dp:\\n            return min(dp.values())\\n        return -1\\n    \\n    def upper_bound(self, B, target):\\n        l=0\\n        r=len(B)\\n        while l<r:\\n            mid=l+(r-l)//2\\n            if B[mid]<=target:\\n                l=mid+1\\n            else:\\n                r=mid\\n        return l\\n                \\n                \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n            return min(\\n                A[n - 1] if min_last_value_given_operations(n - 1, ops) < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        if not arr1: return 0\\n        if not arr2: return arr1 == sorted(arr1)\\n        \\n        arr2 = sorted(list(set(arr2)), reverse=True)\\n        \\n        n, m, inf = len(arr1), len(arr2), float('inf')\\n        f = [[inf]*(n+1) for _ in range(n)]\\n        for i in range(n+1):\\n            f[0][i] = min(arr1[0],arr2[-1])\\n        f[0][0] = arr1[0]\\n        for i in range(1,n):\\n            found_k = 0\\n            for j in range(n+1):\\n                if f[i-1][j] < arr1[i]: f[i][j] = arr1[i]\\n                if not j: continue\\n                va = f[i-1][j-1]\\n                if not found_k:\\n                    if arr2[0] > va:\\n                        l,r = 0,m-1\\n                        while l < r:\\n                            mid = l+r+1 >> 1\\n                            if arr2[mid] > va:\\n                                l = mid\\n                            else: \\n                                r = mid - 1\\n                        k = l\\n                        found_k = 1\\n                        f[i][j] = min(f[i][j],arr2[k])\\n                else:\\n                    while k+1 < m and arr2[k+1] > va: k += 1\\n                    f[i][j] = min(f[i][j],arr2[k])\\n        for i,v in enumerate(f[-1]): \\n            if v < inf: return i\\n        return -1\\n                \\n                \\n                \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            prev_with_op = min_last_value_given_operations(n - 1, ops - 1)\\n            b = find_larger_value_in_B(prev_with_op)\\n            # dp(n - 1, ops) <= dp(n - 1, ops - 1)\\n            # if dp(n - 1, ops - 1) < A[n - 1] => dp(n - 1, ops) < A[n - 1]\\n            if prev_with_op < A[n - 1]:\\n                return min(A[n - 1], b)\\n            elif b <= A[n - 1]:\\n                return b\\n            elif min_last_value_given_operations(n - 1, ops) < A[n - 1]:\\n                return A[n - 1]\\n            return b\\n\\n        for ops in range(min(len(A), len(B)) + 1):\\n            if min_last_value_given_operations(len(A), ops) < float('inf'):\\n                return ops\\n\\n        return -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"def binsearch(arr,x):\\n    if(arr[0]>x):\\n        return 0\\n    l=0\\n    h=len(arr)-1\\n    ret=-1\\n    while(l<=h):\\n        mid=(l+h)//2\\n        if(arr[mid]<=x):\\n            l=mid+1\\n        elif(arr[mid]>x):\\n            ret=mid\\n            h=mid-1\\n    return ret\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m=len(arr2)\\n        n=len(arr1)\\n        dp={}\\n        # print(binsearch(arr2,0))\\n        def dfs(arr1,arr2,left,curr,dp):\\n            if(curr>=len(arr1)):\\n                return 0\\n            if((curr,left) in dp):\\n                return dp[(curr,left)]\\n            res1=sys.maxsize\\n            res2=0\\n            if(arr1[curr]>left):\\n                res1=dfs(arr1,arr2,arr1[curr],curr+1,dp)\\n            mid=binsearch(arr2,left)\\n            if(mid==-1):\\n                res2=sys.maxsize-1\\n            else:\\n                res2=dfs(arr1,arr2,arr2[mid],curr+1,dp)\\n            dp[(curr,left)]=min(res1,1+res2)\\n            return dp[(curr,left)]\\n        x=dfs(arr1,arr2,-sys.maxsize,0,dp)\\n        if(x>=(sys.maxsize-1)):\\n            return -1\\n        return x\\n            \\n\", \"def binsearch(arr,x):\\n    if(arr[0]>x):\\n        return 0\\n    l=0\\n    h=len(arr)-1\\n    ret=-1\\n    while(l<=h):\\n        mid=(l+h)//2\\n        if(arr[mid]<=x):\\n            l=mid+1\\n        elif(arr[mid]>x):\\n            ret=mid\\n            h=mid-1\\n    return ret\\nclass Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m=len(arr2)\\n        n=len(arr1)\\n        dp={}\\n        def dfs(arr1,arr2,left,curr,dp):\\n            if(curr>=len(arr1)):\\n                return 0\\n            if((curr,left) in dp):\\n                return dp[(curr,left)]\\n            res1=sys.maxsize\\n            res2=0\\n            if(arr1[curr]>left):\\n                res1=dfs(arr1,arr2,arr1[curr],curr+1,dp)\\n            mid=binsearch(arr2,left)\\n            if(mid==-1):\\n                res2=sys.maxsize-1\\n            else:\\n                res2=dfs(arr1,arr2,arr2[mid],curr+1,dp)\\n            dp[(curr,left)]=min(res1,1+res2)\\n            return dp[(curr,left)]\\n        x=dfs(arr1,arr2,-sys.maxsize,0,dp)\\n        if(x==(sys.maxsize)):\\n            return -1\\n        return x\\n            \\n\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n        b2idx = {b: i for i, b in enumerate(B)}\\n        \\n        # @lru_cache(None)\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            if val in b2idx:\\n                return B[b2idx[val] + 1]\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            elif n == 0:\\n                return float('-inf')\\n            elif ops > n:\\n                return min_last_value_given_operations(n, n)\\n\\n            skip_op = min_last_value_given_operations(n - 1, ops)\\n            if skip_op == float('inf'):\\n                return float('inf')\\n            return min(\\n                A[n - 1] if skip_op < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        last_success = -1\\n        for ops in range(min(len(A), len(B)), -1, -1):\\n            if min_last_value_given_operations(len(A), ops) == float('inf'):\\n                break\\n            last_success = ops\\n\\n        return last_success\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        arr2.sort()\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        \\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=n:\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,m-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        arr2.sort()\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans\\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            if i>=len(arr1):\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                    # pick any index from arr2 starting j\\n                idx=bsearch(j,len(arr2)-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                \\n                    \\n#                 for k in range(j,len(arr2)):\\n#                     # can we use binary search here\\n#                     # we got to find out the minumum value in arr2 which is greater than prev\\n                    \\n#                     if arr2[k]>prev:\\n#                         # we can probably use binary search here\\n#                         # get the first index which is strictly greater than prev\\n#                         ans=min(ans,1+helper(i+1,k+1,arr2[k]))\\n#                         break\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n                \\n                \\n        \\n        \\n\", \"import bisect\\nclass Solution:\\n    def recurse(self,arr1,arr2,m,n,idx,prev,dp):\\n        if idx>=n:\\n            return 0\\n        k=bisect.bisect_right(arr2,prev)\\n        if dp[idx][k]!=-1:\\n            return dp[idx][k]\\n        c1=self.recurse(arr1,arr2,m,n,idx+1,arr1[idx],dp) if arr1[idx]>prev else 2000\\n        c2=1+self.recurse(arr1,arr2,m,n,idx+1,arr2[k],dp) if k<m else 2000\\n        dp[idx][k]=min(c1,c2)\\n        return dp[idx][k]\\n     \\n\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        dp=[[-1 for i in range(2001)] for j in range(2001)]\\n        arr2.sort()\\n        self.recurse(arr1,arr2,len(arr2),len(arr1),0,-10**9,dp)\\n        if dp[0][0]>=2000:\\n            return -1\\n        else:\\n            return dp[0][0]\\n      \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # Maintain two dp array: 1 for keep arr1[i], 1 for swap arr1[i] with arr2[j]\\n        # Convert arr2 to sorted unique set\\n        a = arr1\\n        b = sorted(list(set(arr2)))\\n        n, m = len(a), len(b)\\n        keep = [0, math.inf]  # keep[i] = keep a[i]\\n        swap = [[1] * m, [math.inf] * m]  # swap[i][j] = a[i] replaced with b[j]\\n\\n        prev, curr = 1, 0\\n        for i in range(1, n):\\n            prev, curr = curr, prev\\n            # must init every time\\n            swap[curr] = [math.inf] * m\\n            keep[curr] = math.inf\\n            # keep[i] case 1: a[i] is bigger than previous keep[i-1]\\n            if a[i] > a[i - 1]:\\n                keep[curr] = keep[prev]\\n            for j in range(m):\\n                # keep[i] case 2: a[i] is bigger then previous swap value.\\n                if a[i] > b[j]:\\n                    keep[curr] = min(keep[curr], swap[prev][j])\\n                # swap case 1: when a[i-1] keeps\\n                if b[j] > a[i - 1]:\\n                    swap[curr][j] = min(swap[curr][j], 1 + keep[prev])\\n                # Swap case 2: when a[i-1] swapped\\n                if j > 0:\\n                    # only need to check last swap[i-1][j-1] since swap[][j] decreases when j increases.\\n                    swap[curr][j] = min(swap[curr][j], 1 + swap[prev][j - 1])\\n\\n        res = min(keep[curr], swap[curr][m - 1])\\n        return res if res < math.inf else -1\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        #10:04\\n        # arr2=list(set(arr2))\\n        n=len(arr1)\\n        arr2=list(set(arr2))\\n        arr2.sort()\\n        m=len(arr2)\\n        # we might need fix at point 0\\n        # hence we will always compare it will\\n        def bsearch(left,right,val):\\n            ans=-1\\n            while left<=right:\\n                mid=left+(right-left)//2\\n                if arr2[mid]>val:\\n                    ans=mid\\n                    right=mid-1\\n                else:\\n                    left=mid+1\\n            return ans \\n        \\n        @lru_cache(None)\\n        def helper(i,j,prev):\\n            nonlocal n,m\\n            if i>=n:\\n                # arr1 is increasing, we have reached so far\\n                return 0\\n            else:\\n                ans=float('inf')\\n                if arr1[i]>prev:\\n                    # no need of replacement\\n                    ans=min(ans,helper(i+1,j,arr1[i]))\\n                # pick first index from arr2 starting j which is greater than prev\\n                idx=bsearch(j,m-1,prev)\\n                if idx!=-1:\\n                    ans=min(ans,1+helper(i+1,idx+1,arr2[idx]))\\n                return ans\\n        ans=helper(0,0,-1)\\n        return -1 if ans==float('inf') else ans\\n\\n    \\n                \\n                \\n        \\n        \\n\", \"import bisect\\nclass Solution:\\n    def recurse(self,arr1,arr2,m,n,idx,prev,dp):\\n        if idx>=n:\\n            return 0\\n        k=bisect.bisect_right(arr2,prev)\\n        if dp[idx][k]!=-1:\\n            return dp[idx][k]\\n        c1=self.recurse(arr1,arr2,m,n,idx+1,arr1[idx],dp) if arr1[idx]>prev else 2000\\n        c2=1+self.recurse(arr1,arr2,m,n,idx+1,arr2[k],dp) if k<m else 2000\\n        dp[idx][k]=min(c1,c2)\\n        return dp[idx][k]\\n     \\n\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        dp=[[-1 for i in range(2001)] for j in range(2001)]\\n        arr2.sort()\\n        ans=self.recurse(arr1,arr2,len(arr2),len(arr1),0,-10**9,dp)\\n        if ans>=2000:\\n            return -1\\n        else:\\n            return ans\\n      \\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        # Maintain two dp array: 1 for keep arr1[i], 1 for swap arr1[i] with arr2[j]\\n        # Convert arr2 to sorted unique set\\n        a = arr1\\n        b = sorted(list(set(arr2)))\\n        n, m = len(a), len(b)\\n        keep = [math.inf] * n  # keep[i] = keep a[i]\\n        swap = [[math.inf] * m for _ in range(n)]  # swap[i][j] = a[i] replaced with b[j]\\n        # init\\n        keep[0] = 0  # keep a[0], no swap\\n        swap[0] = [1] * m  # can be swapped with any a[j]\\n\\n        for i in range(1, n):\\n            # keep[i] case 1: a[i] is bigger than previous keep[i-1]\\n            if a[i] > a[i - 1]:\\n                keep[i] = keep[i - 1]\\n            for j in range(m):\\n                # keep[i] case 2: a[i] is bigger then previous swap value.\\n                if a[i] > b[j]:\\n                    keep[i] = min(keep[i], swap[i - 1][j])\\n                # swap case 1: when a[i-1] keeps\\n                if b[j] > a[i - 1]:\\n                    swap[i][j] = min(swap[i][j], 1 + keep[i - 1])\\n                # Swap case 2: when a[i-1] swapped\\n                if j > 0:\\n                    # only need to check last swap[i-1][j-1] since swap[][j] decreases when j increases.\\n                    swap[i][j] = min(swap[i][j], 1 + swap[i - 1][j - 1])\\n\\n        res = min(keep[n - 1], swap[n - 1][m - 1])\\n        return res if res < math.inf else -1        \", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        m = len(arr1)\\n        n = len(arr2)\\n        newarr2 = [arr2[0]]\\n        for i in range(1, n):\\n            if arr2[i] != newarr2[-1]:\\n                newarr2.append(arr2[i])\\n                \\n        arr2 = newarr2\\n        n = len(arr2)\\n\\n        IL = 10 ** 9 + 7\\n        \\n        dp = [[IL for j in range(n + 1)] for i in range(m)]\\n        \\n        dp[0][n] = 0\\n        for i in range(n):\\n            if arr2[i] < arr1[0]:\\n                dp[0][i] = 1\\n            else:\\n                break\\n        \\n        for i in range(1, m):\\n            idx = 0\\n            if arr2[0] > arr1[i - 1]:\\n                dp[i][0] = dp[i - 1][n] + 1\\n            \\n            for k in range(1, n):\\n                a = dp[i - 1][n] + 1 if dp[i - 1][n] != IL and arr1[i - 1] < arr2[k] else IL\\n                b = dp[i - 1][idx] + 1 if dp[i - 1][idx] != IL else IL\\n                dp[i][k] = min(a, b)\\n                if dp[i - 1][k] < dp[i - 1][idx]:\\n                    idx = k\\n                    \\n            if dp[i - 1][n] != IL and arr1[i] > arr1[i -1]:\\n                dp[i][n] = dp[i - 1][n]\\n            \\n            for k in range(n):\\n                if dp[i - 1][k] != IL and arr2[k] < arr1[i]:\\n                    dp[i][n] = min(dp[i][n], dp[i - 1][k])\\n                    \\n        m = min(dp[m - 1])\\n        \\n        return m if m != IL else -1\\n        \\n                \\n\", \"class Solution:    \\n    def binary_search_right(self, arr, l, r, t):\\n        while l < r:\\n            m = l + (r - l) // 2\\n            if arr[m] <= t:\\n                l = m + 1\\n            else:\\n                r = m\\n        return l\\n    \\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2.sort()\\n        \\n        dp = {}\\n        def help(i1, i2):\\n            if i1 == len(arr1):\\n                return 0\\n            \\n            if i1 != 0:\\n                key = (i1, i2, arr1[i1 - 1])\\n            else:\\n                key = (i1, i2, None)\\n            \\n            if key in dp:\\n                return dp[key]\\n            \\n            # print(key)\\n            \\n            # all possible results\\n            pos = []\\n            \\n            # don't make changes here\\n            if i1 == 0 or arr1[i1 - 1] < arr1[i1]:\\n                res = help(i1 + 1, i2)\\n                if res != -1:\\n                    pos.append(res)\\n                    \\n            if i1 != 0:\\n                # make change\\n                i2 = self.binary_search_right(arr2, i2, len(arr2), arr1[i1 - 1])\\n                if i2 != len(arr2):\\n                    tmp = arr1[i1]\\n                    arr1[i1] = arr2[i2]\\n                    res = help(i1 + 1, i2 + 1)\\n                    if res != -1:\\n                        pos.append(res + 1)\\n                    arr1[i1] = tmp\\n            else:\\n                # make change\\n                if i2 < len(arr2) and arr2[i2] < arr1[i1]:\\n                    tmp = arr1[i1]\\n                    arr1[i1] = arr2[i2]\\n                    res = help(i1 + 1, i2 + 1)\\n                    if res != -1:\\n                        pos.append(res + 1)\\n                    arr1[i1] = tmp\\n            \\n            if len(pos) == 0:\\n                dp[key] = -1\\n            else:\\n                dp[key] = min(pos)\\n            return dp[key]\\n        \\n        return help(0, 0)\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        \\n        # dp[i][k] means the minimum number we can have at ith position with k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                if dp[i - 1][k] < arr1[i]:\\n                    # not assign\\n                    dp[i][k] = arr1[i]\\n                \\n                num = self.helper(arr2, dp[i - 1][k - 1])\\n                if num > dp[i - 1][k - 1]:\\n                    dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n\\n    def helper(self, arr, val):\\n        # find in arr the smallest number that is larger than val\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            else:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        arr2 = sorted(set(arr2))\\n        size1, size2 = len(arr1), len(arr2)\\n        Inf = float('inf')\\n        keep = [Inf] * size1\\n        keep[0] = 0\\n        swap = [[Inf] * size2 for _ in range(size1)]\\n        swap[0] = [1] * size2\\n        for i in range(1, size1):\\n            minKeep = minSwap = Inf\\n            for j in range(size2):\\n                if arr1[i] > arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                if arr1[i] > arr2[j]:\\n                    minKeep = min(minKeep, swap[i-1][j])\\n                if arr2[j] > arr1[i-1]:\\n                    swap[i][j] = keep[i-1] + 1\\n                if j > 0: # arr2[j] > arr2[j-1] is always True\\n                    minSwap = min(minSwap, swap[i-1][j-1] + 1)\\n                keep[i] = min(keep[i], minKeep)\\n                swap[i][j] = min(swap[i][j], minSwap)\\n        res = min(min(swap[-1]), keep[-1])\\n        return -1 if res == Inf else res\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        \\n        # dp[i][k] means the minimum number we can have at ith position with k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                if dp[i - 1][k] < arr1[i]:\\n                    # not assign\\n                    dp[i][k] = arr1[i]\\n                \\n                if k >= 1:\\n                    num = self.helper(arr2, dp[i - 1][k - 1])\\n                    if num > dp[i - 1][k - 1]:\\n                        dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n\\n    def helper(self, arr, val):\\n        # find in arr the smallest number that is larger than val\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            else:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        n = len(arr1)\\n        arr1.insert(0, -1)\\n        arr2.sort()\\n        # dp[i][k] means the minimum number we can get at arr1[i] using k operations\\n        dp = [[sys.maxsize for k in range(n + 1)] for i in range(n + 1)]\\n        dp[0][0] = -1\\n        \\n        for i in range(1, n + 1):\\n            for k in range(i + 1):\\n                # not changing for arr1[i]\\n                if dp[i - 1][k] < arr1[i]:\\n                    dp[i][k] = arr1[i]\\n                \\n                # changing for arr1[i]\\n                # find the smallested number in arr2 that is larger than dp[i - 1][k - 1]\\n                if k >= 1:\\n                    num = self.helper(arr2, dp[i - 1][k - 1])\\n                    if num > dp[i - 1][k - 1]:\\n                        dp[i][k] = min(dp[i][k], num)\\n        \\n        ans = sys.maxsize\\n        for k in range(0, n + 1):\\n            if dp[n][k] < sys.maxsize:\\n                ans = min(ans, k)\\n        return ans if ans < sys.maxsize else -1\\n    \\n    def helper(self, arr, val):\\n        start, end = 0, len(arr) - 1\\n        while start + 1 < end:\\n            mid = start + (end - start) // 2\\n            if arr[mid] <= val:\\n                start = mid\\n            elif arr[mid] > val:\\n                end = mid\\n        \\n        if arr[start] > val:\\n            return arr[start]\\n        return arr[end]\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        set2 = set(arr2)\\n        arr2 = sorted(set2)\\n        N, M = len(arr1), len(arr2)\\n        to_index = {}\\n        left = 0\\n        for x in sorted(arr1):\\n            while left < M and arr2[left] <= x:\\n                left += 1\\n            to_index[x] = left - 1 if left < M else M\\n        \\n        print(to_index, arr2)\\n        dp = [[[None, None] for _ in range(M + 1)] for _ in range(N)]\\n        \\n        def solve(i, j, k):\\n            if i == N: return 0\\n            if j > M or (j == M and k == 1): return N+1\\n            if dp[i][j][k] is None:\\n                if i == 0: \\n                    dp[i][j][k] = min(solve(i+1, to_index[arr1[0]], 0), 1 + solve(i+1, 0, 1))\\n                else:\\n                    result = N + 1\\n                    if j < M - 1: \\n                        result = 1 + solve(i+1, j+1, 1)\\n                    if k == 0 and arr1[i] > arr1[i-1]:\\n                        result = min(result, solve(i+1, to_index[arr1[i]], 0))\\n                    if k == 1 and arr1[i] > arr2[j]:\\n                        result = min(result, solve(i+1, to_index[arr1[i]], 0))\\n                    dp[i][j][k] = result\\n            return dp[i][j][k]\\n        result = solve(0, 0, 0)\\n        # for i in range(N):\\n        #     for j in range(M):\\n        #         for k in [0, 1]:\\n        #             print(dp[i][j][k], i, j, k)\\n        return result if result <= N else -1\", \"import bisect\\nfrom functools import lru_cache\\n\\n\\nclass Solution:\\n    def makeArrayIncreasing(self, A: List[int], B: List[int]) -> int:\\n        B = sorted(set(B))\\n\\n        def find_larger_value_in_B(val):\\n            if val >= B[-1]:\\n                return float('inf')\\n            return B[bisect.bisect_right(B, val)]\\n\\n        @lru_cache(None)\\n        def min_last_value_given_operations(n, ops):\\n            if ops < 0:\\n                return float('inf')\\n            if n == 0:\\n                return float('-inf')\\n                # return A[n - 1] if first_n_of_A_is_sorted(n) else float('inf')\\n\\n            ops = min(n, ops)\\n            return min(\\n                A[n - 1] if min_last_value_given_operations(n - 1, ops) < A[n - 1] else float('inf'),\\n                find_larger_value_in_B(min_last_value_given_operations(n - 1, ops - 1)),\\n            )\\n\\n        for ops in range(min(len(A), len(B)) + 1):\\n            if min_last_value_given_operations(len(A), ops) < float('inf'):\\n                return ops\\n\\n        return -1\\n\", \"class Solution:\\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\\n        ## https://www.youtube.com/watch?v=8ttxdMCU2GE\\n        m = len(arr1)\\n        ## remove dulpicates and sort arr2\\n        arr2 = sorted(list(set(arr2)))\\n        # print(arr2)\\n        n = len(arr2)\\n        swap = [[float('inf') for j in range(n)] for i in range(m)]\\n        keep = [float('inf') for i in range(m)]\\n        \\n        ## initialization\\n        keep[0] = 0\\n        for j in range(n):\\n            swap[0][j] = 1\\n            \\n        for i in range(1, m):\\n            min_keep = float('inf')\\n            min_swap = float('inf')\\n            for j in range(n):\\n                ## two variables to help compute case 3 & 4\\n                ## case 4: the last two elements of current valid array are both from arr2\\n                if j>0:\\n                    min_swap = min(min_swap, swap[i-1][j-1]+1)\\n\\n                ## case 3: the second to last element is replaced by the previous element arr2[j-1] or earlier elements\\n                if arr1[i]>arr2[j]:\\n                    min_keep = min(min_keep, swap[i-1][j])\\n                    \\n                ## case 1: no need to swap; keep arr1[i]\\n                if arr1[i]>arr1[i-1]:\\n                    keep[i] = keep[i-1]\\n                \\n                ## case 2: ## replace arr1[i] by arr2[j]\\n                if arr2[j]>arr1[i-1]:\\n                    swap[i][j] = keep[i-1] + 1\\n                \\n                ## update\\n                swap[i][j] = min(swap[i][j], min_swap)\\n                keep[i] = min(keep[i], min_keep)\\n                \\n        # for i in range(m):\\n        #     print(keep[i], swap[i])\\n                \\n        res = min(min(swap[m-1]), keep[m-1])\\n        if res == float('inf'):\\n            return -1\\n        else:\\n            return res\\n\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                5,
                3,
                6,
                7
            ],
            [
                1,
                6,
                3,
                3
            ]
        ],
        "output": -1,
        "halu_type": "Identification Hallucination",
        "fn_name": "makeArrayIncreasing",
        "starter_code": "\nclass Solution:\n    def makeArrayIncreasing(self, arr1: List[int], arr2: List[int]) -> int:\n        ",
        "url": "https://leetcode.com/problems/make-array-strictly-increasing/"
    },
    {
        "id": 1260,
        "task_id": 2631,
        "test_case_id": 1,
        "question": "There are a total of n courses you have to take, labeled from 0 to n-1.\n\nSome courses may have prerequisites, for example to take course 0 you have to first take course 1, which is expressed as a pair: [0,1]\n\nGiven the total number of courses and a list of prerequisite pairs, is it possible for you to finish all courses?\n\nExample 1:\n\n\nInput: 2, [[1,0]] \nOutput: true\nExplanation: There are a total of 2 courses to take. \n             To take course 1 you should have finished course 0. So it is possible.\n\nExample 2:\n\n\nInput: 2, [[1,0],[0,1]]\nOutput: false\nExplanation: There are a total of 2 courses to take. \n             To take course 1 you should have finished course 0, and to take course 0 you should\n             also have finished course 1. So it is impossible.\n\n\nNote:\n\n\n       The input prerequisites is a graph represented by a list of edges, not adjacency matrices. Read more about how a graph is represented.\n       You may assume that there are no duplicate edges in the input prerequisites.",
        "solutions": "[\"class Solution(object):\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         graph = [[] for _ in range(numCourses)]\\n         visited = [0 for _ in range(numCourses)]\\n         # create graph\\n         for pair in prerequisites:\\n             x, y = pair\\n             graph[x].append(y)\\n         # visit each node\\n         for i in range(numCourses):\\n             if not self.dfs(graph, visited, i):\\n                 return False\\n         return True\\n     \\n     def dfs(self, graph, visited, i):\\n         # if ith node is marked as being visited, then a cycle is found\\n         if visited[i] == -1:\\n             return False\\n         # if it is done visted, then do not visit again\\n         if visited[i] == 1:\\n             return True\\n         # mark as being visited\\n         visited[i] = -1\\n         # visit all the neighbours\\n         for j in graph[i]:\\n             if not self.dfs(graph, visited, j):\\n                 return False\\n         # after visit all the neighbours, mark it as done visited\\n         visited[i] = 1\\n         return True\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         V = numCourses\\n         \\n         # Store outgoing edges\\n         edges = []\\n         \\n         for _ in range(V):\\n             edges.append(set())\\n         \\n         for p in prerequisites:\\n             # Edge goes from v1 to v2\\n             v2, v1 = p\\n             edges[v1].add(v2)\\n         \\n         checked = set()\\n         \\n         def detect_cycle(x, visited):\\n             visited.add(x)\\n             checked.add(x)\\n             \\n             for v in edges[x]:\\n                 if v in visited:\\n                     return True\\n                 if detect_cycle(v, visited):\\n                     return True\\n             visited.remove(x)\\n             return False\\n         \\n         for v in range(V):\\n             if v in checked:\\n                 continue\\n             if detect_cycle(v, set()):\\n                 return False\\n         \\n         return True\\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         \\n         # sol1: topo sort\\n         N = numCourses\\n         indegree = [0] * N\\n         O = [set() for _ in range(N)]\\n         for b, a in prerequisites:\\n             O[a].add(b)\\n             indegree[b] += 1\\n         \\n         stack = [c for c in range(N) if indegree[c] == 0]\\n         done = 0\\n         while stack:\\n             i = stack.pop()\\n             done += 1\\n             for j in O[i]:\\n                 indegree[j] -= 1\\n                 if indegree[j] == 0:\\n                     stack.append(j)\\n         return done == N\\n         \\n         # sol2: UF\\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         pre_count = [0] * numCourses\\n         next_courses = [[] for _ in range(numCourses)]\\n         \\n         for cur, pre in prerequisites:\\n             next_courses[pre].append(cur)\\n             pre_count[cur] += 1\\n         \\n         q = []\\n         \\n         # always chose courses without prereq\\n         for i, count in enumerate(pre_count):\\n             if count == 0:\\n                 q.append(i)\\n         \\n         finish_num = 0\\n         # take those courses which can be taken after finishing prereq\\n         while q:\\n             course = q.pop(0)\\n             finish_num += 1\\n             for next_course in next_courses[course]:\\n                 pre_count[next_course] -= 1\\n                 if pre_count[next_course]  == 0:\\n                     q.append(next_course)\\n         \\n         return finish_num == numCourses\\n             \\n         \\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         graph = [[] for _ in range(numCourses)]\\n         visit = [0 for _ in range(numCourses)]\\n         for x, y in prerequisites:\\n             graph[x].append(y)\\n             \\n         ## states: 0 = unkonwn, 1 == visiting, 2 = visited\\n         def dfs(c,v):\\n             if v[c] == 2: return False\\n             if v[c] == 1: return True\\n             v[c]=1 #lable as visiting, if it does not visited eventually, it will cause an exception\\n             for i in graph[c]:\\n                 if dfs(i,v): return True\\n                 \\n             v[c]=2 # change to visited after done visiting\\n             return False\\n         \\n         \\n         for j in range(numCourses):\\n             if dfs(j,visit): return False\\n         return True\\n             \\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"       \\n         in_degrees = collections.defaultdict(int)        \\n         edges = collections.defaultdict(list)\\n \\n         for i in range(0, numCourses):\\n             in_degrees[i] = 0            \\n         \\n         for prerequisty in prerequisites:\\n             start = prerequisty[1]\\n             end = prerequisty[0]\\n             edges[start].append(end)\\n             in_degrees[end] += 1\\n \\n         queue = collections.deque()\\n         \\n         for i in range(0, numCourses):\\n             if in_degrees[i] == 0:\\n                 queue.append(i)\\n             \\n         count = 0\\n         while len(queue) > 0:\\n             node = queue.popleft()\\n             count += 1\\n             \\n             for neighbor in edges[node]:\\n                 in_degrees[neighbor] -= 1\\n                 if in_degrees[neighbor] == 0:\\n                     queue.append(neighbor)\\n                     \\n         return count == numCourses        \", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         course_map = {i: [] for i in range(numCourses)}\\n         degree = [0 for i in range(numCourses)]\\n         q = []\\n         counter = 0\\n         for c,p in prerequisites:\\n             course_map[c].append(p)\\n             degree[p] += 1\\n         \\n         for i in range(numCourses):\\n             if degree[i] == 0:\\n                 q.append(i)\\n                 counter += 1\\n         while q:\\n             cur = q.pop(0)\\n             for p in course_map[cur]:\\n                 degree[p] -= 1\\n                 if degree[p] == 0:\\n                     q.append(p)\\n                     counter += 1\\n         \\n         return counter == numCourses\\n\", \"class Solution:\\n     def canFinish(self, n, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         def ok(node):\\n             if seens[node] == -1:\\n                 return False\\n             if not seens[node]:\\n                 seens[node] = -1\\n                 for pre in pres[node]:\\n                     if not ok(pre):\\n                         return False\\n                 seens[node] = 1\\n             return True\\n         \\n         pres = [[] for _ in range(n)]\\n         for cur, pre in prerequisites:\\n             pres[cur].append(pre)\\n         \\n         seens = [0] * n\\n         for node in range(n):\\n             if not ok(node):\\n                 return False\\n         return True\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         from collections import deque\\n         class Vertex(object):\\n             def __init__(self):\\n                 self.inDegree = 0\\n                 self.visited = False\\n                 self.adjList = []\\n                 self.invAdjList = []\\n             def info(self):\\n                 print(self.adjList)\\n                 print(self.invAdjList, self.inDegree)\\n                 print(self.visited)\\n         #initialization\\n         vertices = []\\n         for _ in range(numCourses):\\n             vertices.append(Vertex())\\n \\n         for i in prerequisites:\\n             pre = i[0]\\n             post = i[1]\\n             vertices[pre].adjList.append(post)\\n             vertices[post].inDegree += 1\\n             vertices[post].invAdjList.append(pre)\\n \\n         #BFS\\n         q = deque()\\n         coursesLearned = []\\n         for i in range(numCourses):\\n             if vertices[i].inDegree == 0:\\n                 q.append(i)\\n \\n         while q:\\n             course = q.popleft()\\n             coursesLearned.append(course)\\n             vertices[course].visited = True\\n             for c2 in vertices[course].adjList:\\n                 vertices[c2].inDegree -= 1\\n                 if vertices[c2].inDegree == 0:\\n                     q.append(c2)\\n         return len(coursesLearned) == numCourses\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         \\n         N = numCourses\\n         \\n #         # sol1: topo sort\\n #         indegree = [0] * N\\n         O = [set() for _ in range(N)]\\n         for b, a in prerequisites:\\n             O[a].add(b)\\n #             indegree[b] += 1\\n             \\n #         q = [c for c in range(N) if indegree[c] == 0]\\n #         done = 0\\n #         while q:\\n #             c = q.pop()\\n #             done += 1\\n #             for post in O[c]:\\n #                 indegree[post] -= 1\\n #                 if indegree[post] == 0:\\n #                     q.append(post)\\n #         return done == N # CAUTION\\n         \\n         # sol2: dfs\\n         color = [0] * N\\n         def dfs(i):\\n             if color[i] == -1:\\n                 return False\\n             elif color[i] == 1:\\n                 return True\\n             color[i] = -1\\n             if any(not dfs(j) for j in O[i]):\\n                 return False\\n             color[i] = 1\\n             return True\\n         \\n         return all(map(dfs, range(N)))\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         N = numCourses\\n         \\n #         # sol1: topo sort\\n #         pre = [0] * N\\n #         O = [set() for _ in range(N)]\\n #         for b, a in prerequisites:\\n #             O[a].add(b)\\n #             pre[b] += 1\\n             \\n #         stack = [c for c in range(N) if not pre[c]]\\n #         done = 0\\n #         while stack:\\n #             c = stack.pop()\\n #             done += 1\\n #             for post in O[c]:\\n #                 pre[post] -= 1\\n #                 if pre[post] == 0:\\n #                     stack.append(post)\\n #         return done == N\\n         \\n         # sol2: dfs\\n         O = [set() for _ in range(N)]\\n         for b, a in prerequisites:\\n             O[a].add(b) # it's O, not I\\n         visit = [0] * N\\n             \\n         def dfs(i):\\n             if visit[i] == -1:\\n                 return False\\n             elif visit[i] == 1:\\n                 return True\\n             \\n             visit[i] = -1\\n             if any(not dfs(j) for j in O[i]): # \\\"not\\\" dfs(j)\\n                 return False\\n             visit[i] = 1\\n             return True\\n         \\n         return all(map(dfs, range(N)))\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         edges = {i: [] for i in range(numCourses)}\\n         degree = [0 for i in range(numCourses)]\\n         \\n         for k, v in prerequisites:\\n             edges[v].append(k)\\n             degree[k] += 1\\n         \\n         q = []\\n         for i in range(numCourses):\\n             if degree[i] == 0:\\n                 q.append(i)\\n         \\n         count = 0\\n         while q:\\n             node = q.pop()\\n             count += 1\\n             for nn in edges[node]:\\n                 degree[nn] -= 1\\n                 if degree[nn] == 0:\\n                     q.append(nn)\\n         return count == numCourses\\n\"]",
        "difficulty": "interview",
        "input": [
            2,
            [
                [
                    1,
                    0
                ]
            ]
        ],
        "output": true,
        "halu_type": "Identification Hallucination",
        "fn_name": "canFinish",
        "starter_code": "\nclass Solution:\n    def canFinish(self, numCourses: int, prerequisites: List[List[int]]) -> bool:\n        ",
        "url": "https://leetcode.com/problems/course-schedule/"
    },
    {
        "id": 1261,
        "task_id": 2631,
        "test_case_id": 2,
        "question": "There are a total of n courses you have to take, labeled from 0 to n-1.\n\nSome courses may have prerequisites, for example to take course 0 you have to first take course 1, which is expressed as a pair: [0,1]\n\nGiven the total number of courses and a list of prerequisite pairs, is it possible for you to finish all courses?\n\nExample 1:\n\n\nInput: 2, [[1,0]] \nOutput: true\nExplanation: There are a total of 2 courses to take. \n             To take course 1 you should have finished course 0. So it is possible.\n\nExample 2:\n\n\nInput: 2, [[1,0],[0,1]]\nOutput: false\nExplanation: There are a total of 2 courses to take. \n             To take course 1 you should have finished course 0, and to take course 0 you should\n             also have finished course 1. So it is impossible.\n\n\nNote:\n\n\n       The input prerequisites is a graph represented by a list of edges, not adjacency matrices. Read more about how a graph is represented.\n       You may assume that there are no duplicate edges in the input prerequisites.",
        "solutions": "[\"class Solution(object):\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         graph = [[] for _ in range(numCourses)]\\n         visited = [0 for _ in range(numCourses)]\\n         # create graph\\n         for pair in prerequisites:\\n             x, y = pair\\n             graph[x].append(y)\\n         # visit each node\\n         for i in range(numCourses):\\n             if not self.dfs(graph, visited, i):\\n                 return False\\n         return True\\n     \\n     def dfs(self, graph, visited, i):\\n         # if ith node is marked as being visited, then a cycle is found\\n         if visited[i] == -1:\\n             return False\\n         # if it is done visted, then do not visit again\\n         if visited[i] == 1:\\n             return True\\n         # mark as being visited\\n         visited[i] = -1\\n         # visit all the neighbours\\n         for j in graph[i]:\\n             if not self.dfs(graph, visited, j):\\n                 return False\\n         # after visit all the neighbours, mark it as done visited\\n         visited[i] = 1\\n         return True\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         V = numCourses\\n         \\n         # Store outgoing edges\\n         edges = []\\n         \\n         for _ in range(V):\\n             edges.append(set())\\n         \\n         for p in prerequisites:\\n             # Edge goes from v1 to v2\\n             v2, v1 = p\\n             edges[v1].add(v2)\\n         \\n         checked = set()\\n         \\n         def detect_cycle(x, visited):\\n             visited.add(x)\\n             checked.add(x)\\n             \\n             for v in edges[x]:\\n                 if v in visited:\\n                     return True\\n                 if detect_cycle(v, visited):\\n                     return True\\n             visited.remove(x)\\n             return False\\n         \\n         for v in range(V):\\n             if v in checked:\\n                 continue\\n             if detect_cycle(v, set()):\\n                 return False\\n         \\n         return True\\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         \\n         # sol1: topo sort\\n         N = numCourses\\n         indegree = [0] * N\\n         O = [set() for _ in range(N)]\\n         for b, a in prerequisites:\\n             O[a].add(b)\\n             indegree[b] += 1\\n         \\n         stack = [c for c in range(N) if indegree[c] == 0]\\n         done = 0\\n         while stack:\\n             i = stack.pop()\\n             done += 1\\n             for j in O[i]:\\n                 indegree[j] -= 1\\n                 if indegree[j] == 0:\\n                     stack.append(j)\\n         return done == N\\n         \\n         # sol2: UF\\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         pre_count = [0] * numCourses\\n         next_courses = [[] for _ in range(numCourses)]\\n         \\n         for cur, pre in prerequisites:\\n             next_courses[pre].append(cur)\\n             pre_count[cur] += 1\\n         \\n         q = []\\n         \\n         # always chose courses without prereq\\n         for i, count in enumerate(pre_count):\\n             if count == 0:\\n                 q.append(i)\\n         \\n         finish_num = 0\\n         # take those courses which can be taken after finishing prereq\\n         while q:\\n             course = q.pop(0)\\n             finish_num += 1\\n             for next_course in next_courses[course]:\\n                 pre_count[next_course] -= 1\\n                 if pre_count[next_course]  == 0:\\n                     q.append(next_course)\\n         \\n         return finish_num == numCourses\\n             \\n         \\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         graph = [[] for _ in range(numCourses)]\\n         visit = [0 for _ in range(numCourses)]\\n         for x, y in prerequisites:\\n             graph[x].append(y)\\n             \\n         ## states: 0 = unkonwn, 1 == visiting, 2 = visited\\n         def dfs(c,v):\\n             if v[c] == 2: return False\\n             if v[c] == 1: return True\\n             v[c]=1 #lable as visiting, if it does not visited eventually, it will cause an exception\\n             for i in graph[c]:\\n                 if dfs(i,v): return True\\n                 \\n             v[c]=2 # change to visited after done visiting\\n             return False\\n         \\n         \\n         for j in range(numCourses):\\n             if dfs(j,visit): return False\\n         return True\\n             \\n\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"       \\n         in_degrees = collections.defaultdict(int)        \\n         edges = collections.defaultdict(list)\\n \\n         for i in range(0, numCourses):\\n             in_degrees[i] = 0            \\n         \\n         for prerequisty in prerequisites:\\n             start = prerequisty[1]\\n             end = prerequisty[0]\\n             edges[start].append(end)\\n             in_degrees[end] += 1\\n \\n         queue = collections.deque()\\n         \\n         for i in range(0, numCourses):\\n             if in_degrees[i] == 0:\\n                 queue.append(i)\\n             \\n         count = 0\\n         while len(queue) > 0:\\n             node = queue.popleft()\\n             count += 1\\n             \\n             for neighbor in edges[node]:\\n                 in_degrees[neighbor] -= 1\\n                 if in_degrees[neighbor] == 0:\\n                     queue.append(neighbor)\\n                     \\n         return count == numCourses        \", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         course_map = {i: [] for i in range(numCourses)}\\n         degree = [0 for i in range(numCourses)]\\n         q = []\\n         counter = 0\\n         for c,p in prerequisites:\\n             course_map[c].append(p)\\n             degree[p] += 1\\n         \\n         for i in range(numCourses):\\n             if degree[i] == 0:\\n                 q.append(i)\\n                 counter += 1\\n         while q:\\n             cur = q.pop(0)\\n             for p in course_map[cur]:\\n                 degree[p] -= 1\\n                 if degree[p] == 0:\\n                     q.append(p)\\n                     counter += 1\\n         \\n         return counter == numCourses\\n\", \"class Solution:\\n     def canFinish(self, n, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         def ok(node):\\n             if seens[node] == -1:\\n                 return False\\n             if not seens[node]:\\n                 seens[node] = -1\\n                 for pre in pres[node]:\\n                     if not ok(pre):\\n                         return False\\n                 seens[node] = 1\\n             return True\\n         \\n         pres = [[] for _ in range(n)]\\n         for cur, pre in prerequisites:\\n             pres[cur].append(pre)\\n         \\n         seens = [0] * n\\n         for node in range(n):\\n             if not ok(node):\\n                 return False\\n         return True\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         from collections import deque\\n         class Vertex(object):\\n             def __init__(self):\\n                 self.inDegree = 0\\n                 self.visited = False\\n                 self.adjList = []\\n                 self.invAdjList = []\\n             def info(self):\\n                 print(self.adjList)\\n                 print(self.invAdjList, self.inDegree)\\n                 print(self.visited)\\n         #initialization\\n         vertices = []\\n         for _ in range(numCourses):\\n             vertices.append(Vertex())\\n \\n         for i in prerequisites:\\n             pre = i[0]\\n             post = i[1]\\n             vertices[pre].adjList.append(post)\\n             vertices[post].inDegree += 1\\n             vertices[post].invAdjList.append(pre)\\n \\n         #BFS\\n         q = deque()\\n         coursesLearned = []\\n         for i in range(numCourses):\\n             if vertices[i].inDegree == 0:\\n                 q.append(i)\\n \\n         while q:\\n             course = q.popleft()\\n             coursesLearned.append(course)\\n             vertices[course].visited = True\\n             for c2 in vertices[course].adjList:\\n                 vertices[c2].inDegree -= 1\\n                 if vertices[c2].inDegree == 0:\\n                     q.append(c2)\\n         return len(coursesLearned) == numCourses\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         \\n         N = numCourses\\n         \\n #         # sol1: topo sort\\n #         indegree = [0] * N\\n         O = [set() for _ in range(N)]\\n         for b, a in prerequisites:\\n             O[a].add(b)\\n #             indegree[b] += 1\\n             \\n #         q = [c for c in range(N) if indegree[c] == 0]\\n #         done = 0\\n #         while q:\\n #             c = q.pop()\\n #             done += 1\\n #             for post in O[c]:\\n #                 indegree[post] -= 1\\n #                 if indegree[post] == 0:\\n #                     q.append(post)\\n #         return done == N # CAUTION\\n         \\n         # sol2: dfs\\n         color = [0] * N\\n         def dfs(i):\\n             if color[i] == -1:\\n                 return False\\n             elif color[i] == 1:\\n                 return True\\n             color[i] = -1\\n             if any(not dfs(j) for j in O[i]):\\n                 return False\\n             color[i] = 1\\n             return True\\n         \\n         return all(map(dfs, range(N)))\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         N = numCourses\\n         \\n #         # sol1: topo sort\\n #         pre = [0] * N\\n #         O = [set() for _ in range(N)]\\n #         for b, a in prerequisites:\\n #             O[a].add(b)\\n #             pre[b] += 1\\n             \\n #         stack = [c for c in range(N) if not pre[c]]\\n #         done = 0\\n #         while stack:\\n #             c = stack.pop()\\n #             done += 1\\n #             for post in O[c]:\\n #                 pre[post] -= 1\\n #                 if pre[post] == 0:\\n #                     stack.append(post)\\n #         return done == N\\n         \\n         # sol2: dfs\\n         O = [set() for _ in range(N)]\\n         for b, a in prerequisites:\\n             O[a].add(b) # it's O, not I\\n         visit = [0] * N\\n             \\n         def dfs(i):\\n             if visit[i] == -1:\\n                 return False\\n             elif visit[i] == 1:\\n                 return True\\n             \\n             visit[i] = -1\\n             if any(not dfs(j) for j in O[i]): # \\\"not\\\" dfs(j)\\n                 return False\\n             visit[i] = 1\\n             return True\\n         \\n         return all(map(dfs, range(N)))\", \"class Solution:\\n     def canFinish(self, numCourses, prerequisites):\\n         \\\"\\\"\\\"\\n         :type numCourses: int\\n         :type prerequisites: List[List[int]]\\n         :rtype: bool\\n         \\\"\\\"\\\"\\n         edges = {i: [] for i in range(numCourses)}\\n         degree = [0 for i in range(numCourses)]\\n         \\n         for k, v in prerequisites:\\n             edges[v].append(k)\\n             degree[k] += 1\\n         \\n         q = []\\n         for i in range(numCourses):\\n             if degree[i] == 0:\\n                 q.append(i)\\n         \\n         count = 0\\n         while q:\\n             node = q.pop()\\n             count += 1\\n             for nn in edges[node]:\\n                 degree[nn] -= 1\\n                 if degree[nn] == 0:\\n                     q.append(nn)\\n         return count == numCourses\\n\"]",
        "difficulty": "interview",
        "input": [
            2,
            [
                [
                    1,
                    0
                ],
                [
                    0,
                    1
                ]
            ]
        ],
        "output": false,
        "halu_type": "Identification Hallucination",
        "fn_name": "canFinish",
        "starter_code": "\nclass Solution:\n    def canFinish(self, numCourses: int, prerequisites: List[List[int]]) -> bool:\n        ",
        "url": "https://leetcode.com/problems/course-schedule/"
    },
    {
        "id": 1262,
        "task_id": 2632,
        "test_case_id": 1,
        "question": "Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.\n\nNote: You can only move either down or right at any point in time.\n\nExample:\n\n\nInput:\n[\n  [1,3,1],\n  [1,5,1],\n  [4,2,1]\n]\nOutput: 7\nExplanation: Because the path 1→3→1→1→1 minimizes the sum.",
        "solutions": "[\"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m, n = len(grid), len(grid[0])\\n         dp = [0] + [float('inf')] * (n-1)\\n         for i in range(m):\\n             dp[0] = dp[0] + grid[i][0]\\n             for j in range(1, n):\\n                 dp[j] = min(dp[j], dp[j-1]) + grid[i][j]\\n         return dp[-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)\\n         n = len(grid[0])\\n         dp = [grid[0][j] for j in range(n)]\\n         \\n         for j in range(1, n):\\n             dp[j] += dp[j-1]\\n         for i in range(1, m):\\n             for j in range(n):\\n                 if j == 0:\\n                     dp[j] += grid[i][j]\\n                 else:\\n                     dp[j] = min(grid[i][j]+dp[j-1], grid[i][j]+dp[j])\\n         return dp[-1]\\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         M = len(grid)\\n         if M == 0:\\n             return 0\\n         N = len(grid[0])\\n         if N == 0:\\n             return 0\\n         \\n         INF = float('inf')\\n         mem = {}\\n         \\n         # O(M*N)\\n         def min_path(i,j):\\n             if i == M-1 and j == N-1:\\n                 return grid[i][j]\\n             if (i,j) in mem:\\n                 return mem[(i,j)]\\n             \\n             min_sum = INF\\n             if i < M-1:\\n                 min_sum = min_path(i+1, j)\\n             if j < N-1:\\n                 min_sum = min(min_sum, min_path(i, j+1))\\n             \\n             mem[(i,j)] = grid[i][j] + min_sum\\n             return mem[(i,j)]\\n         \\n         return min_path(0, 0)\\n             \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)-1\\n         n = len(grid[0])-1\\n         dic = dict()\\n         s = self.minPathSumHelper(grid, 0, 0, m,n, dic)\\n         return s\\n     \\n     def minPathSumHelper(self, grid, i,j, m,n, dic):\\n         \\n         if (i, j, m, n) in dic:\\n             return dic[(i, j, m, n)]\\n         \\n         if i>m or j>n:\\n             return 0\\n         elif i==m:\\n             dic[(i, j, m, n)] = self.minPathSumHelper(grid, i, j+1, m, n, dic)+grid[i][j]\\n             return dic[(i, j, m, n)]\\n         elif j==n:\\n             dic[(i, j, m, n)] = self.minPathSumHelper(grid, i+1, j, m, n, dic)+grid[i][j]\\n             return dic[(i, j, m, n)]\\n         else:\\n             dic[(i, j, m, n)] = min(self.minPathSumHelper(grid, i+1, j, m,n, dic), self.minPathSumHelper(grid, i, j+1, m, n, dic))+grid[i][j]\\n             return dic[(i, j, m, n)]\\n         \\n\", \"class Solution(object):\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         max_row = len(grid) - 1  # rows\\n         max_col = len(grid[0]) - 1  # columns\\n         helper_grid = [[0]*(len(grid[0])) for _ in range(len(grid))]\\n         helper_grid[max_row][max_col] = grid[max_row][max_col]\\n         # fill max row and max column\\n         for i in range(max_col - 1, -1, -1):\\n             helper_grid[max_row][i] = grid[max_row][i] + helper_grid[max_row][i + 1]\\n         for i in range(max_row - 1, -1, -1):\\n             helper_grid[i][max_col] = grid[i][max_col] + helper_grid[i + 1][max_col]\\n \\n         # fill the rest\\n         for col in range(max_col - 1, -1, -1):\\n             for row in range(max_row - 1, -1, -1):\\n                 helper_grid[row][col] = grid[row][col] + min(helper_grid[row+1][col], helper_grid[row][col+1])\\n         # for row in helper_grid:\\n         #     print(row)\\n         return helper_grid[0][0]\\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not grid:\\n             return 0\\n         row_count = len(grid)\\n         col_count = len(grid[0])\\n         for i in range(1, row_count):\\n             grid[i][0] += grid[i-1][0]\\n         for i in range(1, col_count):\\n             grid[0][i] += grid[0][i-1]\\n         for row in range(1, row_count):\\n             for col in range(1, col_count):\\n                 grid[row][col] += min(grid[row-1][col], grid[row][col-1])\\n         return grid[-1][-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not grid:\\n             return 0\\n         row,col = len(grid),len(grid[0])\\n         dp = [0 for _ in range(col)]\\n         dp[0] = grid[0][0]\\n \\n         for i in range(1,col):\\n             dp[i] = dp[i-1] + grid[0][i]\\n \\n         for i in range(1,row):\\n             for j in range(0,col):\\n                 dp[j] = dp[j] + grid[i][j] if j==0 else min(dp[j-1],dp[j])+grid[i][j]\\n \\n         return dp[-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not grid:\\n             return 0\\n         row_count = len(grid)\\n         col_count = len(grid[0])\\n         dp = [[0 for _ in range(col_count)] for _ in range(row_count)]\\n         dp[0][0] = grid[0][0]\\n         if row_count == col_count and col_count == 1:\\n             return dp[-1][-1]\\n         for row in range(row_count):\\n             for col in range(col_count):\\n                 if row == 0 and col >= 1:\\n                     dp[row][col] = dp[row][col-1] + grid[row][col]\\n                 elif col == 0 and row >= 1:\\n                     dp[row][col] = dp[row-1][col] + grid[row][col]\\n                 else:\\n                     dp[row][col] = min(dp[row-1][col], dp[row][col-1]) + grid[row][col]\\n         return dp[-1][-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)\\n         n = len(grid[0])\\n         s = [[0 for j in range(n)] for i in range(m)]\\n         s[0][0] = grid[0][0]\\n         \\n         for i in range(1, m):\\n             s[i][0] = s[i-1][0] + grid[i][0]\\n \\n         for j in range(1, n):\\n             s[0][j] = s[0][j-1] + grid[0][j]\\n \\n         for i in range(1, m):\\n             for j in range(1, n):\\n                 s[i][j] = min(s[i-1][j]+grid[i][j], s[i][j-1]+grid[i][j])\\n         \\n \\n         return s[m-1][n-1]\\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         height = len(grid)\\n         width = len(grid[0])\\n         step_num = (height + width) - 2\\n         for step in range(1, step_num+1):\\n             for row in range(height):\\n                 col = step - row\\n                 if 0 <= row < height and 0 <= col < width:\\n                     if not row:\\n                         grid[row][col] += grid[row][col-1]\\n                     elif not col:\\n                         grid[row][col] += grid[row-1][col]\\n                     else:\\n                         grid[row][col] += min(grid[row][col-1], grid[row-1][col])\\n         return grid[-1][-1]\", \"class Solution:\\n     def minPathSum(self, grid): \\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         xx=len(grid)-1\\n         yy=len(grid[0])-1\\n         gridv=[[0 for j in range(len(grid[0]))] for i in range(len(grid))]\\n         gridv[xx][yy]=grid[xx][yy]\\n         for i in range(xx,-1,-1):\\n             for j in range(yy,-1,-1):\\n                 if i==xx:\\n                     if j==yy:\\n                          gridv[i][j]=grid[xx][yy]\\n                     else:\\n                          gridv[i][j]=grid[i][j]+gridv[i][j+1]\\n                 elif j==yy:\\n                     if i==xx:\\n                          gridv[i][j]=grid[xx][yy]\\n                     else:\\n                          gridv[i][j]=grid[i][j]+gridv[i+1][j]\\n                 else:            \\n                     gridv[i][j]=min(gridv[i+1][j]+grid[i][j],gridv[i][j+1]+grid[i][j])\\n         print(grid)\\n         print(gridv)\\n         return gridv[0][0]\\n     \\n         \\\"\\\"\\\"\\\"\\n         if(len(grid)==1 or len(grid[0])==1):\\n             return sum(sum(grid,[]))\\n         if\\n         return grid[0][0]+min(self.minPathSum(grid[1:],x,y+1),self.minPathSum([grids[1:] for grids in grid],x+1,y))\\n         \\\"\\\"\\\"\\\"\\\"\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         col = len(grid[0])\\n         row = len(grid)\\n         minSum = [[0 for x in range(col)] for y in range(row)]\\n         \\n         for i in range(row):\\n             for j in range(col):\\n                 add = 0\\n                 if i-1 >= 0 and j-1 >= 0:\\n                     add = min(minSum[i-1][j], minSum[i][j-1])\\n                 elif i-1 >= 0:\\n                     add = minSum[i-1][j]\\n                 elif j-1 >= 0:\\n                     add = minSum[i][j-1]\\n                 minSum[i][j] = grid[i][j] + add\\n         \\n         return minSum[-1][-1]\\n                 \\n         \\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)\\n         n = len(grid[0])\\n         if m == 0 or n == 0: return 0\\n         \\n         memory = [[0 for _ in range(n)] for _ in range(m)]\\n         \\n         def minSum(grid,x,y,n,m):\\n             if x==0 and y==0: return grid[0][0]\\n             if x<0 or y<0: return float(\\\"inf\\\")\\n             if memory[y][x] > 0: return memory[y][x]\\n             memory[y][x] = grid[y][x] + min(minSum(grid,x-1,y,n,m),minSum(grid,x,y-1,n,m))\\n             return memory[y][x]\\n             \\n         return minSum(grid,n-1,m-1,n,m)\\n     \\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    3,
                    1
                ],
                [
                    1,
                    5,
                    1
                ],
                [
                    4,
                    2,
                    1
                ]
            ]
        ],
        "output": 7,
        "halu_type": "Identification Hallucination",
        "fn_name": "minPathSum",
        "starter_code": "\nclass Solution:\n    def minPathSum(self, grid: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/minimum-path-sum/"
    },
    {
        "id": 1263,
        "task_id": 2632,
        "test_case_id": 2,
        "question": "Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.\n\nNote: You can only move either down or right at any point in time.\n\nExample:\n\n\nInput:\n[\n  [1,3,1],\n  [1,5,1],\n  [4,2,1]\n]\nOutput: 7\nExplanation: Because the path 1→3→1→1→1 minimizes the sum.",
        "solutions": "[\"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m, n = len(grid), len(grid[0])\\n         dp = [0] + [float('inf')] * (n-1)\\n         for i in range(m):\\n             dp[0] = dp[0] + grid[i][0]\\n             for j in range(1, n):\\n                 dp[j] = min(dp[j], dp[j-1]) + grid[i][j]\\n         return dp[-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)\\n         n = len(grid[0])\\n         dp = [grid[0][j] for j in range(n)]\\n         \\n         for j in range(1, n):\\n             dp[j] += dp[j-1]\\n         for i in range(1, m):\\n             for j in range(n):\\n                 if j == 0:\\n                     dp[j] += grid[i][j]\\n                 else:\\n                     dp[j] = min(grid[i][j]+dp[j-1], grid[i][j]+dp[j])\\n         return dp[-1]\\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         M = len(grid)\\n         if M == 0:\\n             return 0\\n         N = len(grid[0])\\n         if N == 0:\\n             return 0\\n         \\n         INF = float('inf')\\n         mem = {}\\n         \\n         # O(M*N)\\n         def min_path(i,j):\\n             if i == M-1 and j == N-1:\\n                 return grid[i][j]\\n             if (i,j) in mem:\\n                 return mem[(i,j)]\\n             \\n             min_sum = INF\\n             if i < M-1:\\n                 min_sum = min_path(i+1, j)\\n             if j < N-1:\\n                 min_sum = min(min_sum, min_path(i, j+1))\\n             \\n             mem[(i,j)] = grid[i][j] + min_sum\\n             return mem[(i,j)]\\n         \\n         return min_path(0, 0)\\n             \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n         \\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)-1\\n         n = len(grid[0])-1\\n         dic = dict()\\n         s = self.minPathSumHelper(grid, 0, 0, m,n, dic)\\n         return s\\n     \\n     def minPathSumHelper(self, grid, i,j, m,n, dic):\\n         \\n         if (i, j, m, n) in dic:\\n             return dic[(i, j, m, n)]\\n         \\n         if i>m or j>n:\\n             return 0\\n         elif i==m:\\n             dic[(i, j, m, n)] = self.minPathSumHelper(grid, i, j+1, m, n, dic)+grid[i][j]\\n             return dic[(i, j, m, n)]\\n         elif j==n:\\n             dic[(i, j, m, n)] = self.minPathSumHelper(grid, i+1, j, m, n, dic)+grid[i][j]\\n             return dic[(i, j, m, n)]\\n         else:\\n             dic[(i, j, m, n)] = min(self.minPathSumHelper(grid, i+1, j, m,n, dic), self.minPathSumHelper(grid, i, j+1, m, n, dic))+grid[i][j]\\n             return dic[(i, j, m, n)]\\n         \\n\", \"class Solution(object):\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         max_row = len(grid) - 1  # rows\\n         max_col = len(grid[0]) - 1  # columns\\n         helper_grid = [[0]*(len(grid[0])) for _ in range(len(grid))]\\n         helper_grid[max_row][max_col] = grid[max_row][max_col]\\n         # fill max row and max column\\n         for i in range(max_col - 1, -1, -1):\\n             helper_grid[max_row][i] = grid[max_row][i] + helper_grid[max_row][i + 1]\\n         for i in range(max_row - 1, -1, -1):\\n             helper_grid[i][max_col] = grid[i][max_col] + helper_grid[i + 1][max_col]\\n \\n         # fill the rest\\n         for col in range(max_col - 1, -1, -1):\\n             for row in range(max_row - 1, -1, -1):\\n                 helper_grid[row][col] = grid[row][col] + min(helper_grid[row+1][col], helper_grid[row][col+1])\\n         # for row in helper_grid:\\n         #     print(row)\\n         return helper_grid[0][0]\\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not grid:\\n             return 0\\n         row_count = len(grid)\\n         col_count = len(grid[0])\\n         for i in range(1, row_count):\\n             grid[i][0] += grid[i-1][0]\\n         for i in range(1, col_count):\\n             grid[0][i] += grid[0][i-1]\\n         for row in range(1, row_count):\\n             for col in range(1, col_count):\\n                 grid[row][col] += min(grid[row-1][col], grid[row][col-1])\\n         return grid[-1][-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not grid:\\n             return 0\\n         row,col = len(grid),len(grid[0])\\n         dp = [0 for _ in range(col)]\\n         dp[0] = grid[0][0]\\n \\n         for i in range(1,col):\\n             dp[i] = dp[i-1] + grid[0][i]\\n \\n         for i in range(1,row):\\n             for j in range(0,col):\\n                 dp[j] = dp[j] + grid[i][j] if j==0 else min(dp[j-1],dp[j])+grid[i][j]\\n \\n         return dp[-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not grid:\\n             return 0\\n         row_count = len(grid)\\n         col_count = len(grid[0])\\n         dp = [[0 for _ in range(col_count)] for _ in range(row_count)]\\n         dp[0][0] = grid[0][0]\\n         if row_count == col_count and col_count == 1:\\n             return dp[-1][-1]\\n         for row in range(row_count):\\n             for col in range(col_count):\\n                 if row == 0 and col >= 1:\\n                     dp[row][col] = dp[row][col-1] + grid[row][col]\\n                 elif col == 0 and row >= 1:\\n                     dp[row][col] = dp[row-1][col] + grid[row][col]\\n                 else:\\n                     dp[row][col] = min(dp[row-1][col], dp[row][col-1]) + grid[row][col]\\n         return dp[-1][-1]\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)\\n         n = len(grid[0])\\n         s = [[0 for j in range(n)] for i in range(m)]\\n         s[0][0] = grid[0][0]\\n         \\n         for i in range(1, m):\\n             s[i][0] = s[i-1][0] + grid[i][0]\\n \\n         for j in range(1, n):\\n             s[0][j] = s[0][j-1] + grid[0][j]\\n \\n         for i in range(1, m):\\n             for j in range(1, n):\\n                 s[i][j] = min(s[i-1][j]+grid[i][j], s[i][j-1]+grid[i][j])\\n         \\n \\n         return s[m-1][n-1]\\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         height = len(grid)\\n         width = len(grid[0])\\n         step_num = (height + width) - 2\\n         for step in range(1, step_num+1):\\n             for row in range(height):\\n                 col = step - row\\n                 if 0 <= row < height and 0 <= col < width:\\n                     if not row:\\n                         grid[row][col] += grid[row][col-1]\\n                     elif not col:\\n                         grid[row][col] += grid[row-1][col]\\n                     else:\\n                         grid[row][col] += min(grid[row][col-1], grid[row-1][col])\\n         return grid[-1][-1]\", \"class Solution:\\n     def minPathSum(self, grid): \\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         xx=len(grid)-1\\n         yy=len(grid[0])-1\\n         gridv=[[0 for j in range(len(grid[0]))] for i in range(len(grid))]\\n         gridv[xx][yy]=grid[xx][yy]\\n         for i in range(xx,-1,-1):\\n             for j in range(yy,-1,-1):\\n                 if i==xx:\\n                     if j==yy:\\n                          gridv[i][j]=grid[xx][yy]\\n                     else:\\n                          gridv[i][j]=grid[i][j]+gridv[i][j+1]\\n                 elif j==yy:\\n                     if i==xx:\\n                          gridv[i][j]=grid[xx][yy]\\n                     else:\\n                          gridv[i][j]=grid[i][j]+gridv[i+1][j]\\n                 else:            \\n                     gridv[i][j]=min(gridv[i+1][j]+grid[i][j],gridv[i][j+1]+grid[i][j])\\n         print(grid)\\n         print(gridv)\\n         return gridv[0][0]\\n     \\n         \\\"\\\"\\\"\\\"\\n         if(len(grid)==1 or len(grid[0])==1):\\n             return sum(sum(grid,[]))\\n         if\\n         return grid[0][0]+min(self.minPathSum(grid[1:],x,y+1),self.minPathSum([grids[1:] for grids in grid],x+1,y))\\n         \\\"\\\"\\\"\\\"\\\"\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         col = len(grid[0])\\n         row = len(grid)\\n         minSum = [[0 for x in range(col)] for y in range(row)]\\n         \\n         for i in range(row):\\n             for j in range(col):\\n                 add = 0\\n                 if i-1 >= 0 and j-1 >= 0:\\n                     add = min(minSum[i-1][j], minSum[i][j-1])\\n                 elif i-1 >= 0:\\n                     add = minSum[i-1][j]\\n                 elif j-1 >= 0:\\n                     add = minSum[i][j-1]\\n                 minSum[i][j] = grid[i][j] + add\\n         \\n         return minSum[-1][-1]\\n                 \\n         \\n\", \"class Solution:\\n     def minPathSum(self, grid):\\n         \\\"\\\"\\\"\\n         :type grid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(grid)\\n         n = len(grid[0])\\n         if m == 0 or n == 0: return 0\\n         \\n         memory = [[0 for _ in range(n)] for _ in range(m)]\\n         \\n         def minSum(grid,x,y,n,m):\\n             if x==0 and y==0: return grid[0][0]\\n             if x<0 or y<0: return float(\\\"inf\\\")\\n             if memory[y][x] > 0: return memory[y][x]\\n             memory[y][x] = grid[y][x] + min(minSum(grid,x-1,y,n,m),minSum(grid,x,y-1,n,m))\\n             return memory[y][x]\\n             \\n         return minSum(grid,n-1,m-1,n,m)\\n     \\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    2,
                    3
                ],
                [
                    4,
                    5,
                    6
                ]
            ]
        ],
        "output": 12,
        "halu_type": "Identification Hallucination",
        "fn_name": "minPathSum",
        "starter_code": "\nclass Solution:\n    def minPathSum(self, grid: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/minimum-path-sum/"
    },
    {
        "id": 1264,
        "task_id": 2634,
        "test_case_id": 1,
        "question": "Given a set of distinct integers, nums, return all possible subsets (the power set).\n\nNote: The solution set must not contain duplicate subsets.\n\nExample:\n\n\nInput: nums = [1,2,3]\nOutput:\n[\n  [3],\n  [1],\n  [2],\n  [1,2,3],\n  [1,3],\n  [2,3],\n  [1,2],\n  []\n]",
        "solutions": "[\"class Solution:\\n     def subsets(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         subsets = [[]]\\n         for v in nums:\\n             extra = []\\n             for s in subsets:\\n                 extra.append(s+[v])\\n             subsets += extra\\n         return subsets\", \"class Solution:\\n     def subsets(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def mask(x):\\n             return 1 << x\\n         \\n         exp = 1 << len(nums)\\n         ans = []\\n         for count in range(exp):\\n             new = []\\n             i = 0\\n             while mask(i) <= count:\\n                 if mask(i) & count:\\n                     new.append(nums[i])\\n                 i += 1\\n             ans.append(new)\\n         return ans\", \"class Solution:\\n     def subsets(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = [[]]\\n         for num in nums:\\n             result += self.get_subset(result, num)\\n         return result\\n         \\n     def get_subset(self, subsets, item):\\n         \\n         new_subset_with_item = []\\n         for subset in subsets:\\n             new_subset_with_item.append(subset + [item])\\n         \\n         return new_subset_with_item\\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                2,
                3
            ]
        ],
        "output": [
            [],
            [
                1
            ],
            [
                2
            ],
            [
                1,
                2
            ],
            [
                3
            ],
            [
                1,
                3
            ],
            [
                2,
                3
            ],
            [
                1,
                2,
                3
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "subsets",
        "starter_code": "\nclass Solution:\n    def subsets(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/subsets/"
    },
    {
        "id": 1265,
        "task_id": 2634,
        "test_case_id": 2,
        "question": "Given a set of distinct integers, nums, return all possible subsets (the power set).\n\nNote: The solution set must not contain duplicate subsets.\n\nExample:\n\n\nInput: nums = [1,2,3]\nOutput:\n[\n  [3],\n  [1],\n  [2],\n  [1,2,3],\n  [1,3],\n  [2,3],\n  [1,2],\n  []\n]",
        "solutions": "[\"class Solution:\\n     def subsets(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         subsets = [[]]\\n         for v in nums:\\n             extra = []\\n             for s in subsets:\\n                 extra.append(s+[v])\\n             subsets += extra\\n         return subsets\", \"class Solution:\\n     def subsets(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def mask(x):\\n             return 1 << x\\n         \\n         exp = 1 << len(nums)\\n         ans = []\\n         for count in range(exp):\\n             new = []\\n             i = 0\\n             while mask(i) <= count:\\n                 if mask(i) & count:\\n                     new.append(nums[i])\\n                 i += 1\\n             ans.append(new)\\n         return ans\", \"class Solution:\\n     def subsets(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = [[]]\\n         for num in nums:\\n             result += self.get_subset(result, num)\\n         return result\\n         \\n     def get_subset(self, subsets, item):\\n         \\n         new_subset_with_item = []\\n         for subset in subsets:\\n             new_subset_with_item.append(subset + [item])\\n         \\n         return new_subset_with_item\\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                0
            ]
        ],
        "output": [
            [],
            [
                0
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "subsets",
        "starter_code": "\nclass Solution:\n    def subsets(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/subsets/"
    },
    {
        "id": 1266,
        "task_id": 2635,
        "test_case_id": 1,
        "question": "Given a matrix of m x n elements (m rows, n columns), return all elements of the matrix in spiral order.\n\nExample 1:\n\n\nInput:\n[\n [ 1, 2, 3 ],\n [ 4, 5, 6 ],\n [ 7, 8, 9 ]\n]\nOutput: [1,2,3,6,9,8,7,4,5]\n\n\nExample 2:\n\nInput:\n[\n  [1, 2, 3, 4],\n  [5, 6, 7, 8],\n  [9,10,11,12]\n]\nOutput: [1,2,3,4,8,12,11,10,9,5,6,7]",
        "solutions": "[\"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         if not matrix:\\n             return result\\n         top = left = 0\\n         bottom, right = len(matrix) - 1, len(matrix[0]) - 1\\n         while top < bottom and left < right:\\n             for i in range(left, right):\\n                 result.append(matrix[top][i])\\n             for i in range(top, bottom):\\n                 result.append(matrix[i][right])\\n             for i in range(right, left, -1):\\n                 result.append(matrix[bottom][i])\\n             for i in range(bottom, top, -1):\\n                 result.append(matrix[i][left])\\n             left += 1\\n             right -= 1\\n             top += 1\\n             bottom -= 1\\n         if left == right and top == bottom:\\n             result.append(matrix[left][top])\\n         if left == right and top != bottom:\\n             for i in range(top, bottom+1):\\n                 result.append(matrix[i][left])\\n         if left != right and top == bottom:\\n             for i in range(left, right+1):\\n                 result.append(matrix[top][i])\\n         return result\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return []\\n         def layer(width, height, top, left):\\n             if not (width > 0 and height > 0):\\n                 return\\n             for i in range(left, left+width):\\n                 yield matrix[top][i]\\n             if height != 1:\\n                 for i in range(top+1, top+height):\\n                     yield matrix[i][left+width-1]\\n                 for i in range(left+width-2, left-1, -1):\\n                     yield matrix[top+height-1][i]\\n                 if width != 1:\\n                     for i in range(top+height-2, top, -1):\\n                         yield matrix[i][left]\\n             yield from layer(width-2,height-2, top+1, left+1)\\n         return list(layer(len(matrix[0]), len(matrix), 0, 0))\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         m = matrix\\n         ml = len(m)\\n         if ml == 0:\\n             return []\\n         nl = len(m[0])\\n         ms, me, ns, ne = 0, ml - 1, 0, nl - 1 \\n         ans = []\\n         while ns <= ne or ms <= me: \\n             #print(ms,me,ns,ne)\\n             if ms <= me: \\n                 for j in range(ns, ne + 1): \\n                     ans.append(m[ms][j])\\n                 ms += 1\\n             #print(ms,me,ns,ne)\\n             if ne >= ns: \\n                 for j in range(ms, me + 1): \\n                     ans.append(m[j][ne])\\n                 ne -= 1\\n             #print(ms,me,ns,ne)\\n             if me >= ms: \\n                 for j in range(ne, ns - 1, -1):\\n                     ans.append(m[me][j])\\n                 me -= 1\\n             #print(ms,me,ns,ne)\\n             if ns <= ne: \\n                 for j in range(me, ms - 1, -1):\\n                     ans.append(m[j][ns])\\n                 ns += 1\\n             #print(ms,me,ns,ne)\\n         return ans \", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return matrix\\n         res = []\\n         row = len(matrix)\\n         col = len(matrix[0])\\n         left = 0\\n         right = col - 1\\n         down = row - 1\\n         up = 0\\n         direction = 0\\n         while True:\\n             if direction == 0:\\n                 for i in range(left, right + 1):\\n                     res.append(matrix[up][i])\\n                 up += 1\\n             if direction == 1:\\n                 for i in range(up, down + 1):\\n                     res.append(matrix[i][right])\\n                 right -= 1\\n             if direction == 2:\\n                 for i in reversed(range(left, right+1)):\\n                     res.append(matrix[down][i])\\n                 down -= 1\\n             if direction == 3:\\n                 for i in reversed(range(up, down + 1)):\\n                     res.append(matrix[i][left])\\n                 left += 1\\n             direction = (direction + 1) % 4\\n             if left > right or up > down:\\n                 return res\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = []\\n         if not matrix:\\n             return res\\n         \\n         left, right, top, bottom = 0, len(matrix[0])-1, 0, len(matrix)-1\\n         \\n         while left <= right and top <= bottom:\\n             res += [matrix[top][i] for i in range(left,right+1)]\\n             res += [matrix[i][right] for i in range(top+1,bottom)]\\n             res += [matrix[bottom][i] for i in range(right, left-1,-1) if top < bottom]\\n             res += [matrix[i][left] for i in range(bottom-1,top,-1) if left < right]\\n             left+=1\\n             right-=1\\n             top+=1\\n             bottom-=1\\n         return res\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if len(matrix) == 0:\\n             return []\\n         level = 0\\n         r = []\\n         while len(r) < len(matrix)*len(matrix[0]):\\n             if level%4 == 0:\\n                 r += matrix[level//4][level//4: len(matrix[0]) - level//4]\\n             elif level%4 == 1:\\n                 side_index = len(matrix[0]) - (level)//4 - 1\\n                 for i in range(level//4 + 1,len(matrix) - level//4):\\n                     r.append(matrix[i][side_index])\\n             elif level%4 == 2:\\n                 r += matrix[len(matrix) -1 - level//4][level//4:len(matrix[0]) - level//4 -1][::-1]\\n             elif level%4 == 3:\\n                 side_index = level//4\\n                 for i in range((len(matrix) -1 - level//4 -1),level//4 ,-1):\\n                     r.append(matrix[i][side_index])\\n             level +=1\\n         return r\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ans = []\\n         \\n         if len(matrix) is 0:\\n             return ans\\n         \\n         import numpy as np\\n         npa = np.array(matrix, dtype = int)\\n         \\n         r_s = c_s = 0\\n         r_e = len(matrix)\\n         c_e = len(matrix[0])\\n         \\n         go_row = True\\n         forward = True\\n         row_index = r_s\\n         col_index = c_e - 1\\n         \\n         while(r_s < r_e and c_s < c_e):\\n             if go_row:\\n                 # Traverse row\\n                 s = npa[row_index,c_s:c_e].tolist()\\n                 if forward:\\n                     r_s += 1\\n                     row_index = r_e -1\\n                 else:\\n                     r_e -= 1\\n                     row_index = r_s\\n                     s.reverse()\\n                 ans.extend(s)\\n                 \\n                 \\n             else:\\n                 # traverse col\\n                 s = npa[r_s:r_e, col_index].tolist()\\n                 if forward:\\n                     c_e -= 1\\n                     col_index = c_s\\n                 else:\\n                     c_s += 1\\n                     col_index = c_e - 1\\n                     s.reverse()\\n                 ans.extend(s)\\n                 forward = not forward\\n                                 \\n             go_row = not go_row\\n         \\n         return ans\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         m = matrix\\n         ml = len(m)\\n         if ml == 0:\\n             return []\\n         nl = len(m[0])\\n         ms, me, ns, ne = 0, ml - 1, 0, nl - 1 \\n         ans = []\\n         while ns <= ne or ms <= me: \\n             #print(ms,me,ns,ne)\\n             if ms <= me: \\n                 for j in range(ns, ne + 1): \\n                     ans.append(m[ms][j])\\n                 ms += 1\\n             #print(ms,me,ns,ne)\\n             if ne >= ns: \\n                 for j in range(ms, me + 1): \\n                     ans.append(m[j][ne])\\n                 ne -= 1\\n             #print(ms,me,ns,ne)\\n             if me >= ms: \\n                 for j in range(ne, ns - 1, -1):\\n                     ans.append(m[me][j])\\n                 me -= 1\\n             #print(ms,me,ns,ne)\\n             if ns <= ne: \\n                 for j in range(me, ms - 1, -1):\\n                     ans.append(m[j][ns])\\n                 ns += 1\\n             #print(ms,me,ns,ne)\\n         return ans \", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         # 0 is right, 1 is down, 2 is left, 3 is up\\n         state = 0\\n         next_direction = [1, 2, 3, 0]\\n         r = []\\n         \\n         while len(matrix) > 0:\\n             if state == 0:\\n                 r += matrix[0]\\n                 matrix = matrix[1:]\\n                 state = next_direction[state]\\n             elif state == 1:\\n                 r += [m[len(matrix[0]) - 1] for m in matrix]\\n                 matrix = [m[:len(matrix[0]) - 1] for m in matrix]\\n                 state = 2\\n             elif state == 2:\\n                 r += reversed(matrix[-1])\\n                 matrix = matrix[:-1]\\n                 state = 3\\n             elif state == 3:\\n                 r += [matrix[j][0] for j in range(len(matrix) - 1, -1, -1)]\\n                 matrix = [m[1:] for m in matrix]\\n                 state = 0\\n                \\n             # Covers the case of a list of empty lists (eg. [[],[],[]])\\n             if len(matrix) >= 1 and not matrix[0]:\\n                 matrix = []\\n                 \\n         return r\\n         \\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return matrix\\n         ans=matrix[0]\\n         m=len(matrix)\\n         n=len(matrix[0])\\n         row_m, column_m=n-1, m-1\\n         row_dir, column_dir=-1, 1\\n         i0, j0=0, n-1\\n         while  column_m>0:\\n             ans.extend([matrix[i0+i*column_dir][j0] for i in range(1, column_m+1)])\\n             i0+=column_m*column_dir\\n             column_m-=1\\n             column_dir*=-1\\n                 \\n             if row_m==0:\\n                 return ans\\n             else:\\n                 ans.extend([matrix[i0][j0+row_dir*i] for i in range(1, row_m+1)])\\n                 j0+=row_m*row_dir\\n                 row_m-=1\\n                 row_dir*=-1  \\n         return ans\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         #O(n^2) time; O(1) space\\n         if not matrix:\\n             return []\\n         \\n         ans = []\\n         rowSize = len(matrix); colSize = len(matrix[0])       \\n         rowStart=0; colStart=0; rowEnd = rowSize-1; colEnd = colSize-1        \\n         \\n         while rowStart <= rowEnd and colStart <= colEnd:\\n             \\n             for colIndex in range(colStart,colEnd+1):\\n                 ans.append(matrix[rowStart][colIndex])\\n             rowStart += 1;\\n             \\n             for rowIndex in range(rowStart,rowEnd+1):\\n                 ans.append(matrix[rowIndex][colEnd])\\n             colEnd -= 1;\\n             \\n             if rowStart > rowEnd or colStart > colEnd: break                \\n             \\n             for colIndex in range(colEnd,colStart-1,-1):\\n                 ans.append(matrix[rowEnd][colIndex])\\n             rowEnd -= 1;                           \\n                 \\n             for rowIndex in range(rowEnd,rowStart-1,-1):\\n                 ans.append(matrix[rowIndex][colStart])\\n             colStart += 1;                                     \\n             \\n         return ans\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix or not matrix[0]:\\n             return []\\n         \\n         v = [[False] * len(matrix[0]) for _ in range(len(matrix))]\\n         res = []\\n         i = 0; j = 0\\n         m = len(matrix)\\n         n = len(matrix[0])\\n         count = 0\\n         dest = m * n\\n         way = 0  # 0> 1v 2< 3^\\n         offset = [(0, 1), (1, 0), (0, -1), (-1, 0)]\\n         while count < dest:\\n             res.append(matrix[i][j])\\n             count += 1\\n             v[i][j] = True\\n             if not (0 <= i + offset[way][0] < m) or not (0 <= j + offset[way][1] < n) or \\\\\\n                 v[i + offset[way][0]][j + offset[way][1]]:\\n                     way = (way + 1) % 4\\n             i += offset[way][0]\\n             j += offset[way][1]\\n         return res\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         m = len(matrix)\\n         if m == 0:\\n             return []\\n         n = len(matrix[0])\\n     \\n \\n         visit = set()\\n         result = []\\n \\n         pos = [0, 0]\\n         visit.add(tuple(pos))\\n         result.append(matrix[pos[0]][pos[1]])\\n \\n         while len(visit) < m*n:\\n \\n             for i in range(pos[1]+1, n):\\n                 \\n                 if (pos[0], i) in visit:\\n                     break\\n                 pos[1] = i\\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n \\n \\n             for i in range(pos[0]+1, m):\\n                 \\n                 if (i, pos[1]) in visit:\\n                     break\\n                 pos[0] = i\\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n \\n             for i in range(pos[1]-1, -1, -1):\\n                 \\n                 if (pos[0], i) in visit:\\n                     break\\n \\n                 pos[1] = i\\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n \\n             for i in range(pos[0]-1, -1, -1):\\n \\n                 if (i, pos[1]) in visit:\\n                     break\\n \\n                 pos[0] = i\\n                 \\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n                 \\n         return result\\n                 \\n         return result\\n             \\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         output_arr = []\\n     \\n         while matrix:\\n     \\n             output_arr.extend(matrix.pop(0))\\n         \\n             tmp_arr = []\\n             for i in zip(*matrix):\\n                 tmp_arr.append(list(i))\\n         \\n             matrix = tmp_arr[::-1]\\n \\n         return output_arr\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return []\\n         left = top = 0\\n         right = len(matrix[0]) - 1\\n         bottom = len(matrix) - 1\\n \\n         result = []\\n         while left < right and top < bottom:\\n             for i in range(left, right):\\n                 result.append(matrix[top][i])\\n             for i in range(top, bottom):\\n                 result.append(matrix[i][right])\\n             for i in range(right, left, -1):\\n                 result.append(matrix[bottom][i])\\n             for i in range(bottom, top, -1):\\n                 result.append(matrix[i][left])\\n             left += 1\\n             right -= 1\\n             top += 1\\n             bottom -= 1\\n         if left == right and top == bottom:\\n             result.append(matrix[top][left])\\n         elif left == right:\\n             for i in range(top, bottom + 1):\\n                 result.append(matrix[i][left])\\n         elif top == bottom:\\n             for i in range(left, right + 1):\\n                 result.append(matrix[top][i])\\n         return result\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if len(matrix) == 0:\\n             return []\\n         if len(matrix) == 1:\\n             return matrix[0]\\n         if len(matrix[0]) == 1:\\n             result = []\\n             for m in matrix:\\n                 result.append(m[0])\\n             return result\\n         \\n         \\n         boxTop = -1\\n         boxRight = len(matrix[0])\\n         boxLeft = -1\\n         boxBottom = len(matrix)\\n         \\n         \\n         count = 0\\n         direction = [0, 1]\\n         ptr = [0, 0]\\n         predict = [0, 0]\\n         result = []\\n         \\n         while count < len(matrix) * len(matrix[0]):\\n             count += 1\\n             result.append(matrix[ptr[0]][ptr[1]])\\n             ptr[0] += direction[0]\\n             ptr[1] += direction[1]\\n             \\n             predict[0] = ptr[0] + direction[0]\\n             predict[1] = ptr[1] + direction[1]\\n             \\n             \\n             if direction == [0, 1] and predict[1] >= boxRight:\\n                 direction = [1, 0]\\n                 boxTop += 1\\n             elif direction == [1,0] and predict[0] >= boxBottom:\\n                 direction = [0, -1]\\n                 boxRight -= 1\\n             elif direction == [0, -1] and predict[1] <= boxLeft:\\n                 direction = [-1, 0]\\n                 boxBottom -= 1\\n             elif direction == [-1, 0] and predict[0] <= boxTop:\\n                 direction = [0, 1]\\n                 boxLeft += 1\\n             \\n         return result\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    2,
                    3
                ],
                [
                    4,
                    5,
                    6
                ],
                [
                    7,
                    8,
                    9
                ]
            ]
        ],
        "output": [
            1,
            2,
            3,
            6,
            9,
            8,
            7,
            4,
            5
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "spiralOrder",
        "starter_code": "\nclass Solution:\n    def spiralOrder(self, matrix: List[List[int]]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/spiral-matrix/"
    },
    {
        "id": 1267,
        "task_id": 2635,
        "test_case_id": 2,
        "question": "Given a matrix of m x n elements (m rows, n columns), return all elements of the matrix in spiral order.\n\nExample 1:\n\n\nInput:\n[\n [ 1, 2, 3 ],\n [ 4, 5, 6 ],\n [ 7, 8, 9 ]\n]\nOutput: [1,2,3,6,9,8,7,4,5]\n\n\nExample 2:\n\nInput:\n[\n  [1, 2, 3, 4],\n  [5, 6, 7, 8],\n  [9,10,11,12]\n]\nOutput: [1,2,3,4,8,12,11,10,9,5,6,7]",
        "solutions": "[\"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         if not matrix:\\n             return result\\n         top = left = 0\\n         bottom, right = len(matrix) - 1, len(matrix[0]) - 1\\n         while top < bottom and left < right:\\n             for i in range(left, right):\\n                 result.append(matrix[top][i])\\n             for i in range(top, bottom):\\n                 result.append(matrix[i][right])\\n             for i in range(right, left, -1):\\n                 result.append(matrix[bottom][i])\\n             for i in range(bottom, top, -1):\\n                 result.append(matrix[i][left])\\n             left += 1\\n             right -= 1\\n             top += 1\\n             bottom -= 1\\n         if left == right and top == bottom:\\n             result.append(matrix[left][top])\\n         if left == right and top != bottom:\\n             for i in range(top, bottom+1):\\n                 result.append(matrix[i][left])\\n         if left != right and top == bottom:\\n             for i in range(left, right+1):\\n                 result.append(matrix[top][i])\\n         return result\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return []\\n         def layer(width, height, top, left):\\n             if not (width > 0 and height > 0):\\n                 return\\n             for i in range(left, left+width):\\n                 yield matrix[top][i]\\n             if height != 1:\\n                 for i in range(top+1, top+height):\\n                     yield matrix[i][left+width-1]\\n                 for i in range(left+width-2, left-1, -1):\\n                     yield matrix[top+height-1][i]\\n                 if width != 1:\\n                     for i in range(top+height-2, top, -1):\\n                         yield matrix[i][left]\\n             yield from layer(width-2,height-2, top+1, left+1)\\n         return list(layer(len(matrix[0]), len(matrix), 0, 0))\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         m = matrix\\n         ml = len(m)\\n         if ml == 0:\\n             return []\\n         nl = len(m[0])\\n         ms, me, ns, ne = 0, ml - 1, 0, nl - 1 \\n         ans = []\\n         while ns <= ne or ms <= me: \\n             #print(ms,me,ns,ne)\\n             if ms <= me: \\n                 for j in range(ns, ne + 1): \\n                     ans.append(m[ms][j])\\n                 ms += 1\\n             #print(ms,me,ns,ne)\\n             if ne >= ns: \\n                 for j in range(ms, me + 1): \\n                     ans.append(m[j][ne])\\n                 ne -= 1\\n             #print(ms,me,ns,ne)\\n             if me >= ms: \\n                 for j in range(ne, ns - 1, -1):\\n                     ans.append(m[me][j])\\n                 me -= 1\\n             #print(ms,me,ns,ne)\\n             if ns <= ne: \\n                 for j in range(me, ms - 1, -1):\\n                     ans.append(m[j][ns])\\n                 ns += 1\\n             #print(ms,me,ns,ne)\\n         return ans \", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return matrix\\n         res = []\\n         row = len(matrix)\\n         col = len(matrix[0])\\n         left = 0\\n         right = col - 1\\n         down = row - 1\\n         up = 0\\n         direction = 0\\n         while True:\\n             if direction == 0:\\n                 for i in range(left, right + 1):\\n                     res.append(matrix[up][i])\\n                 up += 1\\n             if direction == 1:\\n                 for i in range(up, down + 1):\\n                     res.append(matrix[i][right])\\n                 right -= 1\\n             if direction == 2:\\n                 for i in reversed(range(left, right+1)):\\n                     res.append(matrix[down][i])\\n                 down -= 1\\n             if direction == 3:\\n                 for i in reversed(range(up, down + 1)):\\n                     res.append(matrix[i][left])\\n                 left += 1\\n             direction = (direction + 1) % 4\\n             if left > right or up > down:\\n                 return res\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = []\\n         if not matrix:\\n             return res\\n         \\n         left, right, top, bottom = 0, len(matrix[0])-1, 0, len(matrix)-1\\n         \\n         while left <= right and top <= bottom:\\n             res += [matrix[top][i] for i in range(left,right+1)]\\n             res += [matrix[i][right] for i in range(top+1,bottom)]\\n             res += [matrix[bottom][i] for i in range(right, left-1,-1) if top < bottom]\\n             res += [matrix[i][left] for i in range(bottom-1,top,-1) if left < right]\\n             left+=1\\n             right-=1\\n             top+=1\\n             bottom-=1\\n         return res\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if len(matrix) == 0:\\n             return []\\n         level = 0\\n         r = []\\n         while len(r) < len(matrix)*len(matrix[0]):\\n             if level%4 == 0:\\n                 r += matrix[level//4][level//4: len(matrix[0]) - level//4]\\n             elif level%4 == 1:\\n                 side_index = len(matrix[0]) - (level)//4 - 1\\n                 for i in range(level//4 + 1,len(matrix) - level//4):\\n                     r.append(matrix[i][side_index])\\n             elif level%4 == 2:\\n                 r += matrix[len(matrix) -1 - level//4][level//4:len(matrix[0]) - level//4 -1][::-1]\\n             elif level%4 == 3:\\n                 side_index = level//4\\n                 for i in range((len(matrix) -1 - level//4 -1),level//4 ,-1):\\n                     r.append(matrix[i][side_index])\\n             level +=1\\n         return r\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ans = []\\n         \\n         if len(matrix) is 0:\\n             return ans\\n         \\n         import numpy as np\\n         npa = np.array(matrix, dtype = int)\\n         \\n         r_s = c_s = 0\\n         r_e = len(matrix)\\n         c_e = len(matrix[0])\\n         \\n         go_row = True\\n         forward = True\\n         row_index = r_s\\n         col_index = c_e - 1\\n         \\n         while(r_s < r_e and c_s < c_e):\\n             if go_row:\\n                 # Traverse row\\n                 s = npa[row_index,c_s:c_e].tolist()\\n                 if forward:\\n                     r_s += 1\\n                     row_index = r_e -1\\n                 else:\\n                     r_e -= 1\\n                     row_index = r_s\\n                     s.reverse()\\n                 ans.extend(s)\\n                 \\n                 \\n             else:\\n                 # traverse col\\n                 s = npa[r_s:r_e, col_index].tolist()\\n                 if forward:\\n                     c_e -= 1\\n                     col_index = c_s\\n                 else:\\n                     c_s += 1\\n                     col_index = c_e - 1\\n                     s.reverse()\\n                 ans.extend(s)\\n                 forward = not forward\\n                                 \\n             go_row = not go_row\\n         \\n         return ans\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         m = matrix\\n         ml = len(m)\\n         if ml == 0:\\n             return []\\n         nl = len(m[0])\\n         ms, me, ns, ne = 0, ml - 1, 0, nl - 1 \\n         ans = []\\n         while ns <= ne or ms <= me: \\n             #print(ms,me,ns,ne)\\n             if ms <= me: \\n                 for j in range(ns, ne + 1): \\n                     ans.append(m[ms][j])\\n                 ms += 1\\n             #print(ms,me,ns,ne)\\n             if ne >= ns: \\n                 for j in range(ms, me + 1): \\n                     ans.append(m[j][ne])\\n                 ne -= 1\\n             #print(ms,me,ns,ne)\\n             if me >= ms: \\n                 for j in range(ne, ns - 1, -1):\\n                     ans.append(m[me][j])\\n                 me -= 1\\n             #print(ms,me,ns,ne)\\n             if ns <= ne: \\n                 for j in range(me, ms - 1, -1):\\n                     ans.append(m[j][ns])\\n                 ns += 1\\n             #print(ms,me,ns,ne)\\n         return ans \", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         # 0 is right, 1 is down, 2 is left, 3 is up\\n         state = 0\\n         next_direction = [1, 2, 3, 0]\\n         r = []\\n         \\n         while len(matrix) > 0:\\n             if state == 0:\\n                 r += matrix[0]\\n                 matrix = matrix[1:]\\n                 state = next_direction[state]\\n             elif state == 1:\\n                 r += [m[len(matrix[0]) - 1] for m in matrix]\\n                 matrix = [m[:len(matrix[0]) - 1] for m in matrix]\\n                 state = 2\\n             elif state == 2:\\n                 r += reversed(matrix[-1])\\n                 matrix = matrix[:-1]\\n                 state = 3\\n             elif state == 3:\\n                 r += [matrix[j][0] for j in range(len(matrix) - 1, -1, -1)]\\n                 matrix = [m[1:] for m in matrix]\\n                 state = 0\\n                \\n             # Covers the case of a list of empty lists (eg. [[],[],[]])\\n             if len(matrix) >= 1 and not matrix[0]:\\n                 matrix = []\\n                 \\n         return r\\n         \\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return matrix\\n         ans=matrix[0]\\n         m=len(matrix)\\n         n=len(matrix[0])\\n         row_m, column_m=n-1, m-1\\n         row_dir, column_dir=-1, 1\\n         i0, j0=0, n-1\\n         while  column_m>0:\\n             ans.extend([matrix[i0+i*column_dir][j0] for i in range(1, column_m+1)])\\n             i0+=column_m*column_dir\\n             column_m-=1\\n             column_dir*=-1\\n                 \\n             if row_m==0:\\n                 return ans\\n             else:\\n                 ans.extend([matrix[i0][j0+row_dir*i] for i in range(1, row_m+1)])\\n                 j0+=row_m*row_dir\\n                 row_m-=1\\n                 row_dir*=-1  \\n         return ans\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         #O(n^2) time; O(1) space\\n         if not matrix:\\n             return []\\n         \\n         ans = []\\n         rowSize = len(matrix); colSize = len(matrix[0])       \\n         rowStart=0; colStart=0; rowEnd = rowSize-1; colEnd = colSize-1        \\n         \\n         while rowStart <= rowEnd and colStart <= colEnd:\\n             \\n             for colIndex in range(colStart,colEnd+1):\\n                 ans.append(matrix[rowStart][colIndex])\\n             rowStart += 1;\\n             \\n             for rowIndex in range(rowStart,rowEnd+1):\\n                 ans.append(matrix[rowIndex][colEnd])\\n             colEnd -= 1;\\n             \\n             if rowStart > rowEnd or colStart > colEnd: break                \\n             \\n             for colIndex in range(colEnd,colStart-1,-1):\\n                 ans.append(matrix[rowEnd][colIndex])\\n             rowEnd -= 1;                           \\n                 \\n             for rowIndex in range(rowEnd,rowStart-1,-1):\\n                 ans.append(matrix[rowIndex][colStart])\\n             colStart += 1;                                     \\n             \\n         return ans\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix or not matrix[0]:\\n             return []\\n         \\n         v = [[False] * len(matrix[0]) for _ in range(len(matrix))]\\n         res = []\\n         i = 0; j = 0\\n         m = len(matrix)\\n         n = len(matrix[0])\\n         count = 0\\n         dest = m * n\\n         way = 0  # 0> 1v 2< 3^\\n         offset = [(0, 1), (1, 0), (0, -1), (-1, 0)]\\n         while count < dest:\\n             res.append(matrix[i][j])\\n             count += 1\\n             v[i][j] = True\\n             if not (0 <= i + offset[way][0] < m) or not (0 <= j + offset[way][1] < n) or \\\\\\n                 v[i + offset[way][0]][j + offset[way][1]]:\\n                     way = (way + 1) % 4\\n             i += offset[way][0]\\n             j += offset[way][1]\\n         return res\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         m = len(matrix)\\n         if m == 0:\\n             return []\\n         n = len(matrix[0])\\n     \\n \\n         visit = set()\\n         result = []\\n \\n         pos = [0, 0]\\n         visit.add(tuple(pos))\\n         result.append(matrix[pos[0]][pos[1]])\\n \\n         while len(visit) < m*n:\\n \\n             for i in range(pos[1]+1, n):\\n                 \\n                 if (pos[0], i) in visit:\\n                     break\\n                 pos[1] = i\\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n \\n \\n             for i in range(pos[0]+1, m):\\n                 \\n                 if (i, pos[1]) in visit:\\n                     break\\n                 pos[0] = i\\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n \\n             for i in range(pos[1]-1, -1, -1):\\n                 \\n                 if (pos[0], i) in visit:\\n                     break\\n \\n                 pos[1] = i\\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n \\n             for i in range(pos[0]-1, -1, -1):\\n \\n                 if (i, pos[1]) in visit:\\n                     break\\n \\n                 pos[0] = i\\n                 \\n                 visit.add(tuple(pos))\\n                 result.append(matrix[pos[0]][pos[1]])\\n                 \\n         return result\\n                 \\n         return result\\n             \\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         output_arr = []\\n     \\n         while matrix:\\n     \\n             output_arr.extend(matrix.pop(0))\\n         \\n             tmp_arr = []\\n             for i in zip(*matrix):\\n                 tmp_arr.append(list(i))\\n         \\n             matrix = tmp_arr[::-1]\\n \\n         return output_arr\\n\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return []\\n         left = top = 0\\n         right = len(matrix[0]) - 1\\n         bottom = len(matrix) - 1\\n \\n         result = []\\n         while left < right and top < bottom:\\n             for i in range(left, right):\\n                 result.append(matrix[top][i])\\n             for i in range(top, bottom):\\n                 result.append(matrix[i][right])\\n             for i in range(right, left, -1):\\n                 result.append(matrix[bottom][i])\\n             for i in range(bottom, top, -1):\\n                 result.append(matrix[i][left])\\n             left += 1\\n             right -= 1\\n             top += 1\\n             bottom -= 1\\n         if left == right and top == bottom:\\n             result.append(matrix[top][left])\\n         elif left == right:\\n             for i in range(top, bottom + 1):\\n                 result.append(matrix[i][left])\\n         elif top == bottom:\\n             for i in range(left, right + 1):\\n                 result.append(matrix[top][i])\\n         return result\", \"class Solution:\\n     def spiralOrder(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[int]]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if len(matrix) == 0:\\n             return []\\n         if len(matrix) == 1:\\n             return matrix[0]\\n         if len(matrix[0]) == 1:\\n             result = []\\n             for m in matrix:\\n                 result.append(m[0])\\n             return result\\n         \\n         \\n         boxTop = -1\\n         boxRight = len(matrix[0])\\n         boxLeft = -1\\n         boxBottom = len(matrix)\\n         \\n         \\n         count = 0\\n         direction = [0, 1]\\n         ptr = [0, 0]\\n         predict = [0, 0]\\n         result = []\\n         \\n         while count < len(matrix) * len(matrix[0]):\\n             count += 1\\n             result.append(matrix[ptr[0]][ptr[1]])\\n             ptr[0] += direction[0]\\n             ptr[1] += direction[1]\\n             \\n             predict[0] = ptr[0] + direction[0]\\n             predict[1] = ptr[1] + direction[1]\\n             \\n             \\n             if direction == [0, 1] and predict[1] >= boxRight:\\n                 direction = [1, 0]\\n                 boxTop += 1\\n             elif direction == [1,0] and predict[0] >= boxBottom:\\n                 direction = [0, -1]\\n                 boxRight -= 1\\n             elif direction == [0, -1] and predict[1] <= boxLeft:\\n                 direction = [-1, 0]\\n                 boxBottom -= 1\\n             elif direction == [-1, 0] and predict[0] <= boxTop:\\n                 direction = [0, 1]\\n                 boxLeft += 1\\n             \\n         return result\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    2,
                    3,
                    4
                ],
                [
                    5,
                    6,
                    7,
                    8
                ],
                [
                    9,
                    10,
                    11,
                    12
                ]
            ]
        ],
        "output": [
            1,
            2,
            3,
            4,
            8,
            12,
            11,
            10,
            9,
            5,
            6,
            7
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "spiralOrder",
        "starter_code": "\nclass Solution:\n    def spiralOrder(self, matrix: List[List[int]]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/spiral-matrix/"
    },
    {
        "id": 1268,
        "task_id": 2637,
        "test_case_id": 1,
        "question": "Given a collection of numbers that might contain duplicates, return all possible unique permutations.\n\nExample:\n\n\nInput: [1,1,2]\nOutput:\n[\n  [1,1,2],\n  [1,2,1],\n  [2,1,1]\n]",
        "solutions": "[\"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return []\\n         \\n         nums.sort()\\n         n = len(nums)\\n         res = [nums[:]]\\n         i = n-1\\n         while i > 0:\\n             if nums[i-1] < nums[i]:\\n                 j = n-1\\n                 while nums[j] <= nums[i-1]:\\n                     j -= 1\\n                 nums[i-1], nums[j] = nums[j], nums[i-1]\\n                 nums[i:] = sorted(nums[i:])\\n                 res.append(nums[:])\\n                 i = n-1\\n             else:\\n                 i -= 1\\n         \\n         return res\\n         \\n         \\n         \\n\", \"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(nums):\\n             if not nums:\\n                 return [[]]\\n             dic = set()\\n             new = []\\n             for i in range(len(nums)):\\n                 if nums[i] not in dic:\\n                     dic.add(nums[i])\\n                     new += [[nums[i]] + item for item in dfs(nums[: i] + nums[i + 1 :])]\\n             return new\\n         \\n         return dfs(nums)\", \"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res=[]\\n         nums.sort()\\n         self.dfs(nums,[],res,[False] * len(nums))\\n         return res\\n     \\n     def dfs(self,nums,path,res,used):\\n         if len(path)==len(nums):\\n             res.append(path)\\n         else:\\n             for i in range(len(nums)):\\n                 if used[i] or (i>0 and nums[i] == nums[i-1] and not used[i-1]):\\n                     continue\\n                 used[i] = True\\n                 self.dfs(nums,path+[nums[i]],res,used)\\n                 used[i]=False\", \"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         nums.sort()\\n         def swap(a, i, j):\\n             temp = a[i]\\n             a[i] = a[j]\\n             a[j] = temp\\n         \\n         def helper(index, path):\\n             if index == len(nums) - 1:\\n                 res.append(path.copy())\\n             for i in range(index, len(nums)):\\n                 if i != index and path[i] == path[index]:\\n                     continue\\n                 swap(path, index, i)\\n                 helper(index + 1, path.copy())\\n             \\n         helper(0, nums)\\n         return res\\n\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                1,
                2
            ]
        ],
        "output": [
            [
                1,
                1,
                2
            ],
            [
                1,
                2,
                1
            ],
            [
                2,
                1,
                1
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "permuteUnique",
        "starter_code": "\nclass Solution:\n    def permuteUnique(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/permutations-ii/"
    },
    {
        "id": 1269,
        "task_id": 2637,
        "test_case_id": 2,
        "question": "Given a collection of numbers that might contain duplicates, return all possible unique permutations.\n\nExample:\n\n\nInput: [1,1,2]\nOutput:\n[\n  [1,1,2],\n  [1,2,1],\n  [2,1,1]\n]",
        "solutions": "[\"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return []\\n         \\n         nums.sort()\\n         n = len(nums)\\n         res = [nums[:]]\\n         i = n-1\\n         while i > 0:\\n             if nums[i-1] < nums[i]:\\n                 j = n-1\\n                 while nums[j] <= nums[i-1]:\\n                     j -= 1\\n                 nums[i-1], nums[j] = nums[j], nums[i-1]\\n                 nums[i:] = sorted(nums[i:])\\n                 res.append(nums[:])\\n                 i = n-1\\n             else:\\n                 i -= 1\\n         \\n         return res\\n         \\n         \\n         \\n\", \"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(nums):\\n             if not nums:\\n                 return [[]]\\n             dic = set()\\n             new = []\\n             for i in range(len(nums)):\\n                 if nums[i] not in dic:\\n                     dic.add(nums[i])\\n                     new += [[nums[i]] + item for item in dfs(nums[: i] + nums[i + 1 :])]\\n             return new\\n         \\n         return dfs(nums)\", \"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res=[]\\n         nums.sort()\\n         self.dfs(nums,[],res,[False] * len(nums))\\n         return res\\n     \\n     def dfs(self,nums,path,res,used):\\n         if len(path)==len(nums):\\n             res.append(path)\\n         else:\\n             for i in range(len(nums)):\\n                 if used[i] or (i>0 and nums[i] == nums[i-1] and not used[i-1]):\\n                     continue\\n                 used[i] = True\\n                 self.dfs(nums,path+[nums[i]],res,used)\\n                 used[i]=False\", \"class Solution:\\n     def permuteUnique(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         nums.sort()\\n         def swap(a, i, j):\\n             temp = a[i]\\n             a[i] = a[j]\\n             a[j] = temp\\n         \\n         def helper(index, path):\\n             if index == len(nums) - 1:\\n                 res.append(path.copy())\\n             for i in range(index, len(nums)):\\n                 if i != index and path[i] == path[index]:\\n                     continue\\n                 swap(path, index, i)\\n                 helper(index + 1, path.copy())\\n             \\n         helper(0, nums)\\n         return res\\n\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                2,
                3
            ]
        ],
        "output": [
            [
                1,
                2,
                3
            ],
            [
                1,
                3,
                2
            ],
            [
                2,
                1,
                3
            ],
            [
                2,
                3,
                1
            ],
            [
                3,
                1,
                2
            ],
            [
                3,
                2,
                1
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "permuteUnique",
        "starter_code": "\nclass Solution:\n    def permuteUnique(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/permutations-ii/"
    },
    {
        "id": 1270,
        "task_id": 2748,
        "test_case_id": 1,
        "question": "Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent.\n\nA mapping of digit to letters (just like on the telephone buttons) is given below. Note that 1 does not map to any letters.\n\n\n\nExample:\n\n\nInput: \"23\"\nOutput: [\"ad\", \"ae\", \"af\", \"bd\", \"be\", \"bf\", \"cd\", \"ce\", \"cf\"].\n\n\nNote:\n\nAlthough the above answer is in lexicographical order, your answer could be in any order you want.",
        "solutions": "[\"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         def dfs(digits, current, result):\\n             if not digits:\\n                 result.append(current)\\n                 return\\n             for c in dic[digits[0]]:\\n                 dfs(digits[1:], current + c, result)\\n         \\n         \\n         if not digits:\\n             return []\\n         \\n         dic = { '2' : \\\"abc\\\", '3': \\\"def\\\", '4':\\\"ghi\\\", \\n                '5':\\\"jkl\\\", '6':\\\"mno\\\", '7':\\\"pqrs\\\", '8':\\\"tuv\\\",'9': \\\"wxyz\\\" }\\n         result = []\\n         dfs(digits, \\\"\\\", result)\\n         return result\\n         \\n         \\n         \\n \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return []\\n         MAPPING = ('0', '1', 'abc', 'def', 'ghi', 'jkl', 'mno', 'pqrs', 'tuv', 'wxyz')\\n         def directed_combinations(i, partial):\\n             if i == len(digits):\\n                 result.append(''.join(partial))\\n                 return\\n             \\n             for c in MAPPING[int(digits[i])]:\\n                 directed_combinations(i + 1, partial + [c])\\n         \\n         result = []\\n         directed_combinations(0, [])\\n         return result\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         dic = {'2':['a','b','c'], '3':['d','e','f'], '4':['g','h','i'], '5':['j','k','l'], \\n                '6':['m','n','o'], '7':['p','q','r','s'], '8':['t','u','v'], '9':['w','x','y','z']}\\n         \\n         if '0' in digits:\\n             digits = digits.replace('0','')\\n         if '1' in digits:\\n             digits = digits.replace('1','')\\n         \\n         count = 0\\n         for a in '23456789':\\n             if a not in digits:\\n                 count += 1\\n         if count == 8:\\n             return []\\n         \\n     \\n         res = []\\n         \\n         temp = ['']\\n         i = 0\\n         while i < len(digits):\\n             for a in temp:\\n                 for b in dic[digits[i]]:\\n                     res.append(a+b)\\n             temp = res\\n             res = []\\n             i += 1\\n         return temp\\n         \\n         \\n         \\n         \\n         \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if '' == digits: \\n             return []\\n         \\n         kvmaps = {\\n             '2': 'abc',\\n             '3': 'def',\\n             '4': 'ghi',\\n             '5': 'jkl',\\n             '6': 'mno',\\n             '7': 'pqrs',\\n             '8': 'tuv',\\n             '9': 'wxyz'\\n         }\\n         rst = ['']\\n         for digit in digits:\\n             temp = [s + c for s in rst for c in kvmaps[digit]]\\n             rst = temp\\n         \\n         return rst\\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if digits==None or len(digits) ==0:\\n             return []\\n         res=[]\\n         cur=\\\"\\\"\\n         dmap={}\\n         dmap[\\\"0\\\"]=[\\\" \\\"]\\n         dmap[\\\"1\\\"]=[]\\n         dmap[\\\"2\\\"]=[\\\"a\\\",\\\"b\\\",\\\"c\\\"]\\n         dmap[\\\"3\\\"]=[\\\"d\\\",\\\"e\\\",\\\"f\\\"]\\n         dmap[\\\"4\\\"]=[\\\"g\\\",\\\"h\\\",\\\"i\\\"]\\n         dmap[\\\"5\\\"]=[\\\"j\\\",\\\"k\\\",\\\"l\\\"]\\n         dmap[\\\"6\\\"]=[\\\"m\\\",\\\"n\\\",\\\"o\\\"]\\n         dmap[\\\"7\\\"]=[\\\"p\\\",\\\"q\\\",\\\"r\\\",\\\"s\\\"]\\n         dmap[\\\"8\\\"]=[\\\"t\\\",\\\"u\\\",\\\"v\\\"]\\n         dmap[\\\"9\\\"]=[\\\"w\\\",\\\"x\\\",\\\"y\\\",\\\"z\\\"]\\n         self.dfs(digits,dmap,0,cur,res)\\n         return res\\n     \\n     def dfs(self,digits,dmap,idx,cur,res):\\n         if idx ==len(digits):\\n             res.append(cur)\\n             return\\n         else:\\n             c=digits[idx]\\n             for i in dmap[c]:\\n                 self.dfs(digits,dmap,idx+1,cur+i,res)\\n\", \"from functools import reduce\\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         d = {}\\n         d[\\\"1\\\"] = \\\"\\\"\\n         d[\\\"2\\\"] = \\\"abc\\\"\\n         d[\\\"3\\\"] = \\\"def\\\"\\n         d[\\\"4\\\"] = \\\"ghi\\\"\\n         d[\\\"5\\\"] = \\\"jkl\\\"\\n         d[\\\"6\\\"] = \\\"mno\\\"\\n         d[\\\"7\\\"] = \\\"pqrs\\\"\\n         d[\\\"8\\\"] = \\\"tuv\\\"\\n         d[\\\"9\\\"] = \\\"wxyz\\\"\\n         d[\\\"0\\\"] = \\\" \\\"\\n         \\n         digs = list(map(lambda x: list(d[x]), digits))\\n         \\n         return reduce(alg_mul, digs, [])\\n                 \\n \\n def alg_mul(xs, ys):\\n     if xs == []:\\n         return ys\\n     \\n     ws = []\\n     for x in xs:\\n         for y in ys:\\n             ws.append(x + y)\\n     \\n     return ws\", \"import queue\\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         letterDict = {0:[],1:[],2:['a','b','c'],3:['d','e','f'],4:['g','h','i'],5:['j','k','l'],\\n                       6:['m','n','o'],7:['p','q','r','s'],8:['t','u','v'],9:['w','x','y','z']}\\n         if len(digits) == 0:\\n             return []\\n         res = [\\\"\\\"]\\n         for i in range(len(digits)):\\n             num = int(digits[i])\\n             chars = letterDict[num]\\n             if len(chars) == 0:\\n                 continue\\n             s = []\\n             for j in range(len(chars)):\\n                 for k in range(len(res)):\\n                     s.append(res[k] + chars[j])\\n             res = s\\n         return res\\n                     \", \"import heapq\\n \\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         digit_map = {'1': [\\\"\\\"],\\n                     '2': ['a', 'b', 'c'],\\n                     '3': ['d', 'e', 'f'],\\n                     '4': ['g', 'h', 'i'],\\n                     '5': ['j', 'k', 'l'],\\n                     '6': ['m', 'n', 'o'],\\n                     '7': ['p', 'q', 'r', 's'],\\n                     '8': ['t', 'u', 'v'],\\n                     '9': ['w', 'x', 'y', 'z'],\\n                     '0': [\\\"\\\"]}\\n         combs = []\\n         # Queue will hold a list of tuples mapping to how many characters are already mapped, \\n         # and the current converted string!\\n         letter_q = [(0, \\\"\\\")]\\n         if not digits:\\n             return []\\n         while letter_q:\\n             cur = heapq.heappop(letter_q)\\n             cur_idx = cur[0]\\n             cur_str = cur[1]\\n             next_dig = digits[cur_idx]\\n             for value in digit_map[next_dig]:\\n                 new_digit_str = (cur_idx+1, cur_str+value)\\n                 if cur_idx + 1 >= len(digits):\\n                     combs.append(new_digit_str)\\n                 else:\\n                     heapq.heappush(letter_q, new_digit_str)    \\n         return list(map(lambda x: x[1], combs))\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         key = {2:'abc', 3:'def', 4:'ghi',5:'jkl',6:'mno',7:'pqrs',8:'tuv',9:'wxyz'}\\n     \\n         nonlocal out\\n         out = ['']\\n \\n         def eachLetter(i):\\n             if i < len(digits):\\n                 nonlocal out\\n                 if int(digits[i]) > 1:\\n                     x = []\\n                     for j in out:\\n                         for k in key[int(digits[i])]:\\n                             x.append(j+k)\\n                     out = x\\n \\n                 eachLetter(i+1)\\n \\n         eachLetter(0)\\n         \\n         return [] if out[0] is '' else out\\n\", \"from collections import defaultdict\\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         digit_to_chars = defaultdict(list)\\n         char_int = ord('a')\\n         for i in range(2,10):\\n             num_chars = 3 if i not in {7,9} else 4\\n             for j in range(num_chars):\\n                 digit_to_chars[str(i)].append(chr(char_int))\\n                 char_int += 1\\n         if not digits:\\n             return []\\n         letter_combinations = ['']\\n         for digit in digits:\\n             new_letter_combinations = []\\n             for letter_combination in letter_combinations:\\n                 for character in digit_to_chars[digit]:\\n                     new_letter_combination = letter_combination + character\\n                     new_letter_combinations.append(new_letter_combination)\\n             letter_combinations = new_letter_combinations\\n         return letter_combinations\", \"class Solution:\\n     Dmap = {'2': 'abc', '3': 'def', '4': 'ghi', '5': 'jkl', '6': 'mno',\\n              '7': 'pqrs', '8': 'tuv', '9': 'wxyz'}\\n     \\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if '1' in digits or '0' in digits:\\n             return []\\n         \\n         if len(digits) == 0:\\n             return []\\n         \\n         return self.letterC(digits)\\n \\n         \\n     def letterC(self, digits):\\n         if len(digits) == 0:\\n             return ['']\\n         \\n         res = []\\n         for back in self.letterC(digits[1:]):\\n             for fr in self.Dmap[digits[0]]:\\n                 res.append(fr + back)\\n         \\n         return res\\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.dic={'2':'abc','3':'def','4':'ghi','5':'jkl','6':'mno','7':'pqrs','8':'tuv','9':'wxyz'}\\n         if not digits:\\n             return []\\n         res = []\\n         self.dfs(digits,0,'',res)\\n         return res\\n     \\n     def dfs(self, digits, index, path, res):\\n         if len(path) == len(digits):\\n             res.append(path)\\n             return\\n         for i in range(len(self.dic[digits[index]])):\\n             path_ = path+self.dic[digits[index]][i]\\n             self.dfs(digits,index+1,path_,res)\\n         \\n         \\n\", \"class Solution:\\n     def __init__(self):\\n         self.phone = [\\\"\\\", \\\"\\\", \\\"abc\\\", \\\"def\\\",\\\"ghi\\\", \\\"jkl\\\", \\\"mno\\\", \\\"pqrs\\\",\\\"tuv\\\", \\\"wxyz\\\"]\\n         \\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         if not digits:\\n             return []\\n         \\n         results = []\\n         self.dfs(digits, results, \\\"\\\", 0)\\n         return results\\n     \\n \\n     def dfs(self, digits, results, string, index):\\n         if index == len(digits):\\n             results.append(string)\\n         else: \\n             letters = self.phone[int(digits[index])]    \\n            \\n             for i in range(0, len(letters)):\\n                 self.dfs(digits, results, string + letters[i], index + 1)\\n \\n         \\n     \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         interpret_digit = {\\n             '1': '',\\n             '2': 'abc',\\n             '3': 'def',\\n             '4': 'ghi',\\n             '5': 'jkl',\\n             '6': 'mno',\\n             '7': 'pqrs',\\n             '8': 'tuv',\\n             '9': 'wxyz',\\n             '0': ' '}\\n         all_combinations = [''] if digits else []\\n         for digit in digits:\\n             current_combinations = list()\\n             for letter in interpret_digit[digit]:\\n                 for combination in all_combinations:\\n                     current_combinations.append(combination + letter)\\n             all_combinations = current_combinations\\n         return all_combinations\\n\"]",
        "difficulty": "interview",
        "input": [
            "\"23\""
        ],
        "output": [
            [
                "\"ad\"",
                "\"ae\"",
                "\"af\"",
                "\"bd\"",
                "\"be\"",
                "\"bf\"",
                "\"cd\"",
                "\"ce\"",
                "\"cf\""
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "letterCombinations",
        "starter_code": "\nclass Solution:\n    def letterCombinations(self, digits: str) -> List[str]:\n        ",
        "url": "https://leetcode.com/problems/letter-combinations-of-a-phone-number/"
    },
    {
        "id": 1271,
        "task_id": 2748,
        "test_case_id": 2,
        "question": "Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent.\n\nA mapping of digit to letters (just like on the telephone buttons) is given below. Note that 1 does not map to any letters.\n\n\n\nExample:\n\n\nInput: \"23\"\nOutput: [\"ad\", \"ae\", \"af\", \"bd\", \"be\", \"bf\", \"cd\", \"ce\", \"cf\"].\n\n\nNote:\n\nAlthough the above answer is in lexicographical order, your answer could be in any order you want.",
        "solutions": "[\"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         def dfs(digits, current, result):\\n             if not digits:\\n                 result.append(current)\\n                 return\\n             for c in dic[digits[0]]:\\n                 dfs(digits[1:], current + c, result)\\n         \\n         \\n         if not digits:\\n             return []\\n         \\n         dic = { '2' : \\\"abc\\\", '3': \\\"def\\\", '4':\\\"ghi\\\", \\n                '5':\\\"jkl\\\", '6':\\\"mno\\\", '7':\\\"pqrs\\\", '8':\\\"tuv\\\",'9': \\\"wxyz\\\" }\\n         result = []\\n         dfs(digits, \\\"\\\", result)\\n         return result\\n         \\n         \\n         \\n \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return []\\n         MAPPING = ('0', '1', 'abc', 'def', 'ghi', 'jkl', 'mno', 'pqrs', 'tuv', 'wxyz')\\n         def directed_combinations(i, partial):\\n             if i == len(digits):\\n                 result.append(''.join(partial))\\n                 return\\n             \\n             for c in MAPPING[int(digits[i])]:\\n                 directed_combinations(i + 1, partial + [c])\\n         \\n         result = []\\n         directed_combinations(0, [])\\n         return result\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         dic = {'2':['a','b','c'], '3':['d','e','f'], '4':['g','h','i'], '5':['j','k','l'], \\n                '6':['m','n','o'], '7':['p','q','r','s'], '8':['t','u','v'], '9':['w','x','y','z']}\\n         \\n         if '0' in digits:\\n             digits = digits.replace('0','')\\n         if '1' in digits:\\n             digits = digits.replace('1','')\\n         \\n         count = 0\\n         for a in '23456789':\\n             if a not in digits:\\n                 count += 1\\n         if count == 8:\\n             return []\\n         \\n     \\n         res = []\\n         \\n         temp = ['']\\n         i = 0\\n         while i < len(digits):\\n             for a in temp:\\n                 for b in dic[digits[i]]:\\n                     res.append(a+b)\\n             temp = res\\n             res = []\\n             i += 1\\n         return temp\\n         \\n         \\n         \\n         \\n         \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if '' == digits: \\n             return []\\n         \\n         kvmaps = {\\n             '2': 'abc',\\n             '3': 'def',\\n             '4': 'ghi',\\n             '5': 'jkl',\\n             '6': 'mno',\\n             '7': 'pqrs',\\n             '8': 'tuv',\\n             '9': 'wxyz'\\n         }\\n         rst = ['']\\n         for digit in digits:\\n             temp = [s + c for s in rst for c in kvmaps[digit]]\\n             rst = temp\\n         \\n         return rst\\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if digits==None or len(digits) ==0:\\n             return []\\n         res=[]\\n         cur=\\\"\\\"\\n         dmap={}\\n         dmap[\\\"0\\\"]=[\\\" \\\"]\\n         dmap[\\\"1\\\"]=[]\\n         dmap[\\\"2\\\"]=[\\\"a\\\",\\\"b\\\",\\\"c\\\"]\\n         dmap[\\\"3\\\"]=[\\\"d\\\",\\\"e\\\",\\\"f\\\"]\\n         dmap[\\\"4\\\"]=[\\\"g\\\",\\\"h\\\",\\\"i\\\"]\\n         dmap[\\\"5\\\"]=[\\\"j\\\",\\\"k\\\",\\\"l\\\"]\\n         dmap[\\\"6\\\"]=[\\\"m\\\",\\\"n\\\",\\\"o\\\"]\\n         dmap[\\\"7\\\"]=[\\\"p\\\",\\\"q\\\",\\\"r\\\",\\\"s\\\"]\\n         dmap[\\\"8\\\"]=[\\\"t\\\",\\\"u\\\",\\\"v\\\"]\\n         dmap[\\\"9\\\"]=[\\\"w\\\",\\\"x\\\",\\\"y\\\",\\\"z\\\"]\\n         self.dfs(digits,dmap,0,cur,res)\\n         return res\\n     \\n     def dfs(self,digits,dmap,idx,cur,res):\\n         if idx ==len(digits):\\n             res.append(cur)\\n             return\\n         else:\\n             c=digits[idx]\\n             for i in dmap[c]:\\n                 self.dfs(digits,dmap,idx+1,cur+i,res)\\n\", \"from functools import reduce\\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         d = {}\\n         d[\\\"1\\\"] = \\\"\\\"\\n         d[\\\"2\\\"] = \\\"abc\\\"\\n         d[\\\"3\\\"] = \\\"def\\\"\\n         d[\\\"4\\\"] = \\\"ghi\\\"\\n         d[\\\"5\\\"] = \\\"jkl\\\"\\n         d[\\\"6\\\"] = \\\"mno\\\"\\n         d[\\\"7\\\"] = \\\"pqrs\\\"\\n         d[\\\"8\\\"] = \\\"tuv\\\"\\n         d[\\\"9\\\"] = \\\"wxyz\\\"\\n         d[\\\"0\\\"] = \\\" \\\"\\n         \\n         digs = list(map(lambda x: list(d[x]), digits))\\n         \\n         return reduce(alg_mul, digs, [])\\n                 \\n \\n def alg_mul(xs, ys):\\n     if xs == []:\\n         return ys\\n     \\n     ws = []\\n     for x in xs:\\n         for y in ys:\\n             ws.append(x + y)\\n     \\n     return ws\", \"import queue\\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         letterDict = {0:[],1:[],2:['a','b','c'],3:['d','e','f'],4:['g','h','i'],5:['j','k','l'],\\n                       6:['m','n','o'],7:['p','q','r','s'],8:['t','u','v'],9:['w','x','y','z']}\\n         if len(digits) == 0:\\n             return []\\n         res = [\\\"\\\"]\\n         for i in range(len(digits)):\\n             num = int(digits[i])\\n             chars = letterDict[num]\\n             if len(chars) == 0:\\n                 continue\\n             s = []\\n             for j in range(len(chars)):\\n                 for k in range(len(res)):\\n                     s.append(res[k] + chars[j])\\n             res = s\\n         return res\\n                     \", \"import heapq\\n \\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         digit_map = {'1': [\\\"\\\"],\\n                     '2': ['a', 'b', 'c'],\\n                     '3': ['d', 'e', 'f'],\\n                     '4': ['g', 'h', 'i'],\\n                     '5': ['j', 'k', 'l'],\\n                     '6': ['m', 'n', 'o'],\\n                     '7': ['p', 'q', 'r', 's'],\\n                     '8': ['t', 'u', 'v'],\\n                     '9': ['w', 'x', 'y', 'z'],\\n                     '0': [\\\"\\\"]}\\n         combs = []\\n         # Queue will hold a list of tuples mapping to how many characters are already mapped, \\n         # and the current converted string!\\n         letter_q = [(0, \\\"\\\")]\\n         if not digits:\\n             return []\\n         while letter_q:\\n             cur = heapq.heappop(letter_q)\\n             cur_idx = cur[0]\\n             cur_str = cur[1]\\n             next_dig = digits[cur_idx]\\n             for value in digit_map[next_dig]:\\n                 new_digit_str = (cur_idx+1, cur_str+value)\\n                 if cur_idx + 1 >= len(digits):\\n                     combs.append(new_digit_str)\\n                 else:\\n                     heapq.heappush(letter_q, new_digit_str)    \\n         return list(map(lambda x: x[1], combs))\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         key = {2:'abc', 3:'def', 4:'ghi',5:'jkl',6:'mno',7:'pqrs',8:'tuv',9:'wxyz'}\\n     \\n         nonlocal out\\n         out = ['']\\n \\n         def eachLetter(i):\\n             if i < len(digits):\\n                 nonlocal out\\n                 if int(digits[i]) > 1:\\n                     x = []\\n                     for j in out:\\n                         for k in key[int(digits[i])]:\\n                             x.append(j+k)\\n                     out = x\\n \\n                 eachLetter(i+1)\\n \\n         eachLetter(0)\\n         \\n         return [] if out[0] is '' else out\\n\", \"from collections import defaultdict\\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         digit_to_chars = defaultdict(list)\\n         char_int = ord('a')\\n         for i in range(2,10):\\n             num_chars = 3 if i not in {7,9} else 4\\n             for j in range(num_chars):\\n                 digit_to_chars[str(i)].append(chr(char_int))\\n                 char_int += 1\\n         if not digits:\\n             return []\\n         letter_combinations = ['']\\n         for digit in digits:\\n             new_letter_combinations = []\\n             for letter_combination in letter_combinations:\\n                 for character in digit_to_chars[digit]:\\n                     new_letter_combination = letter_combination + character\\n                     new_letter_combinations.append(new_letter_combination)\\n             letter_combinations = new_letter_combinations\\n         return letter_combinations\", \"class Solution:\\n     Dmap = {'2': 'abc', '3': 'def', '4': 'ghi', '5': 'jkl', '6': 'mno',\\n              '7': 'pqrs', '8': 'tuv', '9': 'wxyz'}\\n     \\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if '1' in digits or '0' in digits:\\n             return []\\n         \\n         if len(digits) == 0:\\n             return []\\n         \\n         return self.letterC(digits)\\n \\n         \\n     def letterC(self, digits):\\n         if len(digits) == 0:\\n             return ['']\\n         \\n         res = []\\n         for back in self.letterC(digits[1:]):\\n             for fr in self.Dmap[digits[0]]:\\n                 res.append(fr + back)\\n         \\n         return res\\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.dic={'2':'abc','3':'def','4':'ghi','5':'jkl','6':'mno','7':'pqrs','8':'tuv','9':'wxyz'}\\n         if not digits:\\n             return []\\n         res = []\\n         self.dfs(digits,0,'',res)\\n         return res\\n     \\n     def dfs(self, digits, index, path, res):\\n         if len(path) == len(digits):\\n             res.append(path)\\n             return\\n         for i in range(len(self.dic[digits[index]])):\\n             path_ = path+self.dic[digits[index]][i]\\n             self.dfs(digits,index+1,path_,res)\\n         \\n         \\n\", \"class Solution:\\n     def __init__(self):\\n         self.phone = [\\\"\\\", \\\"\\\", \\\"abc\\\", \\\"def\\\",\\\"ghi\\\", \\\"jkl\\\", \\\"mno\\\", \\\"pqrs\\\",\\\"tuv\\\", \\\"wxyz\\\"]\\n         \\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         if not digits:\\n             return []\\n         \\n         results = []\\n         self.dfs(digits, results, \\\"\\\", 0)\\n         return results\\n     \\n \\n     def dfs(self, digits, results, string, index):\\n         if index == len(digits):\\n             results.append(string)\\n         else: \\n             letters = self.phone[int(digits[index])]    \\n            \\n             for i in range(0, len(letters)):\\n                 self.dfs(digits, results, string + letters[i], index + 1)\\n \\n         \\n     \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         interpret_digit = {\\n             '1': '',\\n             '2': 'abc',\\n             '3': 'def',\\n             '4': 'ghi',\\n             '5': 'jkl',\\n             '6': 'mno',\\n             '7': 'pqrs',\\n             '8': 'tuv',\\n             '9': 'wxyz',\\n             '0': ' '}\\n         all_combinations = [''] if digits else []\\n         for digit in digits:\\n             current_combinations = list()\\n             for letter in interpret_digit[digit]:\\n                 for combination in all_combinations:\\n                     current_combinations.append(combination + letter)\\n             all_combinations = current_combinations\\n         return all_combinations\\n\"]",
        "difficulty": "interview",
        "input": [
            "\"\""
        ],
        "output": [
            []
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "letterCombinations",
        "starter_code": "\nclass Solution:\n    def letterCombinations(self, digits: str) -> List[str]:\n        ",
        "url": "https://leetcode.com/problems/letter-combinations-of-a-phone-number/"
    },
    {
        "id": 1272,
        "task_id": 2748,
        "test_case_id": 3,
        "question": "Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent.\n\nA mapping of digit to letters (just like on the telephone buttons) is given below. Note that 1 does not map to any letters.\n\n\n\nExample:\n\n\nInput: \"23\"\nOutput: [\"ad\", \"ae\", \"af\", \"bd\", \"be\", \"bf\", \"cd\", \"ce\", \"cf\"].\n\n\nNote:\n\nAlthough the above answer is in lexicographical order, your answer could be in any order you want.",
        "solutions": "[\"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         def dfs(digits, current, result):\\n             if not digits:\\n                 result.append(current)\\n                 return\\n             for c in dic[digits[0]]:\\n                 dfs(digits[1:], current + c, result)\\n         \\n         \\n         if not digits:\\n             return []\\n         \\n         dic = { '2' : \\\"abc\\\", '3': \\\"def\\\", '4':\\\"ghi\\\", \\n                '5':\\\"jkl\\\", '6':\\\"mno\\\", '7':\\\"pqrs\\\", '8':\\\"tuv\\\",'9': \\\"wxyz\\\" }\\n         result = []\\n         dfs(digits, \\\"\\\", result)\\n         return result\\n         \\n         \\n         \\n \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return []\\n         MAPPING = ('0', '1', 'abc', 'def', 'ghi', 'jkl', 'mno', 'pqrs', 'tuv', 'wxyz')\\n         def directed_combinations(i, partial):\\n             if i == len(digits):\\n                 result.append(''.join(partial))\\n                 return\\n             \\n             for c in MAPPING[int(digits[i])]:\\n                 directed_combinations(i + 1, partial + [c])\\n         \\n         result = []\\n         directed_combinations(0, [])\\n         return result\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         dic = {'2':['a','b','c'], '3':['d','e','f'], '4':['g','h','i'], '5':['j','k','l'], \\n                '6':['m','n','o'], '7':['p','q','r','s'], '8':['t','u','v'], '9':['w','x','y','z']}\\n         \\n         if '0' in digits:\\n             digits = digits.replace('0','')\\n         if '1' in digits:\\n             digits = digits.replace('1','')\\n         \\n         count = 0\\n         for a in '23456789':\\n             if a not in digits:\\n                 count += 1\\n         if count == 8:\\n             return []\\n         \\n     \\n         res = []\\n         \\n         temp = ['']\\n         i = 0\\n         while i < len(digits):\\n             for a in temp:\\n                 for b in dic[digits[i]]:\\n                     res.append(a+b)\\n             temp = res\\n             res = []\\n             i += 1\\n         return temp\\n         \\n         \\n         \\n         \\n         \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if '' == digits: \\n             return []\\n         \\n         kvmaps = {\\n             '2': 'abc',\\n             '3': 'def',\\n             '4': 'ghi',\\n             '5': 'jkl',\\n             '6': 'mno',\\n             '7': 'pqrs',\\n             '8': 'tuv',\\n             '9': 'wxyz'\\n         }\\n         rst = ['']\\n         for digit in digits:\\n             temp = [s + c for s in rst for c in kvmaps[digit]]\\n             rst = temp\\n         \\n         return rst\\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if digits==None or len(digits) ==0:\\n             return []\\n         res=[]\\n         cur=\\\"\\\"\\n         dmap={}\\n         dmap[\\\"0\\\"]=[\\\" \\\"]\\n         dmap[\\\"1\\\"]=[]\\n         dmap[\\\"2\\\"]=[\\\"a\\\",\\\"b\\\",\\\"c\\\"]\\n         dmap[\\\"3\\\"]=[\\\"d\\\",\\\"e\\\",\\\"f\\\"]\\n         dmap[\\\"4\\\"]=[\\\"g\\\",\\\"h\\\",\\\"i\\\"]\\n         dmap[\\\"5\\\"]=[\\\"j\\\",\\\"k\\\",\\\"l\\\"]\\n         dmap[\\\"6\\\"]=[\\\"m\\\",\\\"n\\\",\\\"o\\\"]\\n         dmap[\\\"7\\\"]=[\\\"p\\\",\\\"q\\\",\\\"r\\\",\\\"s\\\"]\\n         dmap[\\\"8\\\"]=[\\\"t\\\",\\\"u\\\",\\\"v\\\"]\\n         dmap[\\\"9\\\"]=[\\\"w\\\",\\\"x\\\",\\\"y\\\",\\\"z\\\"]\\n         self.dfs(digits,dmap,0,cur,res)\\n         return res\\n     \\n     def dfs(self,digits,dmap,idx,cur,res):\\n         if idx ==len(digits):\\n             res.append(cur)\\n             return\\n         else:\\n             c=digits[idx]\\n             for i in dmap[c]:\\n                 self.dfs(digits,dmap,idx+1,cur+i,res)\\n\", \"from functools import reduce\\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         d = {}\\n         d[\\\"1\\\"] = \\\"\\\"\\n         d[\\\"2\\\"] = \\\"abc\\\"\\n         d[\\\"3\\\"] = \\\"def\\\"\\n         d[\\\"4\\\"] = \\\"ghi\\\"\\n         d[\\\"5\\\"] = \\\"jkl\\\"\\n         d[\\\"6\\\"] = \\\"mno\\\"\\n         d[\\\"7\\\"] = \\\"pqrs\\\"\\n         d[\\\"8\\\"] = \\\"tuv\\\"\\n         d[\\\"9\\\"] = \\\"wxyz\\\"\\n         d[\\\"0\\\"] = \\\" \\\"\\n         \\n         digs = list(map(lambda x: list(d[x]), digits))\\n         \\n         return reduce(alg_mul, digs, [])\\n                 \\n \\n def alg_mul(xs, ys):\\n     if xs == []:\\n         return ys\\n     \\n     ws = []\\n     for x in xs:\\n         for y in ys:\\n             ws.append(x + y)\\n     \\n     return ws\", \"import queue\\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         letterDict = {0:[],1:[],2:['a','b','c'],3:['d','e','f'],4:['g','h','i'],5:['j','k','l'],\\n                       6:['m','n','o'],7:['p','q','r','s'],8:['t','u','v'],9:['w','x','y','z']}\\n         if len(digits) == 0:\\n             return []\\n         res = [\\\"\\\"]\\n         for i in range(len(digits)):\\n             num = int(digits[i])\\n             chars = letterDict[num]\\n             if len(chars) == 0:\\n                 continue\\n             s = []\\n             for j in range(len(chars)):\\n                 for k in range(len(res)):\\n                     s.append(res[k] + chars[j])\\n             res = s\\n         return res\\n                     \", \"import heapq\\n \\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         digit_map = {'1': [\\\"\\\"],\\n                     '2': ['a', 'b', 'c'],\\n                     '3': ['d', 'e', 'f'],\\n                     '4': ['g', 'h', 'i'],\\n                     '5': ['j', 'k', 'l'],\\n                     '6': ['m', 'n', 'o'],\\n                     '7': ['p', 'q', 'r', 's'],\\n                     '8': ['t', 'u', 'v'],\\n                     '9': ['w', 'x', 'y', 'z'],\\n                     '0': [\\\"\\\"]}\\n         combs = []\\n         # Queue will hold a list of tuples mapping to how many characters are already mapped, \\n         # and the current converted string!\\n         letter_q = [(0, \\\"\\\")]\\n         if not digits:\\n             return []\\n         while letter_q:\\n             cur = heapq.heappop(letter_q)\\n             cur_idx = cur[0]\\n             cur_str = cur[1]\\n             next_dig = digits[cur_idx]\\n             for value in digit_map[next_dig]:\\n                 new_digit_str = (cur_idx+1, cur_str+value)\\n                 if cur_idx + 1 >= len(digits):\\n                     combs.append(new_digit_str)\\n                 else:\\n                     heapq.heappush(letter_q, new_digit_str)    \\n         return list(map(lambda x: x[1], combs))\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         key = {2:'abc', 3:'def', 4:'ghi',5:'jkl',6:'mno',7:'pqrs',8:'tuv',9:'wxyz'}\\n     \\n         nonlocal out\\n         out = ['']\\n \\n         def eachLetter(i):\\n             if i < len(digits):\\n                 nonlocal out\\n                 if int(digits[i]) > 1:\\n                     x = []\\n                     for j in out:\\n                         for k in key[int(digits[i])]:\\n                             x.append(j+k)\\n                     out = x\\n \\n                 eachLetter(i+1)\\n \\n         eachLetter(0)\\n         \\n         return [] if out[0] is '' else out\\n\", \"from collections import defaultdict\\n \\n class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         digit_to_chars = defaultdict(list)\\n         char_int = ord('a')\\n         for i in range(2,10):\\n             num_chars = 3 if i not in {7,9} else 4\\n             for j in range(num_chars):\\n                 digit_to_chars[str(i)].append(chr(char_int))\\n                 char_int += 1\\n         if not digits:\\n             return []\\n         letter_combinations = ['']\\n         for digit in digits:\\n             new_letter_combinations = []\\n             for letter_combination in letter_combinations:\\n                 for character in digit_to_chars[digit]:\\n                     new_letter_combination = letter_combination + character\\n                     new_letter_combinations.append(new_letter_combination)\\n             letter_combinations = new_letter_combinations\\n         return letter_combinations\", \"class Solution:\\n     Dmap = {'2': 'abc', '3': 'def', '4': 'ghi', '5': 'jkl', '6': 'mno',\\n              '7': 'pqrs', '8': 'tuv', '9': 'wxyz'}\\n     \\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if '1' in digits or '0' in digits:\\n             return []\\n         \\n         if len(digits) == 0:\\n             return []\\n         \\n         return self.letterC(digits)\\n \\n         \\n     def letterC(self, digits):\\n         if len(digits) == 0:\\n             return ['']\\n         \\n         res = []\\n         for back in self.letterC(digits[1:]):\\n             for fr in self.Dmap[digits[0]]:\\n                 res.append(fr + back)\\n         \\n         return res\\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.dic={'2':'abc','3':'def','4':'ghi','5':'jkl','6':'mno','7':'pqrs','8':'tuv','9':'wxyz'}\\n         if not digits:\\n             return []\\n         res = []\\n         self.dfs(digits,0,'',res)\\n         return res\\n     \\n     def dfs(self, digits, index, path, res):\\n         if len(path) == len(digits):\\n             res.append(path)\\n             return\\n         for i in range(len(self.dic[digits[index]])):\\n             path_ = path+self.dic[digits[index]][i]\\n             self.dfs(digits,index+1,path_,res)\\n         \\n         \\n\", \"class Solution:\\n     def __init__(self):\\n         self.phone = [\\\"\\\", \\\"\\\", \\\"abc\\\", \\\"def\\\",\\\"ghi\\\", \\\"jkl\\\", \\\"mno\\\", \\\"pqrs\\\",\\\"tuv\\\", \\\"wxyz\\\"]\\n         \\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         if not digits:\\n             return []\\n         \\n         results = []\\n         self.dfs(digits, results, \\\"\\\", 0)\\n         return results\\n     \\n \\n     def dfs(self, digits, results, string, index):\\n         if index == len(digits):\\n             results.append(string)\\n         else: \\n             letters = self.phone[int(digits[index])]    \\n            \\n             for i in range(0, len(letters)):\\n                 self.dfs(digits, results, string + letters[i], index + 1)\\n \\n         \\n     \\n\", \"class Solution:\\n     def letterCombinations(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: str\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         interpret_digit = {\\n             '1': '',\\n             '2': 'abc',\\n             '3': 'def',\\n             '4': 'ghi',\\n             '5': 'jkl',\\n             '6': 'mno',\\n             '7': 'pqrs',\\n             '8': 'tuv',\\n             '9': 'wxyz',\\n             '0': ' '}\\n         all_combinations = [''] if digits else []\\n         for digit in digits:\\n             current_combinations = list()\\n             for letter in interpret_digit[digit]:\\n                 for combination in all_combinations:\\n                     current_combinations.append(combination + letter)\\n             all_combinations = current_combinations\\n         return all_combinations\\n\"]",
        "difficulty": "interview",
        "input": [
            "\"2\""
        ],
        "output": [
            [
                "\"a\"",
                "\"b\"",
                "\"c\""
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "letterCombinations",
        "starter_code": "\nclass Solution:\n    def letterCombinations(self, digits: str) -> List[str]:\n        ",
        "url": "https://leetcode.com/problems/letter-combinations-of-a-phone-number/"
    },
    {
        "id": 1273,
        "task_id": 2882,
        "test_case_id": 1,
        "question": "Given n pairs of parentheses, write a function to generate all combinations of well-formed parentheses.\n\n\n\nFor example, given n = 3, a solution set is:\n\n\n[\n  \"((()))\",\n  \"(()())\",\n  \"(())()\",\n  \"()(())\",\n  \"()()()\"\n]",
        "solutions": "[\"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         left = right = n\\n         result = []\\n         self.generate(left, right, result, '')\\n         return result\\n     def generate(self, left, right, result, string):\\n         if left == 0 and right == 0:\\n             result.append(string)\\n             return\\n         if left:\\n             self.generate(left - 1, right , result, string+'(')\\n         if left < right:\\n             self.generate(left, right - 1, result, string+')')\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.mem = {}\\n         \\n         return self.top_down(n)\\n         \\n         \\n     def merge(self, left, right):\\n         comb = []\\n         for i in left:\\n             comb += [i + j for j in right]\\n         \\n         return list(set(comb))\\n         \\n     def top_down(self, n):\\n         if n == 1:\\n             return ['()']\\n         \\n         if not n-1 in self.mem:\\n             self.mem[n-1] = self.top_down(n-1)\\n         rest = self.mem[n-1]\\n         comb = ['(' + i + ')' for i in rest]\\n         \\n         for i in range(1, n):\\n             \\n             if not i in self.mem:\\n                 self.mem[i] = self.top_down(i)\\n             comb_left = self.mem[i]\\n             if not n-i in self.mem:\\n                 self.mem[n-i] = self.top_down(n-i)\\n             comb_right = self.mem[n-i]\\n             \\n             comb += self.merge(comb_left,comb_right)\\n             \\n         \\n         comb = list(set(comb))\\n         \\n         return comb\\n     \\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n < 1 or type(n) != int:\\n             return []\\n         ret = []\\n         tmp = []\\n         # define a set of struct like {str:\\\"((())\\\", cnt:3, rcnt:2)}\\n         tmp.append({\\\"str\\\": \\\"(\\\", \\\"cnt\\\": 1, \\\"rcnt\\\": 0})\\n         while tmp:\\n             cur = tmp.pop()\\n             if n == cur[\\\"cnt\\\"]:\\n                 ret.append(cur[\\\"str\\\"] + ')'*(n - cur[\\\"rcnt\\\"]))\\n             else:\\n                 if cur[\\\"cnt\\\"] > cur[\\\"rcnt\\\"]:\\n                     tmp.append({\\\"str\\\":cur[\\\"str\\\"]+')', \\\"cnt\\\": cur[\\\"cnt\\\"], \\\"rcnt\\\": cur[\\\"rcnt\\\"] + 1})\\n                 tmp.append({\\\"str\\\":cur[\\\"str\\\"]+'(', \\\"cnt\\\": cur[\\\"cnt\\\"] + 1, \\\"rcnt\\\": cur[\\\"rcnt\\\"]})\\n         return ret\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         stack = []\\n         s = ''\\n         self.dfs(stack, 0, 0, s, n)\\n         return stack\\n \\n     def dfs(self, stack, first, last, s, n):\\n         if last==n:\\n             stack.append(s)\\n         else:\\n             if first < n:\\n                 self.dfs(stack, first+1, last, s+'(', n)\\n             if last < first:\\n                 self.dfs(stack, first, last+1, s+')', n)     \", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         ret = []\\n         def backtrack(left = 0, right = 0, tmp = ''):\\n             if len(tmp) == 2 * n:\\n                 ret.append(tmp)\\n                 return\\n             if left < n:\\n                 backtrack(left + 1, right, tmp + '(')\\n             if left > right:\\n                 backtrack(left, right + 1, tmp + ')')\\n         \\n         backtrack()\\n         return ret\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         # if n==0:\\n         #     return [\\\"\\\"]\\n         # prev = self.generateParenthesis(n-1)\\n         # results = set()\\n         # for r in prev:\\n         #     results.add('('+r+')')\\n         #     results.add('()'+r)\\n         #     results.add(r+'()')\\n         # return sorted(list(set(results)))\\n         \\n         def gen(n,open_par,res,res_set):\\n             print(res)\\n             if len(res)==2*n:\\n                 res_set.append(''.join(res))\\n                 return\\n             if 2*n-len(res)>open_par:\\n                 res.append('(')\\n                 gen(n,open_par+1,res,res_set)\\n                 res.pop()\\n             if open_par>0:\\n                 res.append(')')\\n                 gen(n,open_par-1,res,res_set)\\n                 res.pop()\\n         res_set=[]\\n         gen(n,0,[],res_set)\\n         return res_set\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         result = set()\\n         if n == 0:\\n             result = [\\\"\\\"]\\n         else:\\n             for i in range(1, n + 1):\\n                 temp1 = {\\n                     \\\"(\\\" + x + \\\")\\\" for x in self.generateParenthesis(i - 1)}\\n                 temp2 = {x for x in self.generateParenthesis(n - i)}\\n                 temp = {x + y for x in temp1 for y in temp2} | {x +\\n                                                                 y for x in temp2 for y in temp1}\\n                 result = result | temp\\n \\n             result = sorted(list(result))\\n         return result\\n\", \"class Node(object):\\n     def __init__(self,val,root,n,p):\\n         self.val=val\\n         self.n=n\\n         self.p=p\\n         if root:\\n             self.root=root\\n         else:\\n             self.root=self\\n         self.left=self.leftchild(n,p)\\n         self.right=self.rightchild(n,p)\\n            \\n     def leftchild(self,n,p):\\n         if n==0:\\n             if p==0:                 \\n                 temp=''.join(self.val);\\n                 self.root.val.append(temp)\\n             return None\\n \\n         if n>0:\\n             return Node(self.val+['('],self.root,n-1,p+1)\\n \\n     def rightchild(self,n,p):\\n         if p==0:\\n               return None\\n         else:\\n               return Node(self.val+[')'],self.root,n,p-1)\\n             \\n \\n class Solution:\\n     def generateParenthesis(self, s):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: void Do not return anything, modify nums in-place instead.\\n         \\\"\\\"\\\"\\n         D=Node([],None,s,0)\\n         return D.val\\n \\n         \\n                \", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         rst = []\\n         cake = ''\\n         self.generateRecursion(rst,cake,n,n)\\n         return rst\\n     def generateRecursion(self,rst,cake,left,right):\\n         if right==0:\\n             rst.append(cake)\\n         if left>0:\\n             self.generateRecursion(rst,cake+'(',left-1,right)\\n         if right>left:\\n             self.generateRecursion(rst,cake+')',left,right-1)\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if not n:\\n             return []\\n         cache = [[\\\"(\\\", 1, n-1]]\\n         for _ in range(2*n-1):\\n             for _ in range(len(cache)):\\n                 temp = cache.pop(0)\\n                 if temp[1]:\\n                     toAdd = temp.copy()\\n                     toAdd[0] += ')'\\n                     toAdd[1] -= 1\\n                     cache.append(toAdd)\\n                 if temp[2]:\\n                     temp[0] += '('\\n                     temp[1] += 1\\n                     temp[2] -= 1\\n                     cache.append(temp)\\n         return [items[0] for items in cache]\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.res = []\\n         self.dfs(n, 0, 0, '')\\n         return self.res\\n \\n     def dfs(self, n, left, right, path):\\n         if left > n or right > n:\\n             return\\n \\n         if n == 0:\\n             self.res.append(path)\\n             return\\n \\n         if left == 0:\\n             self.dfs(n, left+1, right, path+'(')\\n \\n         if left == right and left != 0:\\n             self.dfs(n-left, 0, 0, path)\\n \\n         if left > right:\\n             self.dfs(n, left+1, right, path+'(')\\n             self.dfs(n, left, right+1, path+')')\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [\\\"\\\"]\\n         if n == 1:\\n             return [\\\"()\\\"]\\n         results = []\\n \\n         for i in range(1, 2*n, 2):\\n             first = self.generateParenthesis(int((i-1)/2))\\n             second = self.generateParenthesis(int((2*n-i-1)/2))\\n             results += [\\\"(\\\"+u+\\\")\\\"+v for u in first for v in second]\\n         return  results\", \"class Solution:\\n     def parenthesis(self, n):\\n         return \\\"(\\\"*n+\\\")\\\"*n\\n     \\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         elif n == 1:\\n             return [\\\"()\\\"]\\n         \\n         result = []\\n         for i in range(1,n):\\n             r1 = self.generateParenthesis(i)\\n             r2 = self.generateParenthesis(n-i)\\n             for r11 in r1:\\n                 for r21 in r2:\\n                     result.append(r11+r21)\\n                     \\n         r3 = self.generateParenthesis(n-1)\\n         for r31 in r3:\\n             result.append(\\\"({})\\\".format(r31))\\n         \\n         result.sort()\\n         rr = []\\n         last = None\\n         for r in result:\\n             if r == last:\\n                 continue\\n             rr.append(r)\\n             last = r\\n             \\n                 \\n         return rr\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         def balance(str):\\n             lst = list(str)\\n             bal = 0\\n             for char in lst:\\n                 if char == \\\"(\\\":\\n                     bal += 1\\n                 else:\\n                     bal -= 1\\n             return bal\\n         \\n         def left(str):\\n             lst = list(str)\\n             sum = 0\\n             for char in lst:\\n                 if char == \\\"(\\\":\\n                     sum += 1\\n             return n - sum\\n         \\n         def right(str):\\n             lst = list(str)\\n             sum = 0\\n             for char in lst:\\n                 if char == \\\")\\\":\\n                     sum += 1\\n             return n - sum\\n         \\n \\n         prev = [\\\"(\\\"]\\n         length = 1\\n         \\n         while length < 2*n:\\n             current = [] \\n                 \\n             for item in prev:\\n                 if left(item) != 0:\\n                     current.append(item+\\\"(\\\")\\n                     \\n                 if right(item) != 0 and balance(item) > 0:\\n                     current.append(item+\\\")\\\")\\n             length += 1\\n             prev = current\\n         \\n         return prev\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         results = []\\n         \\n         def genparen(prefix, opens, closes):\\n             if len(prefix) == 2*n:\\n                 results.append(prefix)\\n                 return\\n             \\n             if opens < n:\\n                 genparen(prefix+\\\"(\\\", opens+1, closes)\\n             if closes < opens:\\n                 genparen(prefix+\\\")\\\", opens, closes+1)\\n         \\n         genparen('', 0, 0)\\n         return results\\n             \\n         \\n         \\n #         paren_map = {}\\n #         paren_map[1] = set([\\\"()\\\"])\\n         \\n #         for i in range(2, n+1):\\n #             paren_map[i] = self.generate_parens(paren_map[i-1])\\n             \\n #         return list(paren_map[n])\\n             \\n             \\n #     def generate_parens(self, st):\\n #         result = set()\\n         \\n #         for elem in st:\\n #             for i in range(len(elem)):\\n #                 candidate = elem[:i] + \\\"()\\\" + elem[i:]\\n #                 result.add(candidate)\\n         \\n #         return result\\n         \\n\", \"class Solution:\\n     \\n     hashMap = {}\\n     outPut = []\\n     \\n     def helper(self, n, string, length, opened, openUsed):\\n         \\n         ## base case\\n         if length == n*2:\\n             if string not in self.outPut:\\n                 self.outPut.append(string)\\n             return\\n         \\n         ## skip if already tried\\n         if string in self.hashMap:\\n             return\\n         else:\\n             self.hashMap[string] = 1\\n         \\n         if opened == 0:\\n             tempOpened = 1\\n             tempOpenUsed = openUsed + 1\\n             self.helper(n,string+'(',length+1,tempOpened,tempOpenUsed)\\n             \\n         if opened > 0:\\n             tempOpened = opened - 1\\n             self.helper(n,string+')',length+1,tempOpened,openUsed)\\n             \\n         if openUsed < n:\\n             tempOpened = opened + 1\\n             tempOpenUsed = openUsed + 1\\n             self.helper(n,string+'(',length+1,tempOpened,tempOpenUsed)\\n     \\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.hashMap = {}\\n         self.outPut = []\\n         self.helper(n,\\\"\\\",0,0,0)\\n         \\n         return self.outPut\\n         \\n         \\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         # array to keep track of results         \\n         answers = []\\n         for i in range(0,n+1):\\n             answers.append(Solution.catalanHelper(i, answers))\\n             print((i, answers))\\n         \\n         return list(set(answers[n]))\\n         \\n     @staticmethod\\n     # returns parenthesis\\n     def catalanHelper(n, answers):\\n         \\\"\\\"\\\"\\n         : type n : int \\n         : rtype : List[str]\\n         \\\"\\\"\\\"\\n         \\n         if n == 0:\\n             return [\\\"\\\"]\\n         elif n == 1:\\n             return [\\\"()\\\"]\\n         else:\\n             output = []\\n             for i in range(0, n-1):\\n                 front_list = answers[i] \\n                 end_list = answers[n-i-1]\\n                 for front in front_list:\\n                     for end in end_list:\\n                         \\n                         output.append(\\\"(\\\" + front + \\\")\\\" + end )\\n                         output.append(\\\"(\\\" + front  + end +\\\")\\\")\\n \\n             return output\\n         \\n         \\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            3
        ],
        "output": [
            "\"((()))\"",
            "\"(()())\"",
            "\"(())()\"",
            "\"()(())\"",
            "\"()()()\""
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "generateParenthesis",
        "starter_code": "\nclass Solution:\n    def generateParenthesis(self, n: int) -> List[str]:\n        ",
        "url": "https://leetcode.com/problems/generate-parentheses/"
    },
    {
        "id": 1274,
        "task_id": 2882,
        "test_case_id": 2,
        "question": "Given n pairs of parentheses, write a function to generate all combinations of well-formed parentheses.\n\n\n\nFor example, given n = 3, a solution set is:\n\n\n[\n  \"((()))\",\n  \"(()())\",\n  \"(())()\",\n  \"()(())\",\n  \"()()()\"\n]",
        "solutions": "[\"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         left = right = n\\n         result = []\\n         self.generate(left, right, result, '')\\n         return result\\n     def generate(self, left, right, result, string):\\n         if left == 0 and right == 0:\\n             result.append(string)\\n             return\\n         if left:\\n             self.generate(left - 1, right , result, string+'(')\\n         if left < right:\\n             self.generate(left, right - 1, result, string+')')\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.mem = {}\\n         \\n         return self.top_down(n)\\n         \\n         \\n     def merge(self, left, right):\\n         comb = []\\n         for i in left:\\n             comb += [i + j for j in right]\\n         \\n         return list(set(comb))\\n         \\n     def top_down(self, n):\\n         if n == 1:\\n             return ['()']\\n         \\n         if not n-1 in self.mem:\\n             self.mem[n-1] = self.top_down(n-1)\\n         rest = self.mem[n-1]\\n         comb = ['(' + i + ')' for i in rest]\\n         \\n         for i in range(1, n):\\n             \\n             if not i in self.mem:\\n                 self.mem[i] = self.top_down(i)\\n             comb_left = self.mem[i]\\n             if not n-i in self.mem:\\n                 self.mem[n-i] = self.top_down(n-i)\\n             comb_right = self.mem[n-i]\\n             \\n             comb += self.merge(comb_left,comb_right)\\n             \\n         \\n         comb = list(set(comb))\\n         \\n         return comb\\n     \\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n < 1 or type(n) != int:\\n             return []\\n         ret = []\\n         tmp = []\\n         # define a set of struct like {str:\\\"((())\\\", cnt:3, rcnt:2)}\\n         tmp.append({\\\"str\\\": \\\"(\\\", \\\"cnt\\\": 1, \\\"rcnt\\\": 0})\\n         while tmp:\\n             cur = tmp.pop()\\n             if n == cur[\\\"cnt\\\"]:\\n                 ret.append(cur[\\\"str\\\"] + ')'*(n - cur[\\\"rcnt\\\"]))\\n             else:\\n                 if cur[\\\"cnt\\\"] > cur[\\\"rcnt\\\"]:\\n                     tmp.append({\\\"str\\\":cur[\\\"str\\\"]+')', \\\"cnt\\\": cur[\\\"cnt\\\"], \\\"rcnt\\\": cur[\\\"rcnt\\\"] + 1})\\n                 tmp.append({\\\"str\\\":cur[\\\"str\\\"]+'(', \\\"cnt\\\": cur[\\\"cnt\\\"] + 1, \\\"rcnt\\\": cur[\\\"rcnt\\\"]})\\n         return ret\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         stack = []\\n         s = ''\\n         self.dfs(stack, 0, 0, s, n)\\n         return stack\\n \\n     def dfs(self, stack, first, last, s, n):\\n         if last==n:\\n             stack.append(s)\\n         else:\\n             if first < n:\\n                 self.dfs(stack, first+1, last, s+'(', n)\\n             if last < first:\\n                 self.dfs(stack, first, last+1, s+')', n)     \", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         ret = []\\n         def backtrack(left = 0, right = 0, tmp = ''):\\n             if len(tmp) == 2 * n:\\n                 ret.append(tmp)\\n                 return\\n             if left < n:\\n                 backtrack(left + 1, right, tmp + '(')\\n             if left > right:\\n                 backtrack(left, right + 1, tmp + ')')\\n         \\n         backtrack()\\n         return ret\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         # if n==0:\\n         #     return [\\\"\\\"]\\n         # prev = self.generateParenthesis(n-1)\\n         # results = set()\\n         # for r in prev:\\n         #     results.add('('+r+')')\\n         #     results.add('()'+r)\\n         #     results.add(r+'()')\\n         # return sorted(list(set(results)))\\n         \\n         def gen(n,open_par,res,res_set):\\n             print(res)\\n             if len(res)==2*n:\\n                 res_set.append(''.join(res))\\n                 return\\n             if 2*n-len(res)>open_par:\\n                 res.append('(')\\n                 gen(n,open_par+1,res,res_set)\\n                 res.pop()\\n             if open_par>0:\\n                 res.append(')')\\n                 gen(n,open_par-1,res,res_set)\\n                 res.pop()\\n         res_set=[]\\n         gen(n,0,[],res_set)\\n         return res_set\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         result = set()\\n         if n == 0:\\n             result = [\\\"\\\"]\\n         else:\\n             for i in range(1, n + 1):\\n                 temp1 = {\\n                     \\\"(\\\" + x + \\\")\\\" for x in self.generateParenthesis(i - 1)}\\n                 temp2 = {x for x in self.generateParenthesis(n - i)}\\n                 temp = {x + y for x in temp1 for y in temp2} | {x +\\n                                                                 y for x in temp2 for y in temp1}\\n                 result = result | temp\\n \\n             result = sorted(list(result))\\n         return result\\n\", \"class Node(object):\\n     def __init__(self,val,root,n,p):\\n         self.val=val\\n         self.n=n\\n         self.p=p\\n         if root:\\n             self.root=root\\n         else:\\n             self.root=self\\n         self.left=self.leftchild(n,p)\\n         self.right=self.rightchild(n,p)\\n            \\n     def leftchild(self,n,p):\\n         if n==0:\\n             if p==0:                 \\n                 temp=''.join(self.val);\\n                 self.root.val.append(temp)\\n             return None\\n \\n         if n>0:\\n             return Node(self.val+['('],self.root,n-1,p+1)\\n \\n     def rightchild(self,n,p):\\n         if p==0:\\n               return None\\n         else:\\n               return Node(self.val+[')'],self.root,n,p-1)\\n             \\n \\n class Solution:\\n     def generateParenthesis(self, s):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: void Do not return anything, modify nums in-place instead.\\n         \\\"\\\"\\\"\\n         D=Node([],None,s,0)\\n         return D.val\\n \\n         \\n                \", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         rst = []\\n         cake = ''\\n         self.generateRecursion(rst,cake,n,n)\\n         return rst\\n     def generateRecursion(self,rst,cake,left,right):\\n         if right==0:\\n             rst.append(cake)\\n         if left>0:\\n             self.generateRecursion(rst,cake+'(',left-1,right)\\n         if right>left:\\n             self.generateRecursion(rst,cake+')',left,right-1)\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if not n:\\n             return []\\n         cache = [[\\\"(\\\", 1, n-1]]\\n         for _ in range(2*n-1):\\n             for _ in range(len(cache)):\\n                 temp = cache.pop(0)\\n                 if temp[1]:\\n                     toAdd = temp.copy()\\n                     toAdd[0] += ')'\\n                     toAdd[1] -= 1\\n                     cache.append(toAdd)\\n                 if temp[2]:\\n                     temp[0] += '('\\n                     temp[1] += 1\\n                     temp[2] -= 1\\n                     cache.append(temp)\\n         return [items[0] for items in cache]\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.res = []\\n         self.dfs(n, 0, 0, '')\\n         return self.res\\n \\n     def dfs(self, n, left, right, path):\\n         if left > n or right > n:\\n             return\\n \\n         if n == 0:\\n             self.res.append(path)\\n             return\\n \\n         if left == 0:\\n             self.dfs(n, left+1, right, path+'(')\\n \\n         if left == right and left != 0:\\n             self.dfs(n-left, 0, 0, path)\\n \\n         if left > right:\\n             self.dfs(n, left+1, right, path+'(')\\n             self.dfs(n, left, right+1, path+')')\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return [\\\"\\\"]\\n         if n == 1:\\n             return [\\\"()\\\"]\\n         results = []\\n \\n         for i in range(1, 2*n, 2):\\n             first = self.generateParenthesis(int((i-1)/2))\\n             second = self.generateParenthesis(int((2*n-i-1)/2))\\n             results += [\\\"(\\\"+u+\\\")\\\"+v for u in first for v in second]\\n         return  results\", \"class Solution:\\n     def parenthesis(self, n):\\n         return \\\"(\\\"*n+\\\")\\\"*n\\n     \\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         if n == 0:\\n             return []\\n         elif n == 1:\\n             return [\\\"()\\\"]\\n         \\n         result = []\\n         for i in range(1,n):\\n             r1 = self.generateParenthesis(i)\\n             r2 = self.generateParenthesis(n-i)\\n             for r11 in r1:\\n                 for r21 in r2:\\n                     result.append(r11+r21)\\n                     \\n         r3 = self.generateParenthesis(n-1)\\n         for r31 in r3:\\n             result.append(\\\"({})\\\".format(r31))\\n         \\n         result.sort()\\n         rr = []\\n         last = None\\n         for r in result:\\n             if r == last:\\n                 continue\\n             rr.append(r)\\n             last = r\\n             \\n                 \\n         return rr\\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         \\n         def balance(str):\\n             lst = list(str)\\n             bal = 0\\n             for char in lst:\\n                 if char == \\\"(\\\":\\n                     bal += 1\\n                 else:\\n                     bal -= 1\\n             return bal\\n         \\n         def left(str):\\n             lst = list(str)\\n             sum = 0\\n             for char in lst:\\n                 if char == \\\"(\\\":\\n                     sum += 1\\n             return n - sum\\n         \\n         def right(str):\\n             lst = list(str)\\n             sum = 0\\n             for char in lst:\\n                 if char == \\\")\\\":\\n                     sum += 1\\n             return n - sum\\n         \\n \\n         prev = [\\\"(\\\"]\\n         length = 1\\n         \\n         while length < 2*n:\\n             current = [] \\n                 \\n             for item in prev:\\n                 if left(item) != 0:\\n                     current.append(item+\\\"(\\\")\\n                     \\n                 if right(item) != 0 and balance(item) > 0:\\n                     current.append(item+\\\")\\\")\\n             length += 1\\n             prev = current\\n         \\n         return prev\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         results = []\\n         \\n         def genparen(prefix, opens, closes):\\n             if len(prefix) == 2*n:\\n                 results.append(prefix)\\n                 return\\n             \\n             if opens < n:\\n                 genparen(prefix+\\\"(\\\", opens+1, closes)\\n             if closes < opens:\\n                 genparen(prefix+\\\")\\\", opens, closes+1)\\n         \\n         genparen('', 0, 0)\\n         return results\\n             \\n         \\n         \\n #         paren_map = {}\\n #         paren_map[1] = set([\\\"()\\\"])\\n         \\n #         for i in range(2, n+1):\\n #             paren_map[i] = self.generate_parens(paren_map[i-1])\\n             \\n #         return list(paren_map[n])\\n             \\n             \\n #     def generate_parens(self, st):\\n #         result = set()\\n         \\n #         for elem in st:\\n #             for i in range(len(elem)):\\n #                 candidate = elem[:i] + \\\"()\\\" + elem[i:]\\n #                 result.add(candidate)\\n         \\n #         return result\\n         \\n\", \"class Solution:\\n     \\n     hashMap = {}\\n     outPut = []\\n     \\n     def helper(self, n, string, length, opened, openUsed):\\n         \\n         ## base case\\n         if length == n*2:\\n             if string not in self.outPut:\\n                 self.outPut.append(string)\\n             return\\n         \\n         ## skip if already tried\\n         if string in self.hashMap:\\n             return\\n         else:\\n             self.hashMap[string] = 1\\n         \\n         if opened == 0:\\n             tempOpened = 1\\n             tempOpenUsed = openUsed + 1\\n             self.helper(n,string+'(',length+1,tempOpened,tempOpenUsed)\\n             \\n         if opened > 0:\\n             tempOpened = opened - 1\\n             self.helper(n,string+')',length+1,tempOpened,openUsed)\\n             \\n         if openUsed < n:\\n             tempOpened = opened + 1\\n             tempOpenUsed = openUsed + 1\\n             self.helper(n,string+'(',length+1,tempOpened,tempOpenUsed)\\n     \\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         self.hashMap = {}\\n         self.outPut = []\\n         self.helper(n,\\\"\\\",0,0,0)\\n         \\n         return self.outPut\\n         \\n         \\n\", \"class Solution:\\n     def generateParenthesis(self, n):\\n         \\\"\\\"\\\"\\n         :type n: int\\n         :rtype: List[str]\\n         \\\"\\\"\\\"\\n         # array to keep track of results         \\n         answers = []\\n         for i in range(0,n+1):\\n             answers.append(Solution.catalanHelper(i, answers))\\n             print((i, answers))\\n         \\n         return list(set(answers[n]))\\n         \\n     @staticmethod\\n     # returns parenthesis\\n     def catalanHelper(n, answers):\\n         \\\"\\\"\\\"\\n         : type n : int \\n         : rtype : List[str]\\n         \\\"\\\"\\\"\\n         \\n         if n == 0:\\n             return [\\\"\\\"]\\n         elif n == 1:\\n             return [\\\"()\\\"]\\n         else:\\n             output = []\\n             for i in range(0, n-1):\\n                 front_list = answers[i] \\n                 end_list = answers[n-i-1]\\n                 for front in front_list:\\n                     for end in end_list:\\n                         \\n                         output.append(\\\"(\\\" + front + \\\")\\\" + end )\\n                         output.append(\\\"(\\\" + front  + end +\\\")\\\")\\n \\n             return output\\n         \\n         \\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            1
        ],
        "output": [
            "\"()\""
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "generateParenthesis",
        "starter_code": "\nclass Solution:\n    def generateParenthesis(self, n: int) -> List[str]:\n        ",
        "url": "https://leetcode.com/problems/generate-parentheses/"
    },
    {
        "id": 1275,
        "task_id": 4533,
        "test_case_id": 1,
        "question": "Assume you are an awesome parent and want to give your children some cookies. But, you should give each child at most one cookie. Each child i has a greed factor gi, which is the minimum size of a cookie that the child will be content with; and each cookie j has a size sj. If sj >= gi, we can assign the cookie j to the child i, and the child i will be content. Your goal is to maximize the number of your content children and output the maximum number.\n\n\nNote:\nYou may assume the greed factor is always positive. \nYou cannot assign more than one cookie to one child.\n\n\nExample 1:\n\nInput: [1,2,3], [1,1]\n\nOutput: 1\n\nExplanation: You have 3 children and 2 cookies. The greed factors of 3 children are 1, 2, 3. \nAnd even though you have 2 cookies, since their size is both 1, you could only make the child whose greed factor is 1 content.\nYou need to output 1.\n\n\n\nExample 2:\n\nInput: [1,2], [1,2,3]\n\nOutput: 2\n\nExplanation: You have 2 children and 3 cookies. The greed factors of 2 children are 1, 2. \nYou have 3 cookies and their sizes are big enough to gratify all of the children, \nYou need to output 2.",
        "solutions": "[\"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         res = 0\\n         heapq.heapify(g)\\n         s.sort()\\n         for num in s:\\n             if not g:\\n                 break\\n             elif g[0] <= num:\\n                 res += 1\\n                 heapq.heappop(g)\\n         return res\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         res = 0 \\n         Lg,Ls = len(g),len(s)\\n         i=j=0 \\n         while i<Lg and j<Ls:\\n             if s[j] >= g[i]:\\n                 res += 1\\n                 j += 1\\n                 i += 1\\n             else:\\n                 j += 1\\n         return res\\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         i,j = 0,0\\n         happyKids = 0\\n         while i < len(g) and j < len(s):\\n             if s[j] >= g[i]:\\n                 happyKids += 1\\n                 i += 1\\n             j += 1\\n         return happyKids\\n         \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort(), s.sort()\\n         count = 0\\n         i = 0\\n         while count < len(g) and i < len(s):\\n             if s[i] >= g[count]:\\n                 count += 1\\n             i+=1\\n         return count\\n  \\n         \\n         \\n                 \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g = sorted(g)\\n         s = sorted(s)\\n         result = 0\\n         i = 0\\n         j = 0\\n         while i<len(g) and j< len(s):\\n             if g[i]<= s[j]:\\n                 result = result + 1\\n                 i = i+1\\n                 j = j+1\\n             else:\\n                 j = j+1\\n         return result\\n         \\n                 \\n\", \"class Solution:\\n #     def findContentChildren(self, g, s):\\n #         \\\"\\\"\\\"\\n #         :type g: List[int]\\n #         :type s: List[int]\\n #         :rtype: int\\n #         \\\"\\\"\\\"\\n #         total = 1\\n #         child_to_give = []\\n #         child_index = 0\\n #         s.sort()\\n #         g.sort()\\n #         for cookie in s:\\n #             for child_greed in g:\\n #                 print(\\\"child_greed \\\" + str(child_greed))\\n #                 print(\\\"cookie \\\" + str(cookie))\\n #                 if child_greed <= cookie:\\n #                     if child_index not in child_to_give:\\n #                         child_to_give.append(child_index)\\n #                         g.remove(child_greed)  \\n \\n #                 child_index += 1        \\n         \\n #         return len(child_to_give)\\n \\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         s.sort()\\n         g.sort()\\n         total = 1\\n         child_to_give = []\\n         child_index = 0\\n \\n         start_i = 0\\n         for j in range(0, len(g)):\\n             for i in range(start_i, len(s)):      \\n                 if s[i] >= g[j]:\\n                     child_to_give.append(j)\\n                     start_i = i + 1\\n                     break\\n \\n             \\n         print(child_to_give)\\n         return len(child_to_give)\\n     \\n #     def findContentChildrenSlow(self, g, s):\\n #         \\\"\\\"\\\"\\n #         :type g: List[int]\\n #         :type s: List[int]\\n #         :rtype: int\\n #         \\\"\\\"\\\"\\n #         s.sort()\\n #         g.sort()\\n #         total = 1\\n #         child_to_give = []\\n #         child_index = 0\\n #         for child_greed in g:\\n #             # Find children greed less than or equal to cookie value\\n #             for cookie in s:\\n #                 if child_greed <= cookie:\\n #                     # print(cookie)\\n #                     # print(child_greed)\\n #                     if child_index not in child_to_give:\\n #                         child_to_give.append(child_index)\\n #                         # g.remove(child_greed)  \\n #                         s.remove(cookie)\\n #                         # print(g)\\n #                         # print(s)\\n #                         # break\\n #             child_index += 1\\n             \\n #         return len(child_to_give)\\n #         # return 2\\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         print(g)\\n         print(s)\\n         \\n         count_child = 0\\n         count_cookie = 0\\n         \\n         while(count_child < len(g) and count_cookie < len(s)):\\n             if (g[count_child] <= s[count_cookie]):\\n                 count_child += 1\\n             count_cookie += 1\\n             \\n         return count_child\\n     \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         \\n         index, result = 0, 0\\n         while index < len(s) and len(g) > 0:\\n             if s[index] >= g[0]:\\n                 result += 1\\n                 index += 1\\n                 g.remove(g[0])\\n             else:\\n                 index += 1\\n         return result\\n             \\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                2,
                3
            ],
            [
                1,
                1
            ]
        ],
        "output": 0,
        "halu_type": "Identification Hallucination",
        "fn_name": "findContentChildren",
        "starter_code": "\nclass Solution:\n    def findContentChildren(self, g: List[int], s: List[int]) -> int:\n        ",
        "url": "https://leetcode.com/problems/assign-cookies/"
    },
    {
        "id": 1276,
        "task_id": 4533,
        "test_case_id": 2,
        "question": "Assume you are an awesome parent and want to give your children some cookies. But, you should give each child at most one cookie. Each child i has a greed factor gi, which is the minimum size of a cookie that the child will be content with; and each cookie j has a size sj. If sj >= gi, we can assign the cookie j to the child i, and the child i will be content. Your goal is to maximize the number of your content children and output the maximum number.\n\n\nNote:\nYou may assume the greed factor is always positive. \nYou cannot assign more than one cookie to one child.\n\n\nExample 1:\n\nInput: [1,2,3], [1,1]\n\nOutput: 1\n\nExplanation: You have 3 children and 2 cookies. The greed factors of 3 children are 1, 2, 3. \nAnd even though you have 2 cookies, since their size is both 1, you could only make the child whose greed factor is 1 content.\nYou need to output 1.\n\n\n\nExample 2:\n\nInput: [1,2], [1,2,3]\n\nOutput: 2\n\nExplanation: You have 2 children and 3 cookies. The greed factors of 2 children are 1, 2. \nYou have 3 cookies and their sizes are big enough to gratify all of the children, \nYou need to output 2.",
        "solutions": "[\"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         res = 0\\n         heapq.heapify(g)\\n         s.sort()\\n         for num in s:\\n             if not g:\\n                 break\\n             elif g[0] <= num:\\n                 res += 1\\n                 heapq.heappop(g)\\n         return res\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         res = 0 \\n         Lg,Ls = len(g),len(s)\\n         i=j=0 \\n         while i<Lg and j<Ls:\\n             if s[j] >= g[i]:\\n                 res += 1\\n                 j += 1\\n                 i += 1\\n             else:\\n                 j += 1\\n         return res\\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         i,j = 0,0\\n         happyKids = 0\\n         while i < len(g) and j < len(s):\\n             if s[j] >= g[i]:\\n                 happyKids += 1\\n                 i += 1\\n             j += 1\\n         return happyKids\\n         \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort(), s.sort()\\n         count = 0\\n         i = 0\\n         while count < len(g) and i < len(s):\\n             if s[i] >= g[count]:\\n                 count += 1\\n             i+=1\\n         return count\\n  \\n         \\n         \\n                 \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g = sorted(g)\\n         s = sorted(s)\\n         result = 0\\n         i = 0\\n         j = 0\\n         while i<len(g) and j< len(s):\\n             if g[i]<= s[j]:\\n                 result = result + 1\\n                 i = i+1\\n                 j = j+1\\n             else:\\n                 j = j+1\\n         return result\\n         \\n                 \\n\", \"class Solution:\\n #     def findContentChildren(self, g, s):\\n #         \\\"\\\"\\\"\\n #         :type g: List[int]\\n #         :type s: List[int]\\n #         :rtype: int\\n #         \\\"\\\"\\\"\\n #         total = 1\\n #         child_to_give = []\\n #         child_index = 0\\n #         s.sort()\\n #         g.sort()\\n #         for cookie in s:\\n #             for child_greed in g:\\n #                 print(\\\"child_greed \\\" + str(child_greed))\\n #                 print(\\\"cookie \\\" + str(cookie))\\n #                 if child_greed <= cookie:\\n #                     if child_index not in child_to_give:\\n #                         child_to_give.append(child_index)\\n #                         g.remove(child_greed)  \\n \\n #                 child_index += 1        \\n         \\n #         return len(child_to_give)\\n \\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         s.sort()\\n         g.sort()\\n         total = 1\\n         child_to_give = []\\n         child_index = 0\\n \\n         start_i = 0\\n         for j in range(0, len(g)):\\n             for i in range(start_i, len(s)):      \\n                 if s[i] >= g[j]:\\n                     child_to_give.append(j)\\n                     start_i = i + 1\\n                     break\\n \\n             \\n         print(child_to_give)\\n         return len(child_to_give)\\n     \\n #     def findContentChildrenSlow(self, g, s):\\n #         \\\"\\\"\\\"\\n #         :type g: List[int]\\n #         :type s: List[int]\\n #         :rtype: int\\n #         \\\"\\\"\\\"\\n #         s.sort()\\n #         g.sort()\\n #         total = 1\\n #         child_to_give = []\\n #         child_index = 0\\n #         for child_greed in g:\\n #             # Find children greed less than or equal to cookie value\\n #             for cookie in s:\\n #                 if child_greed <= cookie:\\n #                     # print(cookie)\\n #                     # print(child_greed)\\n #                     if child_index not in child_to_give:\\n #                         child_to_give.append(child_index)\\n #                         # g.remove(child_greed)  \\n #                         s.remove(cookie)\\n #                         # print(g)\\n #                         # print(s)\\n #                         # break\\n #             child_index += 1\\n             \\n #         return len(child_to_give)\\n #         # return 2\\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         print(g)\\n         print(s)\\n         \\n         count_child = 0\\n         count_cookie = 0\\n         \\n         while(count_child < len(g) and count_cookie < len(s)):\\n             if (g[count_child] <= s[count_cookie]):\\n                 count_child += 1\\n             count_cookie += 1\\n             \\n         return count_child\\n     \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         \\n         index, result = 0, 0\\n         while index < len(s) and len(g) > 0:\\n             if s[index] >= g[0]:\\n                 result += 1\\n                 index += 1\\n                 g.remove(g[0])\\n             else:\\n                 index += 1\\n         return result\\n             \\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                1,
                2,
                3
            ],
            [
                1,
                1
            ]
        ],
        "output": 1,
        "halu_type": "Identification Hallucination",
        "fn_name": "findContentChildren",
        "starter_code": "\nclass Solution:\n    def findContentChildren(self, g: List[int], s: List[int]) -> int:\n        ",
        "url": "https://leetcode.com/problems/assign-cookies/"
    },
    {
        "id": 1277,
        "task_id": 4533,
        "test_case_id": 3,
        "question": "Assume you are an awesome parent and want to give your children some cookies. But, you should give each child at most one cookie. Each child i has a greed factor gi, which is the minimum size of a cookie that the child will be content with; and each cookie j has a size sj. If sj >= gi, we can assign the cookie j to the child i, and the child i will be content. Your goal is to maximize the number of your content children and output the maximum number.\n\n\nNote:\nYou may assume the greed factor is always positive. \nYou cannot assign more than one cookie to one child.\n\n\nExample 1:\n\nInput: [1,2,3], [1,1]\n\nOutput: 1\n\nExplanation: You have 3 children and 2 cookies. The greed factors of 3 children are 1, 2, 3. \nAnd even though you have 2 cookies, since their size is both 1, you could only make the child whose greed factor is 1 content.\nYou need to output 1.\n\n\n\nExample 2:\n\nInput: [1,2], [1,2,3]\n\nOutput: 2\n\nExplanation: You have 2 children and 3 cookies. The greed factors of 2 children are 1, 2. \nYou have 3 cookies and their sizes are big enough to gratify all of the children, \nYou need to output 2.",
        "solutions": "[\"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         res = 0\\n         heapq.heapify(g)\\n         s.sort()\\n         for num in s:\\n             if not g:\\n                 break\\n             elif g[0] <= num:\\n                 res += 1\\n                 heapq.heappop(g)\\n         return res\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         res = 0 \\n         Lg,Ls = len(g),len(s)\\n         i=j=0 \\n         while i<Lg and j<Ls:\\n             if s[j] >= g[i]:\\n                 res += 1\\n                 j += 1\\n                 i += 1\\n             else:\\n                 j += 1\\n         return res\\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         i,j = 0,0\\n         happyKids = 0\\n         while i < len(g) and j < len(s):\\n             if s[j] >= g[i]:\\n                 happyKids += 1\\n                 i += 1\\n             j += 1\\n         return happyKids\\n         \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort(), s.sort()\\n         count = 0\\n         i = 0\\n         while count < len(g) and i < len(s):\\n             if s[i] >= g[count]:\\n                 count += 1\\n             i+=1\\n         return count\\n  \\n         \\n         \\n                 \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g = sorted(g)\\n         s = sorted(s)\\n         result = 0\\n         i = 0\\n         j = 0\\n         while i<len(g) and j< len(s):\\n             if g[i]<= s[j]:\\n                 result = result + 1\\n                 i = i+1\\n                 j = j+1\\n             else:\\n                 j = j+1\\n         return result\\n         \\n                 \\n\", \"class Solution:\\n #     def findContentChildren(self, g, s):\\n #         \\\"\\\"\\\"\\n #         :type g: List[int]\\n #         :type s: List[int]\\n #         :rtype: int\\n #         \\\"\\\"\\\"\\n #         total = 1\\n #         child_to_give = []\\n #         child_index = 0\\n #         s.sort()\\n #         g.sort()\\n #         for cookie in s:\\n #             for child_greed in g:\\n #                 print(\\\"child_greed \\\" + str(child_greed))\\n #                 print(\\\"cookie \\\" + str(cookie))\\n #                 if child_greed <= cookie:\\n #                     if child_index not in child_to_give:\\n #                         child_to_give.append(child_index)\\n #                         g.remove(child_greed)  \\n \\n #                 child_index += 1        \\n         \\n #         return len(child_to_give)\\n \\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         s.sort()\\n         g.sort()\\n         total = 1\\n         child_to_give = []\\n         child_index = 0\\n \\n         start_i = 0\\n         for j in range(0, len(g)):\\n             for i in range(start_i, len(s)):      \\n                 if s[i] >= g[j]:\\n                     child_to_give.append(j)\\n                     start_i = i + 1\\n                     break\\n \\n             \\n         print(child_to_give)\\n         return len(child_to_give)\\n     \\n #     def findContentChildrenSlow(self, g, s):\\n #         \\\"\\\"\\\"\\n #         :type g: List[int]\\n #         :type s: List[int]\\n #         :rtype: int\\n #         \\\"\\\"\\\"\\n #         s.sort()\\n #         g.sort()\\n #         total = 1\\n #         child_to_give = []\\n #         child_index = 0\\n #         for child_greed in g:\\n #             # Find children greed less than or equal to cookie value\\n #             for cookie in s:\\n #                 if child_greed <= cookie:\\n #                     # print(cookie)\\n #                     # print(child_greed)\\n #                     if child_index not in child_to_give:\\n #                         child_to_give.append(child_index)\\n #                         # g.remove(child_greed)  \\n #                         s.remove(cookie)\\n #                         # print(g)\\n #                         # print(s)\\n #                         # break\\n #             child_index += 1\\n             \\n #         return len(child_to_give)\\n #         # return 2\\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         print(g)\\n         print(s)\\n         \\n         count_child = 0\\n         count_cookie = 0\\n         \\n         while(count_child < len(g) and count_cookie < len(s)):\\n             if (g[count_child] <= s[count_cookie]):\\n                 count_child += 1\\n             count_cookie += 1\\n             \\n         return count_child\\n     \\n\", \"class Solution:\\n     def findContentChildren(self, g, s):\\n         \\\"\\\"\\\"\\n         :type g: List[int]\\n         :type s: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         g.sort()\\n         s.sort()\\n         \\n         index, result = 0, 0\\n         while index < len(s) and len(g) > 0:\\n             if s[index] >= g[0]:\\n                 result += 1\\n                 index += 1\\n                 g.remove(g[0])\\n             else:\\n                 index += 1\\n         return result\\n             \\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                1,
                2
            ],
            [
                1,
                2,
                3
            ]
        ],
        "output": 2,
        "halu_type": "Identification Hallucination",
        "fn_name": "findContentChildren",
        "starter_code": "\nclass Solution:\n    def findContentChildren(self, g: List[int], s: List[int]) -> int:\n        ",
        "url": "https://leetcode.com/problems/assign-cookies/"
    },
    {
        "id": 1278,
        "task_id": 2469,
        "test_case_id": 1,
        "question": "Given an integer array of size n, find all elements that appear more than ⌊ n/3 ⌋ times.\n\nNote: The algorithm should run in linear time and in O(1) space.\n\nExample 1:\n\n\nInput: [3,2,3]\nOutput: [3]\n\nExample 2:\n\n\nInput: [1,1,1,3,3,2,2,2]\nOutput: [1,2]",
        "solutions": "[\"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         if nums == []:\\n             return []\\n         \\n         dct = {}\\n         for el in nums:\\n             if el in dct:\\n                 dct[el] += 1\\n             else:\\n                 dct[el] = 1\\n             \\n         fin = []\\n         for key in dct:\\n             if dct[key] > (len(nums) // 3):\\n                 fin.append(key)\\n         \\n         return fin\\n         \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         d = {}\\n         for i in nums:\\n             if i in d:\\n                 d[i] += 1\\n             else:\\n                 d[i] = 1\\n         \\n         l = []\\n         for i in list(d.keys()):\\n             if d[i] > len(nums) // 3:\\n                 l.append(i)\\n                 \\n         return l\\n                 \\n         \\n         \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return([])\\n         \\n         l = len(nums)\\n         candidate1 = candidate2 = 0\\n         count1 = count2 = 0\\n         \\n         for i in nums:\\n             if candidate1 == i:\\n                 count1 += 1\\n             elif candidate2 == i:\\n                 count2 += 1\\n             elif count1 == 0:\\n                 candidate1 = i\\n                 count1 = 1 \\n             elif count2 == 0:\\n                 candidate2 = i\\n                 count2 = 1\\n             else:\\n                 count1 -= 1\\n                 count2 -= 1\\n             \\n         c1 = nums.count(candidate1)\\n         c2 = nums.count(candidate2)\\n     \\n         res = []\\n         if c1 > l/3:\\n             res.append(candidate1)\\n         if c2 > l/3 and candidate1 != candidate2:\\n             res.append(candidate2)\\n         \\n         return(res)\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if len(nums)<=2:\\n             return list(set(nums))\\n         can,can2 = 0,0\\n         num,num2 = 0,1\\n         for i in range(len(nums)):\\n             if nums[i]==can:\\n                 num+=1\\n             elif nums[i] == can2:\\n                 num2 +=2\\n             elif num ==0:\\n                 num=1\\n                 can = nums[i]\\n             elif num2==0:\\n                 num2=1\\n                 can2 = nums[i]\\n             else:\\n                 num,num2=num-1,num2-1\\n         res = []\\n         if nums.count(can)>int(len(nums)/3):\\n             res.append(can)\\n         if nums.count(can2)>int(len(nums)/3) and can2!=can:\\n             res.append(can2)\\n         return res\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         length=len(nums)\\n         ans=[]\\n         for i in set(nums):\\n             if nums.count(i)>length/3:\\n                 ans.append(i)\\n         return ans\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         n1 = n2 = None\\n         c1 = c2 = 0\\n         \\n         for i in nums:\\n             if n1 == i:\\n                 c1 += 1\\n             elif n2 == i:\\n                 c2 += 1\\n             elif c1 == 0:\\n                 n1, c1 = i, 1\\n             elif c2 == 0:\\n                 n2, c2 = i, 1\\n             else:\\n                 c1, c2 = c1 - 1, c2 -1\\n         return [n for n in [n1, n2] if n is not None and nums.count(n) > len(nums)//3]\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ans = []\\n         \\n         if not nums:\\n             return []\\n         \\n         a1 = a2 = None\\n         c1 = c2 = 0\\n         \\n         for num in nums:\\n             if a1 == num:\\n                 c1 += 1\\n             elif a2 == num:\\n                 c2 += 1\\n             elif c1 == 0:\\n                 a1, c1 = num, 1\\n             elif c2 == 0:\\n                 a2, c2 = num, 1\\n             else:\\n                 c1 -= 1\\n                 c2 -= 1\\n         \\n         c1 = c2 = 0\\n         \\n         for num in nums:\\n             if num == a1:\\n                 c1 += 1\\n             elif num == a2:\\n                 c2 += 1\\n         \\n         for a, c in ((a1, c1), (a2, c2)):\\n             if c > len(nums) // 3:\\n                 ans.append(a)\\n         \\n         return ans\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         dic = {}\\n         res = []\\n         n = len(nums)\\n         for i in range(n):\\n             dic[nums[i]] = dic.get(nums[i], 0) + 1\\n         print(dic)\\n         for num, cnt in dic.items():\\n             if cnt > n/3:\\n                 res.append(num)\\n         return res\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ret = []\\n         least = len(nums) / 3\\n         h = {}\\n         for n in nums:\\n             if n in list(h.keys()):\\n                 h[n] += 1\\n             else:\\n                 h[n] = 1\\n                 \\n             if h[n] > least and n not in ret:\\n                 ret.append(n)\\n                 \\n         return ret\\n             \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         N = len(nums)\\n         N3 = N/3\\n         nums.sort()\\n         result = []\\n         if N < 1:\\n             return nums\\n         \\n         _cnt = 0\\n         _num = nums[0]\\n         \\n         for n in nums:\\n             if n != _num:\\n                 if _cnt > N3:\\n                     result.append(_num)\\n                 _cnt = 1\\n                 _num = n\\n             else:\\n                 _cnt += 1\\n         else:\\n             if _cnt > N3:\\n                 result.append(_num)\\n         \\n         \\n         return result\", \"class Solution:\\n     def majorityElement(self, nums):\\n         ctr = collections.Counter()\\n         for n in nums:\\n             ctr[n] += 1\\n             if len(ctr) == 3:\\n                 ctr -= collections.Counter(set(ctr))\\n         return [n for n in ctr if nums.count(n) > len(nums)//3]\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = dict()\\n         results = []\\n         for num in nums:\\n             d[num] = d.get(num,0) + 1\\n             if d[num] > len(nums) // 3:\\n                 if num not in results:\\n                     results.append(num)\\n         return results\"]",
        "difficulty": "interview",
        "input": [
            [
                3,
                2,
                3
            ]
        ],
        "output": [
            3
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "majorityElement",
        "starter_code": "\nclass Solution:\n    def majorityElement(self, nums: List[int]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/majority-element-ii/"
    },
    {
        "id": 1279,
        "task_id": 2469,
        "test_case_id": 2,
        "question": "Given an integer array of size n, find all elements that appear more than ⌊ n/3 ⌋ times.\n\nNote: The algorithm should run in linear time and in O(1) space.\n\nExample 1:\n\n\nInput: [3,2,3]\nOutput: [3]\n\nExample 2:\n\n\nInput: [1,1,1,3,3,2,2,2]\nOutput: [1,2]",
        "solutions": "[\"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         if nums == []:\\n             return []\\n         \\n         dct = {}\\n         for el in nums:\\n             if el in dct:\\n                 dct[el] += 1\\n             else:\\n                 dct[el] = 1\\n             \\n         fin = []\\n         for key in dct:\\n             if dct[key] > (len(nums) // 3):\\n                 fin.append(key)\\n         \\n         return fin\\n         \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         d = {}\\n         for i in nums:\\n             if i in d:\\n                 d[i] += 1\\n             else:\\n                 d[i] = 1\\n         \\n         l = []\\n         for i in list(d.keys()):\\n             if d[i] > len(nums) // 3:\\n                 l.append(i)\\n                 \\n         return l\\n                 \\n         \\n         \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return([])\\n         \\n         l = len(nums)\\n         candidate1 = candidate2 = 0\\n         count1 = count2 = 0\\n         \\n         for i in nums:\\n             if candidate1 == i:\\n                 count1 += 1\\n             elif candidate2 == i:\\n                 count2 += 1\\n             elif count1 == 0:\\n                 candidate1 = i\\n                 count1 = 1 \\n             elif count2 == 0:\\n                 candidate2 = i\\n                 count2 = 1\\n             else:\\n                 count1 -= 1\\n                 count2 -= 1\\n             \\n         c1 = nums.count(candidate1)\\n         c2 = nums.count(candidate2)\\n     \\n         res = []\\n         if c1 > l/3:\\n             res.append(candidate1)\\n         if c2 > l/3 and candidate1 != candidate2:\\n             res.append(candidate2)\\n         \\n         return(res)\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if len(nums)<=2:\\n             return list(set(nums))\\n         can,can2 = 0,0\\n         num,num2 = 0,1\\n         for i in range(len(nums)):\\n             if nums[i]==can:\\n                 num+=1\\n             elif nums[i] == can2:\\n                 num2 +=2\\n             elif num ==0:\\n                 num=1\\n                 can = nums[i]\\n             elif num2==0:\\n                 num2=1\\n                 can2 = nums[i]\\n             else:\\n                 num,num2=num-1,num2-1\\n         res = []\\n         if nums.count(can)>int(len(nums)/3):\\n             res.append(can)\\n         if nums.count(can2)>int(len(nums)/3) and can2!=can:\\n             res.append(can2)\\n         return res\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         length=len(nums)\\n         ans=[]\\n         for i in set(nums):\\n             if nums.count(i)>length/3:\\n                 ans.append(i)\\n         return ans\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         n1 = n2 = None\\n         c1 = c2 = 0\\n         \\n         for i in nums:\\n             if n1 == i:\\n                 c1 += 1\\n             elif n2 == i:\\n                 c2 += 1\\n             elif c1 == 0:\\n                 n1, c1 = i, 1\\n             elif c2 == 0:\\n                 n2, c2 = i, 1\\n             else:\\n                 c1, c2 = c1 - 1, c2 -1\\n         return [n for n in [n1, n2] if n is not None and nums.count(n) > len(nums)//3]\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ans = []\\n         \\n         if not nums:\\n             return []\\n         \\n         a1 = a2 = None\\n         c1 = c2 = 0\\n         \\n         for num in nums:\\n             if a1 == num:\\n                 c1 += 1\\n             elif a2 == num:\\n                 c2 += 1\\n             elif c1 == 0:\\n                 a1, c1 = num, 1\\n             elif c2 == 0:\\n                 a2, c2 = num, 1\\n             else:\\n                 c1 -= 1\\n                 c2 -= 1\\n         \\n         c1 = c2 = 0\\n         \\n         for num in nums:\\n             if num == a1:\\n                 c1 += 1\\n             elif num == a2:\\n                 c2 += 1\\n         \\n         for a, c in ((a1, c1), (a2, c2)):\\n             if c > len(nums) // 3:\\n                 ans.append(a)\\n         \\n         return ans\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         dic = {}\\n         res = []\\n         n = len(nums)\\n         for i in range(n):\\n             dic[nums[i]] = dic.get(nums[i], 0) + 1\\n         print(dic)\\n         for num, cnt in dic.items():\\n             if cnt > n/3:\\n                 res.append(num)\\n         return res\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ret = []\\n         least = len(nums) / 3\\n         h = {}\\n         for n in nums:\\n             if n in list(h.keys()):\\n                 h[n] += 1\\n             else:\\n                 h[n] = 1\\n                 \\n             if h[n] > least and n not in ret:\\n                 ret.append(n)\\n                 \\n         return ret\\n             \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         N = len(nums)\\n         N3 = N/3\\n         nums.sort()\\n         result = []\\n         if N < 1:\\n             return nums\\n         \\n         _cnt = 0\\n         _num = nums[0]\\n         \\n         for n in nums:\\n             if n != _num:\\n                 if _cnt > N3:\\n                     result.append(_num)\\n                 _cnt = 1\\n                 _num = n\\n             else:\\n                 _cnt += 1\\n         else:\\n             if _cnt > N3:\\n                 result.append(_num)\\n         \\n         \\n         return result\", \"class Solution:\\n     def majorityElement(self, nums):\\n         ctr = collections.Counter()\\n         for n in nums:\\n             ctr[n] += 1\\n             if len(ctr) == 3:\\n                 ctr -= collections.Counter(set(ctr))\\n         return [n for n in ctr if nums.count(n) > len(nums)//3]\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = dict()\\n         results = []\\n         for num in nums:\\n             d[num] = d.get(num,0) + 1\\n             if d[num] > len(nums) // 3:\\n                 if num not in results:\\n                     results.append(num)\\n         return results\"]",
        "difficulty": "interview",
        "input": [
            [
                1
            ]
        ],
        "output": [
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "majorityElement",
        "starter_code": "\nclass Solution:\n    def majorityElement(self, nums: List[int]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/majority-element-ii/"
    },
    {
        "id": 1280,
        "task_id": 2469,
        "test_case_id": 3,
        "question": "Given an integer array of size n, find all elements that appear more than ⌊ n/3 ⌋ times.\n\nNote: The algorithm should run in linear time and in O(1) space.\n\nExample 1:\n\n\nInput: [3,2,3]\nOutput: [3]\n\nExample 2:\n\n\nInput: [1,1,1,3,3,2,2,2]\nOutput: [1,2]",
        "solutions": "[\"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         if nums == []:\\n             return []\\n         \\n         dct = {}\\n         for el in nums:\\n             if el in dct:\\n                 dct[el] += 1\\n             else:\\n                 dct[el] = 1\\n             \\n         fin = []\\n         for key in dct:\\n             if dct[key] > (len(nums) // 3):\\n                 fin.append(key)\\n         \\n         return fin\\n         \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         d = {}\\n         for i in nums:\\n             if i in d:\\n                 d[i] += 1\\n             else:\\n                 d[i] = 1\\n         \\n         l = []\\n         for i in list(d.keys()):\\n             if d[i] > len(nums) // 3:\\n                 l.append(i)\\n                 \\n         return l\\n                 \\n         \\n         \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return([])\\n         \\n         l = len(nums)\\n         candidate1 = candidate2 = 0\\n         count1 = count2 = 0\\n         \\n         for i in nums:\\n             if candidate1 == i:\\n                 count1 += 1\\n             elif candidate2 == i:\\n                 count2 += 1\\n             elif count1 == 0:\\n                 candidate1 = i\\n                 count1 = 1 \\n             elif count2 == 0:\\n                 candidate2 = i\\n                 count2 = 1\\n             else:\\n                 count1 -= 1\\n                 count2 -= 1\\n             \\n         c1 = nums.count(candidate1)\\n         c2 = nums.count(candidate2)\\n     \\n         res = []\\n         if c1 > l/3:\\n             res.append(candidate1)\\n         if c2 > l/3 and candidate1 != candidate2:\\n             res.append(candidate2)\\n         \\n         return(res)\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if len(nums)<=2:\\n             return list(set(nums))\\n         can,can2 = 0,0\\n         num,num2 = 0,1\\n         for i in range(len(nums)):\\n             if nums[i]==can:\\n                 num+=1\\n             elif nums[i] == can2:\\n                 num2 +=2\\n             elif num ==0:\\n                 num=1\\n                 can = nums[i]\\n             elif num2==0:\\n                 num2=1\\n                 can2 = nums[i]\\n             else:\\n                 num,num2=num-1,num2-1\\n         res = []\\n         if nums.count(can)>int(len(nums)/3):\\n             res.append(can)\\n         if nums.count(can2)>int(len(nums)/3) and can2!=can:\\n             res.append(can2)\\n         return res\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         length=len(nums)\\n         ans=[]\\n         for i in set(nums):\\n             if nums.count(i)>length/3:\\n                 ans.append(i)\\n         return ans\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         n1 = n2 = None\\n         c1 = c2 = 0\\n         \\n         for i in nums:\\n             if n1 == i:\\n                 c1 += 1\\n             elif n2 == i:\\n                 c2 += 1\\n             elif c1 == 0:\\n                 n1, c1 = i, 1\\n             elif c2 == 0:\\n                 n2, c2 = i, 1\\n             else:\\n                 c1, c2 = c1 - 1, c2 -1\\n         return [n for n in [n1, n2] if n is not None and nums.count(n) > len(nums)//3]\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ans = []\\n         \\n         if not nums:\\n             return []\\n         \\n         a1 = a2 = None\\n         c1 = c2 = 0\\n         \\n         for num in nums:\\n             if a1 == num:\\n                 c1 += 1\\n             elif a2 == num:\\n                 c2 += 1\\n             elif c1 == 0:\\n                 a1, c1 = num, 1\\n             elif c2 == 0:\\n                 a2, c2 = num, 1\\n             else:\\n                 c1 -= 1\\n                 c2 -= 1\\n         \\n         c1 = c2 = 0\\n         \\n         for num in nums:\\n             if num == a1:\\n                 c1 += 1\\n             elif num == a2:\\n                 c2 += 1\\n         \\n         for a, c in ((a1, c1), (a2, c2)):\\n             if c > len(nums) // 3:\\n                 ans.append(a)\\n         \\n         return ans\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         dic = {}\\n         res = []\\n         n = len(nums)\\n         for i in range(n):\\n             dic[nums[i]] = dic.get(nums[i], 0) + 1\\n         print(dic)\\n         for num, cnt in dic.items():\\n             if cnt > n/3:\\n                 res.append(num)\\n         return res\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         ret = []\\n         least = len(nums) / 3\\n         h = {}\\n         for n in nums:\\n             if n in list(h.keys()):\\n                 h[n] += 1\\n             else:\\n                 h[n] = 1\\n                 \\n             if h[n] > least and n not in ret:\\n                 ret.append(n)\\n                 \\n         return ret\\n             \\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         N = len(nums)\\n         N3 = N/3\\n         nums.sort()\\n         result = []\\n         if N < 1:\\n             return nums\\n         \\n         _cnt = 0\\n         _num = nums[0]\\n         \\n         for n in nums:\\n             if n != _num:\\n                 if _cnt > N3:\\n                     result.append(_num)\\n                 _cnt = 1\\n                 _num = n\\n             else:\\n                 _cnt += 1\\n         else:\\n             if _cnt > N3:\\n                 result.append(_num)\\n         \\n         \\n         return result\", \"class Solution:\\n     def majorityElement(self, nums):\\n         ctr = collections.Counter()\\n         for n in nums:\\n             ctr[n] += 1\\n             if len(ctr) == 3:\\n                 ctr -= collections.Counter(set(ctr))\\n         return [n for n in ctr if nums.count(n) > len(nums)//3]\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def majorityElement(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = dict()\\n         results = []\\n         for num in nums:\\n             d[num] = d.get(num,0) + 1\\n             if d[num] > len(nums) // 3:\\n                 if num not in results:\\n                     results.append(num)\\n         return results\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                2
            ]
        ],
        "output": [
            1,
            2
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "majorityElement",
        "starter_code": "\nclass Solution:\n    def majorityElement(self, nums: List[int]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/majority-element-ii/"
    },
    {
        "id": 1281,
        "task_id": 2627,
        "test_case_id": 1,
        "question": "Given a 2D binary matrix filled with 0's and 1's, find the largest rectangle containing only 1's and return its area.\n\nExample:\n\n\nInput:\n[\n  [\"1\",\"0\",\"1\",\"0\",\"0\"],\n  [\"1\",\"0\",\"1\",\"1\",\"1\"],\n  [\"1\",\"1\",\"1\",\"1\",\"1\"],\n  [\"1\",\"0\",\"0\",\"1\",\"0\"]\n]\nOutput: 6",
        "solutions": "[\"class Solution:\\n     def maximalRectangle(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[str]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not matrix or not matrix[0]:\\n             return 0\\n         n = len(matrix[0])\\n         height = [0] * (n + 1)\\n         ans = 0\\n         for row in matrix:\\n             for i in range(n):\\n                 height[i] = height[i] + 1 if row[i] == '1' else 0\\n             stack = [-1]\\n             for i in range(n + 1):\\n                 while height[i] < height[stack[-1]]:\\n                     h = height[stack.pop()]\\n                     w = i - 1 - stack[-1]\\n                     ans = max(ans, h * w)\\n                 stack.append(i)\\n         return ans\", \"class Solution:\\n     def maximalRectangle(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[str]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return 0\\n         m = len(matrix)\\n         n = len(matrix[0])\\n         ans = 0\\n         heights = [0]*(n+1)\\n \\n         for i in range(m):\\n             for j in range(n):\\n                 heights[j] = heights[j] + 1 if matrix[i][j] == '1' else 0\\n             stack = [-1]\\n             for i in range(n + 1):\\n                 while heights[i] < heights[stack[-1]]:\\n                     h = heights[stack.pop()]\\n                     w = i - stack[-1] - 1\\n                     ans = max(ans, w*h)\\n                 stack.append(i)\\n         return ans\\n                 \\n\", \"class Solution:\\n     \\n     def __init__(self):\\n         self.maxarea = 0\\n         \\n     def maximalRectangle(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[str]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n          \\n         if len(matrix) == 0 or len(matrix[0]) == 0:\\n             return 0\\n         \\n         def maxrect(height):\\n             height.append(0)\\n             stack = [-1]\\n             ans = 0\\n             for i in range(len(height)):\\n                 while height[i] < height[stack[-1]]:\\n                     h = height[stack.pop()]\\n                     w = i - stack[-1] - 1\\n                     ans = max(ans, h * w)\\n                 stack.append(i)\\n             height.pop()\\n             return ans\\n \\n \\n             \\n         \\n         abs_max = 0\\n         height = [0]*len(matrix[0])\\n         for i in range(len(matrix)):\\n             for j in range(len(matrix[0])):\\n                 if matrix[i][j] == \\\"0\\\":\\n                     height[j] = 0\\n                 else:\\n                     height[j] += 1\\n             local_max = maxrect(height)\\n             if local_max > abs_max:\\n                 abs_max = local_max\\n         \\n         return abs_max\\n         \\n         \\n                 \\n         \\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    "1",
                    "0",
                    "1",
                    "0",
                    "0"
                ],
                [
                    "1",
                    "0",
                    "1",
                    "1",
                    "1"
                ],
                [
                    "1",
                    "1",
                    "1",
                    "1",
                    "1"
                ],
                [
                    "1",
                    "0",
                    "0",
                    "1",
                    "0"
                ]
            ]
        ],
        "output": 6,
        "halu_type": "Identification Hallucination",
        "fn_name": "maximalRectangle",
        "starter_code": "\nclass Solution:\n    def maximalRectangle(self, matrix: List[List[str]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/maximal-rectangle/"
    },
    {
        "id": 1282,
        "task_id": 2627,
        "test_case_id": 2,
        "question": "Given a 2D binary matrix filled with 0's and 1's, find the largest rectangle containing only 1's and return its area.\n\nExample:\n\n\nInput:\n[\n  [\"1\",\"0\",\"1\",\"0\",\"0\"],\n  [\"1\",\"0\",\"1\",\"1\",\"1\"],\n  [\"1\",\"1\",\"1\",\"1\",\"1\"],\n  [\"1\",\"0\",\"0\",\"1\",\"0\"]\n]\nOutput: 6",
        "solutions": "[\"class Solution:\\n     def maximalRectangle(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[str]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not matrix or not matrix[0]:\\n             return 0\\n         n = len(matrix[0])\\n         height = [0] * (n + 1)\\n         ans = 0\\n         for row in matrix:\\n             for i in range(n):\\n                 height[i] = height[i] + 1 if row[i] == '1' else 0\\n             stack = [-1]\\n             for i in range(n + 1):\\n                 while height[i] < height[stack[-1]]:\\n                     h = height[stack.pop()]\\n                     w = i - 1 - stack[-1]\\n                     ans = max(ans, h * w)\\n                 stack.append(i)\\n         return ans\", \"class Solution:\\n     def maximalRectangle(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[str]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not matrix:\\n             return 0\\n         m = len(matrix)\\n         n = len(matrix[0])\\n         ans = 0\\n         heights = [0]*(n+1)\\n \\n         for i in range(m):\\n             for j in range(n):\\n                 heights[j] = heights[j] + 1 if matrix[i][j] == '1' else 0\\n             stack = [-1]\\n             for i in range(n + 1):\\n                 while heights[i] < heights[stack[-1]]:\\n                     h = heights[stack.pop()]\\n                     w = i - stack[-1] - 1\\n                     ans = max(ans, w*h)\\n                 stack.append(i)\\n         return ans\\n                 \\n\", \"class Solution:\\n     \\n     def __init__(self):\\n         self.maxarea = 0\\n         \\n     def maximalRectangle(self, matrix):\\n         \\\"\\\"\\\"\\n         :type matrix: List[List[str]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n          \\n         if len(matrix) == 0 or len(matrix[0]) == 0:\\n             return 0\\n         \\n         def maxrect(height):\\n             height.append(0)\\n             stack = [-1]\\n             ans = 0\\n             for i in range(len(height)):\\n                 while height[i] < height[stack[-1]]:\\n                     h = height[stack.pop()]\\n                     w = i - stack[-1] - 1\\n                     ans = max(ans, h * w)\\n                 stack.append(i)\\n             height.pop()\\n             return ans\\n \\n \\n             \\n         \\n         abs_max = 0\\n         height = [0]*len(matrix[0])\\n         for i in range(len(matrix)):\\n             for j in range(len(matrix[0])):\\n                 if matrix[i][j] == \\\"0\\\":\\n                     height[j] = 0\\n                 else:\\n                     height[j] += 1\\n             local_max = maxrect(height)\\n             if local_max > abs_max:\\n                 abs_max = local_max\\n         \\n         return abs_max\\n         \\n         \\n                 \\n         \\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    "0",
                    "0"
                ]
            ]
        ],
        "output": 0,
        "halu_type": "Identification Hallucination",
        "fn_name": "maximalRectangle",
        "starter_code": "\nclass Solution:\n    def maximalRectangle(self, matrix: List[List[str]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/maximal-rectangle/"
    },
    {
        "id": 1283,
        "task_id": 2630,
        "test_case_id": 1,
        "question": "A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).\n\nThe robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).\n\nNow consider if some obstacles are added to the grids. How many unique paths would there be?\n\n\n\nAn obstacle and empty space is marked as 1 and 0 respectively in the grid.\n\nNote: m and n will be at most 100.\n\nExample 1:\n\n\nInput:\n[\n  [0,0,0],\n  [0,1,0],\n  [0,0,0]\n]\nOutput: 2\nExplanation:\nThere is one obstacle in the middle of the 3x3 grid above.\nThere are two ways to reach the bottom-right corner:\n1. Right -> Right -> Down -> Down\n2. Down -> Down -> Right -> Right",
        "solutions": "[\"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid) #row\\n         n = len(obstacleGrid[0]) #col\\n         path = [[0 for j in range(n)] for i in range(m)]\\n         for i in range(m):\\n             if obstacleGrid[i][0] == 0:\\n                 path[i][0] = 1\\n             else:\\n                 break\\n         for i in range(n):\\n             if obstacleGrid[0][i] == 0:\\n                 path[0][i] = 1\\n             else:\\n                 break\\n         for i in range(1,m):\\n             for j in range(1,n):\\n                 if obstacleGrid[i][j] != 1:\\n                     path[i][j] = path[i-1][j] + path[i][j-1]\\n                 else:\\n                     path[i][j] = 0\\n         return path[m-1][n-1]\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         if not obstacleGrid or not obstacleGrid[0]:\\n             return 0\\n \\n         m, n = len(obstacleGrid), len(obstacleGrid[0])\\n \\n         # DP:\\n         #   dp(i, j):   no. of unique paths to obstacleGrid[i][j]\\n         #   bound:\\n         #       dp(0, 0) = 0 if obstacleGrid[0][0] == 1 else 1\\n         #       dp(-1, *) = 0\\n         #       dp(*, -1) =  0\\n         #   progress:\\n         #       dp(i, j) = | 0,             if obstacleGrid[i][j] == 1\\n         #                  | dp(i-1, j) + dp(i, j-1),           else\\n \\n         dp = [0] * n\\n         for i in range(m):\\n             for j in range(n):\\n                 if obstacleGrid[i][j] == 1:\\n                     dp[j] = 0\\n                 else:       \\n                     if i == 0:\\n                         if j == 0:\\n                             dp[j] = 1\\n                         else:\\n                             dp[j] = dp[j-1]\\n                     else:\\n                         if j > 0:\\n                             dp[j] = dp[j] + dp[j-1]\\n                             \\n         \\n         return dp[n-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not obstacleGrid:\\n             return 0\\n         if obstacleGrid[-1][-1] == 1:\\n             return 0\\n         dp = []\\n         for each in obstacleGrid:\\n             temp = each[:]\\n             dp.append(temp)\\n \\n         for i in range(len(dp[0])):\\n             if obstacleGrid[0][i] == 1:\\n                 # dp[0][i] = 0\\n                 break\\n             else:\\n                 dp[0][i] = 1\\n         for j in range(len(dp)):\\n             if obstacleGrid[j][0] == 1:\\n                 break\\n             else:\\n                 dp[j][0] = 1\\n         \\n         # print(dp,obstacleGrid)\\n         \\n         for row in range(1, len(obstacleGrid)):\\n             for col in range(1, len(obstacleGrid[0])):\\n                 if obstacleGrid[row][col] == 0:\\n                     if obstacleGrid[row - 1][col] != 1:\\n                         dp[row][col] += dp[row - 1][col]\\n                     if obstacleGrid[row][col - 1] != 1:\\n                         dp[row][col] += dp[row][col - 1]\\n         print(dp)\\n         return dp[-1][-1]\", \"class Solution:\\n     def paths(self,obstacleGrid,n,m,a,b,memo):\\n         if n>a or m>b:\\n             return 0\\n \\n         if obstacleGrid[n][m] == 1:\\n             return 0\\n \\n         if n==a and m==b:\\n             return 1\\n \\n         if str(n)+\\\" \\\"+str(m) not in memo:\\n             memo[str(n)+\\\" \\\"+str(m)]=self.paths(obstacleGrid,n+1,m,a,b,memo)+self.paths(obstacleGrid,n,m+1,a,b,memo)\\n \\n         return memo[str(n)+\\\" \\\"+str(m)]\\n \\n         \\n         \\n         \\n         \\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         a = len(obstacleGrid)\\n         b= len(obstacleGrid[0])\\n         memo = dict()\\n         return  self.paths(obstacleGrid,0,0,a-1,b-1,memo)\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if len(obstacleGrid)==0 and len(obstacleGrid[0])==0:\\n             return 1\\n         if len(obstacleGrid)==0:\\n             return 0\\n         \\n         \\n         ob=False\\n         for i in range(len(obstacleGrid)):\\n             for j in range(len(obstacleGrid[i])):\\n                 if obstacleGrid[i][j]==1:\\n                     obstacleGrid[i][j]=None\\n                     ob=True\\n                     \\n         for i in range(len(obstacleGrid)):            \\n             if obstacleGrid[i][0]==0:\\n                 obstacleGrid[i][0]=1\\n             if obstacleGrid[i][0]==None:\\n                 break\\n                 \\n         for i in range(len(obstacleGrid[0])):            \\n             if obstacleGrid[0][i]==0:\\n                 obstacleGrid[0][i]=1\\n             if obstacleGrid[0][i]==None:\\n                 break\\n                 \\n         for i in range(1,len(obstacleGrid)):\\n             for j in range(1,len(obstacleGrid[i])):\\n                 if obstacleGrid[i][j]!=None:\\n                     if obstacleGrid[i-1][j]==None and obstacleGrid[i][j-1]==None:\\n                         obstacleGrid[i][j]=None\\n                     elif obstacleGrid[i-1][j]==None and obstacleGrid[i][j-1]!=None:\\n                         obstacleGrid[i][j]=obstacleGrid[i][j-1]\\n                     elif obstacleGrid[i-1][j]!=None and obstacleGrid[i][j-1]==None:\\n                         obstacleGrid[i][j]=obstacleGrid[i-1][j]\\n                     else:\\n                         obstacleGrid[i][j]=obstacleGrid[i-1][j]+obstacleGrid[i][j-1]\\n                 \\n         if len(obstacleGrid)==1 or len(obstacleGrid[0])==1:\\n             if ob:\\n                 return 0\\n             else:\\n                 return 1\\n         \\n         if obstacleGrid[-1][-1]==None or obstacleGrid[0][0]==None:\\n             return 0\\n         \\n         return obstacleGrid[-1][-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         1 1 0\\n         1 1 0\\n         1 1 0\\n         \\\"\\\"\\\"\\n         if not any(obstacleGrid):\\n             return 0\\n \\n         n, m = len(obstacleGrid), len(obstacleGrid[0])\\n         dp = [[1 if i == 0 or j == 0 else 1 for j in range(m)] for i in range(n)]\\n \\n         # Set first column of dp.\\n         obs = False\\n         first_col = [i[0] for i in obstacleGrid]\\n \\n         try:\\n             idx = first_col.index(1)\\n             for i in range(n):\\n                 dp[i][0] = 1 if i < idx else 0\\n         except ValueError:\\n             pass\\n \\n         # Set first row of dp.\\n         try:\\n             first_row = obstacleGrid[0]\\n             idx = first_row.index(1)\\n             dp[0] = [1] * idx + [0] * (m - idx)\\n         except ValueError:\\n             pass\\n \\n         print(obstacleGrid)\\n         for i in range(1, n):\\n             for j in range(1, m):\\n                 print(i, j)\\n                 if obstacleGrid[i][j] == 1:\\n                     dp[i][j] = 0\\n                 else:\\n                     dp[i][j] = dp[i-1][j] + dp[i][j-1]\\n         return dp[n-1][m-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         P = [[0 for j in range(n+1)] for i in range(m+1)]\\n         \\n         P[0][1] = 1\\n         \\n         for i in range(1,m+1):\\n             for j in range(1, n+1):\\n                 if obstacleGrid[i-1][j-1] != 1:\\n                     P[i][j] = P[i-1][j] + P[i][j-1]\\n         return P[m][n]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         ResGrid = [[0 for x in range(n+1)] for x in range(m+1)]\\n         ResGrid[0][1] = 1\\n \\n         for i in range(1, m+1):\\n             for j in range(1, n+1):\\n                 if not obstacleGrid[i-1][j-1]:\\n                     ResGrid[i][j] = ResGrid[i][j-1]+ResGrid[i-1][j]\\n \\n         return ResGrid[m][n]\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         dp = [0] * n  # \\u8fd9\\u91cc\\u5148\\u586b\\u5145\\u4e860\\uff0c\\u8fd9\\u6837\\u5f53i==0\\u65f6\\u4e5f\\u53ef\\u4ee5dp[j] += dp[j-1]\\n         for i in range(m):\\n             for j in range(n):\\n                 if obstacleGrid[i][j] == 1:  # \\u9047\\u5230\\u969c\\u788d\\u7269\\u76f4\\u63a5\\u4e3a0\\n                     dp[j] = 0\\n                 else:  # \\u6ca1\\u6709\\u9047\\u5230\\u969c\\u788d\\u7269\\n                     if i == 0 and j == 0:  # \\u5de6\\u4e0a\\u89d2\\u603b\\u662f\\u4e3a1\\uff0c\\u4e5f\\u4e3a\\u540e\\u9762i!=0\\u800cj==0\\u7684\\u60c5\\u51b5\\u505a\\u94fa\\u57ab\\n                         dp[j] = 1\\n                     elif j != 0:  # \\u56e0\\u4e3aj==0\\u65f6\\uff0c\\u5e94\\u8be5\\u4e3adp[j] += 0\\uff0c\\u7b49\\u4e8e\\u4e0d\\u53d8\\uff0c\\u6240\\u4ee5continue\\n                         dp[j] += dp[j - 1]\\n         return dp[-1]\", \"class Solution:\\n     def paths(self,obstacleGrid,n,m,a,b,memo):\\n         if n>a or m>b:\\n             return 0\\n \\n         if obstacleGrid[n][m] == 1:\\n             return 0\\n \\n         if n==a and m==b:\\n             return 1\\n \\n         if str(n)+\\\" \\\"+str(m) not in memo:\\n             memo[str(n)+\\\" \\\"+str(m)]=self.paths(obstacleGrid,n+1,m,a,b,memo)+self.paths(obstacleGrid,n,m+1,a,b,memo)\\n \\n         return memo[str(n)+\\\" \\\"+str(m)]\\n \\n         \\n         \\n         \\n         \\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         a = len(obstacleGrid)\\n         b= len(obstacleGrid[0])\\n         memo = dict()\\n         return  self.paths(obstacleGrid,0,0,a-1,b-1,memo)\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not len(obstacleGrid) >0 :\\n             return 0\\n             \\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         arr = [[1 for y in range(n)] for x in range(m)]\\n         for x in range(m):\\n             for y in range(n):\\n                 if x == 0:\\n                     arr[x][y] = arr[x][y-1]\\n                 elif y == 0:\\n                     arr[x][y] = arr[x-1][y]\\n                 else:\\n                     arr[x][y] = arr[x-1][y] + arr[x][y-1]\\n                 if obstacleGrid[x][y] == 1:\\n                     arr[x][y] = 0\\n         return arr[-1][-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         # m * n\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         ways = [[0 for i in range(n)] for j in range(m)]\\n         for i in range(m):\\n             for j in range(n):\\n                 if obstacleGrid[i][j] == 1:\\n                     ways[i][j] = 0\\n                 elif i == 0 and j == 0:\\n                     ways[i][j] = 1\\n                 elif i == 0:\\n                     ways[i][j] = ways[i][j-1]\\n                 elif j == 0:\\n                     ways[i][j] = ways[i-1][j]\\n                 else:\\n                     ways[i][j] = ways[i-1][j] + ways[i][j-1]\\n         return ways[m-1][n-1]\\n                     \\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m, n = len(obstacleGrid), len(obstacleGrid[0])\\n         dp = [[1] * n for _ in range(m)]\\n         for i in range(m):\\n             dp[i][0] = 0 if obstacleGrid[i][0] == 1 else dp[i-1][0]\\n         for j in range(n):\\n             dp[0][j] = 0 if obstacleGrid[0][j] == 1 else dp[0][j-1]\\n         \\n         for i in range(1, m):\\n             for j in range(1, n):\\n                 if obstacleGrid[i][j] == 1:\\n                     dp[i][j] = 0\\n                 else:\\n                     dp[i][j] = dp[i-1][j] + dp[i][j-1]\\n         return dp[m-1][n-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         mat = [[0] * n for _ in range(m)]\\n         if not obstacleGrid[0][0]:\\n             mat[0][0] = 1\\n         for row in range(m):\\n             for col in range(n):\\n                 if col == 0 and row == 0:\\n                     mat[row][col] = obstacleGrid[row][col] * -1 + 1\\n                 elif obstacleGrid[row][col] == 1:\\n                     mat[row][col] == 0\\n                 else:\\n                     mat[row][col] = mat[row - 1][col] + mat[row][col - 1]\\n         return mat[m - 1][n - 1]\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         \\n         dp = [[0 for _ in range(n)] for _ in range(m)]\\n         for i in range(n):\\n             if obstacleGrid[0][i] == 0:\\n                 dp[0][i] = 1\\n             else:\\n                 break\\n         for j in range(m):\\n             if obstacleGrid[j][0] == 0:\\n                 dp[j][0] = 1\\n             else:\\n                 break\\n         \\n         if obstacleGrid[m-1][n-1] == 1:\\n             return 0\\n         \\n         print(dp)\\n \\n         for y in range(1, n):\\n             for x in range(1, m):\\n                 if obstacleGrid[x][y] == 1:\\n                     dp[x][y] = 0\\n                 else: \\n                     dp[x][y] = dp[x - 1][y] + dp[x][y - 1]\\n \\n         return dp[m - 1][n - 1]\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    0,
                    0,
                    0
                ],
                [
                    0,
                    1,
                    0
                ],
                [
                    0,
                    0,
                    0
                ]
            ]
        ],
        "output": 2,
        "halu_type": "Identification Hallucination",
        "fn_name": "uniquePathsWithObstacles",
        "starter_code": "\nclass Solution:\n    def uniquePathsWithObstacles(self, obstacleGrid: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/unique-paths-ii/"
    },
    {
        "id": 1284,
        "task_id": 2630,
        "test_case_id": 2,
        "question": "A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).\n\nThe robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).\n\nNow consider if some obstacles are added to the grids. How many unique paths would there be?\n\n\n\nAn obstacle and empty space is marked as 1 and 0 respectively in the grid.\n\nNote: m and n will be at most 100.\n\nExample 1:\n\n\nInput:\n[\n  [0,0,0],\n  [0,1,0],\n  [0,0,0]\n]\nOutput: 2\nExplanation:\nThere is one obstacle in the middle of the 3x3 grid above.\nThere are two ways to reach the bottom-right corner:\n1. Right -> Right -> Down -> Down\n2. Down -> Down -> Right -> Right",
        "solutions": "[\"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid) #row\\n         n = len(obstacleGrid[0]) #col\\n         path = [[0 for j in range(n)] for i in range(m)]\\n         for i in range(m):\\n             if obstacleGrid[i][0] == 0:\\n                 path[i][0] = 1\\n             else:\\n                 break\\n         for i in range(n):\\n             if obstacleGrid[0][i] == 0:\\n                 path[0][i] = 1\\n             else:\\n                 break\\n         for i in range(1,m):\\n             for j in range(1,n):\\n                 if obstacleGrid[i][j] != 1:\\n                     path[i][j] = path[i-1][j] + path[i][j-1]\\n                 else:\\n                     path[i][j] = 0\\n         return path[m-1][n-1]\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         if not obstacleGrid or not obstacleGrid[0]:\\n             return 0\\n \\n         m, n = len(obstacleGrid), len(obstacleGrid[0])\\n \\n         # DP:\\n         #   dp(i, j):   no. of unique paths to obstacleGrid[i][j]\\n         #   bound:\\n         #       dp(0, 0) = 0 if obstacleGrid[0][0] == 1 else 1\\n         #       dp(-1, *) = 0\\n         #       dp(*, -1) =  0\\n         #   progress:\\n         #       dp(i, j) = | 0,             if obstacleGrid[i][j] == 1\\n         #                  | dp(i-1, j) + dp(i, j-1),           else\\n \\n         dp = [0] * n\\n         for i in range(m):\\n             for j in range(n):\\n                 if obstacleGrid[i][j] == 1:\\n                     dp[j] = 0\\n                 else:       \\n                     if i == 0:\\n                         if j == 0:\\n                             dp[j] = 1\\n                         else:\\n                             dp[j] = dp[j-1]\\n                     else:\\n                         if j > 0:\\n                             dp[j] = dp[j] + dp[j-1]\\n                             \\n         \\n         return dp[n-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not obstacleGrid:\\n             return 0\\n         if obstacleGrid[-1][-1] == 1:\\n             return 0\\n         dp = []\\n         for each in obstacleGrid:\\n             temp = each[:]\\n             dp.append(temp)\\n \\n         for i in range(len(dp[0])):\\n             if obstacleGrid[0][i] == 1:\\n                 # dp[0][i] = 0\\n                 break\\n             else:\\n                 dp[0][i] = 1\\n         for j in range(len(dp)):\\n             if obstacleGrid[j][0] == 1:\\n                 break\\n             else:\\n                 dp[j][0] = 1\\n         \\n         # print(dp,obstacleGrid)\\n         \\n         for row in range(1, len(obstacleGrid)):\\n             for col in range(1, len(obstacleGrid[0])):\\n                 if obstacleGrid[row][col] == 0:\\n                     if obstacleGrid[row - 1][col] != 1:\\n                         dp[row][col] += dp[row - 1][col]\\n                     if obstacleGrid[row][col - 1] != 1:\\n                         dp[row][col] += dp[row][col - 1]\\n         print(dp)\\n         return dp[-1][-1]\", \"class Solution:\\n     def paths(self,obstacleGrid,n,m,a,b,memo):\\n         if n>a or m>b:\\n             return 0\\n \\n         if obstacleGrid[n][m] == 1:\\n             return 0\\n \\n         if n==a and m==b:\\n             return 1\\n \\n         if str(n)+\\\" \\\"+str(m) not in memo:\\n             memo[str(n)+\\\" \\\"+str(m)]=self.paths(obstacleGrid,n+1,m,a,b,memo)+self.paths(obstacleGrid,n,m+1,a,b,memo)\\n \\n         return memo[str(n)+\\\" \\\"+str(m)]\\n \\n         \\n         \\n         \\n         \\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         a = len(obstacleGrid)\\n         b= len(obstacleGrid[0])\\n         memo = dict()\\n         return  self.paths(obstacleGrid,0,0,a-1,b-1,memo)\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if len(obstacleGrid)==0 and len(obstacleGrid[0])==0:\\n             return 1\\n         if len(obstacleGrid)==0:\\n             return 0\\n         \\n         \\n         ob=False\\n         for i in range(len(obstacleGrid)):\\n             for j in range(len(obstacleGrid[i])):\\n                 if obstacleGrid[i][j]==1:\\n                     obstacleGrid[i][j]=None\\n                     ob=True\\n                     \\n         for i in range(len(obstacleGrid)):            \\n             if obstacleGrid[i][0]==0:\\n                 obstacleGrid[i][0]=1\\n             if obstacleGrid[i][0]==None:\\n                 break\\n                 \\n         for i in range(len(obstacleGrid[0])):            \\n             if obstacleGrid[0][i]==0:\\n                 obstacleGrid[0][i]=1\\n             if obstacleGrid[0][i]==None:\\n                 break\\n                 \\n         for i in range(1,len(obstacleGrid)):\\n             for j in range(1,len(obstacleGrid[i])):\\n                 if obstacleGrid[i][j]!=None:\\n                     if obstacleGrid[i-1][j]==None and obstacleGrid[i][j-1]==None:\\n                         obstacleGrid[i][j]=None\\n                     elif obstacleGrid[i-1][j]==None and obstacleGrid[i][j-1]!=None:\\n                         obstacleGrid[i][j]=obstacleGrid[i][j-1]\\n                     elif obstacleGrid[i-1][j]!=None and obstacleGrid[i][j-1]==None:\\n                         obstacleGrid[i][j]=obstacleGrid[i-1][j]\\n                     else:\\n                         obstacleGrid[i][j]=obstacleGrid[i-1][j]+obstacleGrid[i][j-1]\\n                 \\n         if len(obstacleGrid)==1 or len(obstacleGrid[0])==1:\\n             if ob:\\n                 return 0\\n             else:\\n                 return 1\\n         \\n         if obstacleGrid[-1][-1]==None or obstacleGrid[0][0]==None:\\n             return 0\\n         \\n         return obstacleGrid[-1][-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         1 1 0\\n         1 1 0\\n         1 1 0\\n         \\\"\\\"\\\"\\n         if not any(obstacleGrid):\\n             return 0\\n \\n         n, m = len(obstacleGrid), len(obstacleGrid[0])\\n         dp = [[1 if i == 0 or j == 0 else 1 for j in range(m)] for i in range(n)]\\n \\n         # Set first column of dp.\\n         obs = False\\n         first_col = [i[0] for i in obstacleGrid]\\n \\n         try:\\n             idx = first_col.index(1)\\n             for i in range(n):\\n                 dp[i][0] = 1 if i < idx else 0\\n         except ValueError:\\n             pass\\n \\n         # Set first row of dp.\\n         try:\\n             first_row = obstacleGrid[0]\\n             idx = first_row.index(1)\\n             dp[0] = [1] * idx + [0] * (m - idx)\\n         except ValueError:\\n             pass\\n \\n         print(obstacleGrid)\\n         for i in range(1, n):\\n             for j in range(1, m):\\n                 print(i, j)\\n                 if obstacleGrid[i][j] == 1:\\n                     dp[i][j] = 0\\n                 else:\\n                     dp[i][j] = dp[i-1][j] + dp[i][j-1]\\n         return dp[n-1][m-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         P = [[0 for j in range(n+1)] for i in range(m+1)]\\n         \\n         P[0][1] = 1\\n         \\n         for i in range(1,m+1):\\n             for j in range(1, n+1):\\n                 if obstacleGrid[i-1][j-1] != 1:\\n                     P[i][j] = P[i-1][j] + P[i][j-1]\\n         return P[m][n]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         ResGrid = [[0 for x in range(n+1)] for x in range(m+1)]\\n         ResGrid[0][1] = 1\\n \\n         for i in range(1, m+1):\\n             for j in range(1, n+1):\\n                 if not obstacleGrid[i-1][j-1]:\\n                     ResGrid[i][j] = ResGrid[i][j-1]+ResGrid[i-1][j]\\n \\n         return ResGrid[m][n]\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         dp = [0] * n  # \\u8fd9\\u91cc\\u5148\\u586b\\u5145\\u4e860\\uff0c\\u8fd9\\u6837\\u5f53i==0\\u65f6\\u4e5f\\u53ef\\u4ee5dp[j] += dp[j-1]\\n         for i in range(m):\\n             for j in range(n):\\n                 if obstacleGrid[i][j] == 1:  # \\u9047\\u5230\\u969c\\u788d\\u7269\\u76f4\\u63a5\\u4e3a0\\n                     dp[j] = 0\\n                 else:  # \\u6ca1\\u6709\\u9047\\u5230\\u969c\\u788d\\u7269\\n                     if i == 0 and j == 0:  # \\u5de6\\u4e0a\\u89d2\\u603b\\u662f\\u4e3a1\\uff0c\\u4e5f\\u4e3a\\u540e\\u9762i!=0\\u800cj==0\\u7684\\u60c5\\u51b5\\u505a\\u94fa\\u57ab\\n                         dp[j] = 1\\n                     elif j != 0:  # \\u56e0\\u4e3aj==0\\u65f6\\uff0c\\u5e94\\u8be5\\u4e3adp[j] += 0\\uff0c\\u7b49\\u4e8e\\u4e0d\\u53d8\\uff0c\\u6240\\u4ee5continue\\n                         dp[j] += dp[j - 1]\\n         return dp[-1]\", \"class Solution:\\n     def paths(self,obstacleGrid,n,m,a,b,memo):\\n         if n>a or m>b:\\n             return 0\\n \\n         if obstacleGrid[n][m] == 1:\\n             return 0\\n \\n         if n==a and m==b:\\n             return 1\\n \\n         if str(n)+\\\" \\\"+str(m) not in memo:\\n             memo[str(n)+\\\" \\\"+str(m)]=self.paths(obstacleGrid,n+1,m,a,b,memo)+self.paths(obstacleGrid,n,m+1,a,b,memo)\\n \\n         return memo[str(n)+\\\" \\\"+str(m)]\\n \\n         \\n         \\n         \\n         \\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         a = len(obstacleGrid)\\n         b= len(obstacleGrid[0])\\n         memo = dict()\\n         return  self.paths(obstacleGrid,0,0,a-1,b-1,memo)\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not len(obstacleGrid) >0 :\\n             return 0\\n             \\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         arr = [[1 for y in range(n)] for x in range(m)]\\n         for x in range(m):\\n             for y in range(n):\\n                 if x == 0:\\n                     arr[x][y] = arr[x][y-1]\\n                 elif y == 0:\\n                     arr[x][y] = arr[x-1][y]\\n                 else:\\n                     arr[x][y] = arr[x-1][y] + arr[x][y-1]\\n                 if obstacleGrid[x][y] == 1:\\n                     arr[x][y] = 0\\n         return arr[-1][-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         # m * n\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         ways = [[0 for i in range(n)] for j in range(m)]\\n         for i in range(m):\\n             for j in range(n):\\n                 if obstacleGrid[i][j] == 1:\\n                     ways[i][j] = 0\\n                 elif i == 0 and j == 0:\\n                     ways[i][j] = 1\\n                 elif i == 0:\\n                     ways[i][j] = ways[i][j-1]\\n                 elif j == 0:\\n                     ways[i][j] = ways[i-1][j]\\n                 else:\\n                     ways[i][j] = ways[i-1][j] + ways[i][j-1]\\n         return ways[m-1][n-1]\\n                     \\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m, n = len(obstacleGrid), len(obstacleGrid[0])\\n         dp = [[1] * n for _ in range(m)]\\n         for i in range(m):\\n             dp[i][0] = 0 if obstacleGrid[i][0] == 1 else dp[i-1][0]\\n         for j in range(n):\\n             dp[0][j] = 0 if obstacleGrid[0][j] == 1 else dp[0][j-1]\\n         \\n         for i in range(1, m):\\n             for j in range(1, n):\\n                 if obstacleGrid[i][j] == 1:\\n                     dp[i][j] = 0\\n                 else:\\n                     dp[i][j] = dp[i-1][j] + dp[i][j-1]\\n         return dp[m-1][n-1]\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         mat = [[0] * n for _ in range(m)]\\n         if not obstacleGrid[0][0]:\\n             mat[0][0] = 1\\n         for row in range(m):\\n             for col in range(n):\\n                 if col == 0 and row == 0:\\n                     mat[row][col] = obstacleGrid[row][col] * -1 + 1\\n                 elif obstacleGrid[row][col] == 1:\\n                     mat[row][col] == 0\\n                 else:\\n                     mat[row][col] = mat[row - 1][col] + mat[row][col - 1]\\n         return mat[m - 1][n - 1]\\n\", \"class Solution:\\n     def uniquePathsWithObstacles(self, obstacleGrid):\\n         \\\"\\\"\\\"\\n         :type obstacleGrid: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(obstacleGrid)\\n         n = len(obstacleGrid[0])\\n         \\n         dp = [[0 for _ in range(n)] for _ in range(m)]\\n         for i in range(n):\\n             if obstacleGrid[0][i] == 0:\\n                 dp[0][i] = 1\\n             else:\\n                 break\\n         for j in range(m):\\n             if obstacleGrid[j][0] == 0:\\n                 dp[j][0] = 1\\n             else:\\n                 break\\n         \\n         if obstacleGrid[m-1][n-1] == 1:\\n             return 0\\n         \\n         print(dp)\\n \\n         for y in range(1, n):\\n             for x in range(1, m):\\n                 if obstacleGrid[x][y] == 1:\\n                     dp[x][y] = 0\\n                 else: \\n                     dp[x][y] = dp[x - 1][y] + dp[x][y - 1]\\n \\n         return dp[m - 1][n - 1]\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    0,
                    1
                ],
                [
                    0,
                    0
                ]
            ]
        ],
        "output": 1,
        "halu_type": "Identification Hallucination",
        "fn_name": "uniquePathsWithObstacles",
        "starter_code": "\nclass Solution:\n    def uniquePathsWithObstacles(self, obstacleGrid: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/unique-paths-ii/"
    },
    {
        "id": 1285,
        "task_id": 2638,
        "test_case_id": 1,
        "question": "Given a triangle, find the minimum path sum from top to bottom. Each step you may move to adjacent numbers on the row below.\n\nFor example, given the following triangle\n\n\n[\n     [2],\n    [3,4],\n   [6,5,7],\n  [4,1,8,3]\n]\n\n\nThe minimum path sum from top to bottom is 11 (i.e., 2 + 3 + 5 + 1 = 11).\n\nNote:\n\nBonus point if you are able to do this using only O(n) extra space, where n is the total number of rows in the triangle.",
        "solutions": "[\"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         length = len(triangle)\\n         for i in range(length - 1, 0, -1):\\n             for j in range(1, len(triangle[i])):\\n                 if triangle[i][j] < triangle[i][j-1]:\\n                     triangle[i-1][j-1] += triangle[i][j]\\n                 else:\\n                     triangle[i-1][j-1] += triangle[i][j - 1]\\n         return triangle[0][0]\", \"class Solution:\\n     \\\"\\\"\\\"\\n     def minmum(self, triangle, i, j):\\n         if (i + 1) == len(triangle):\\n             return triangle[i][j]\\n         return min(self.minmum(triangle, i+1, j), self.minmum(triangle, i+1, j+1)) + triangle[i][j]\\n     \\\"\\\"\\\"\\n     \\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         #return self.minmum(triangle, 0, 0)\\n         \\n         rows = len(triangle)\\n         minPath = triangle[rows-1]\\n         for i in range(rows-1)[::-1]:\\n             for j in range(i+1):\\n                 minPath[j] = min(minPath[j], minPath[j+1]) + triangle[i][j]\\n         return minPath[0]\\n         \\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         for i in range(1,len(triangle)):\\n             for j in range(i+1):\\n                 if j == 0:\\n                     triangle[i][j] += triangle[i-1][0]\\n                 elif j == i:\\n                     triangle[i][j] += triangle[i-1][-1]\\n                 else:\\n                     triangle[i][j] += min(triangle[i-1][j-1], triangle[i-1][j])\\n         return min(triangle[-1])\\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         for i in range(1,len(triangle)):\\n             triangle[i][0] = triangle[i][0] + triangle[i-1][0]\\n             triangle[i][-1] = triangle[i][-1] + triangle[i-1][-1]\\n             for j in range(1,len(triangle[i])-1):\\n                 triangle[i][j] = min(triangle[i-1][j-1], triangle[i-1][j]) + triangle[i][j]\\n                 \\n         return min(triangle[-1])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         length = len(triangle)\\n         dp = []\\n         for i in range(length):\\n             dp.append([])\\n             for j in range(len(triangle[i])):\\n                 dp[i].append(float('inf'))\\n         dp[0][0] = triangle[0][0]\\n         for i in range(1, length):\\n             dp[i][0] = dp[i-1][0] + triangle[i][0]\\n             dp[i][-1] = dp[i-1][-1] + triangle[i][-1]\\n         \\n         for i in range(1, length):\\n             for j in range(1, len(triangle[i]) - 1):\\n                 dp[i][j] = min(dp[i][j], dp[i-1][j] + triangle[i][j], dp[i-1][j-1] + triangle[i][j])\\n         return min(dp[length-1])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         mini = 100000000\\n         level =0 \\n         while level < (len(triangle)-1):\\n             for i in range(len(triangle[level+1])):\\n                 if i==0 :\\n                     triangle[level+1][i] += triangle[level][i]\\n                 elif i == len(triangle[level]):\\n                     triangle[level+1][i] += triangle[level][i-1]\\n                 else :\\n                     triangle[level+1][i] += min(triangle[level][i-1], triangle[level][i])\\n                     \\n             level +=1\\n         \\n         return  min(triangle[len(triangle)-1])\\n             \\n         \\n \\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     '''\\n     def minimumTotal(self, triangle):\\n         cost = [[0 for j in range(len(i))] for i in triangle]\\n         for row in range(len(triangle)):\\n             for col in range(len(triangle[row])):\\n                 if row == 0:\\n                     cost[row][col] = triangle[row][col]\\n                 else:\\n                     if col-1 >= 0 and col < len(triangle[row-1]):\\n                         cost[row][col] = min(cost[row-1][col-1]+triangle[row][col],cost[row-1][col]+triangle[row][col])\\n                     elif col-1 >= 0:\\n                         cost[row][col] = cost[row-1][col-1]+triangle[row][col]\\n                     elif col < len(triangle[row-1]):\\n                         cost[row][col] = cost[row-1][col]+triangle[row][col]\\n         return min(cost[len(triangle)-1])\\n     '''\\n     #O(n) extra space:\\n     def minimumTotal(self, triangle):\\n         cost = [[0 for j in range(len(triangle))] for i in range(2)]\\n         for row in range(len(triangle)):\\n             for col in range(len(triangle[row])):\\n                 if row == 0:\\n                     cost[row][col] = triangle[row][col]\\n                 else:\\n                     if col-1 >= 0 and col < len(triangle[row-1]):\\n                         cost[row%2][col] = min(cost[(row-1)%2][col-1]+triangle[row][col],cost[(row-1)%2][col]+triangle[row][col])\\n                     elif col-1 >= 0:\\n                         cost[row%2][col] = cost[(row-1)%2][col-1]+triangle[row][col]\\n                     elif col < len(triangle[row-1]):\\n                         cost[row%2][col] = cost[(row-1)%2][col]+triangle[row][col]\\n         return min(cost[(len(triangle)-1)%2])\", \"class Solution:\\n     def nodeSum(self, L, U):\\n         ans = [0]*len(U)\\n         for i,v in enumerate(U):\\n             ans[i] = (v+min(L[i],L[i+1]))\\n         return ans\\n     \\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         L = triangle[-1]\\n         for i in range(len(triangle)-1, 0, -1):\\n             L=self.nodeSum(L, triangle[i-1])\\n         return L[0]\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         row = [triangle[0][0]]\\n         for i in range(1, len(triangle)):\\n             new_row = []\\n             for j in range(len(row)+1):\\n                 new_row.append(triangle[i][j]+min(row[j-1] if j-1>=0 else sys.maxsize, row[j] if j<len(row) else sys.maxsize))\\n             row = new_row\\n         return min(row)\\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         res = [0]\\n         for row in triangle:\\n             res = [row[i]+min([res[j] for j in (i-1, i) if 0<=j<len(res)]) for i in range(len(row))]\\n         return min(res)\\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         import math\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         cost = {(-1, -1): math.inf, (-1,0):0, (-1, 1):math.inf}\\n         for i in range(len(triangle)):\\n             cost[(i, -1)] = math.inf\\n             cost[i, len(triangle[i])] = math.inf\\n \\n         for i in range(len(triangle)):\\n             for j in range(len(triangle[i])):\\n                 cost[(i, j)] = triangle[i][j] + min(cost[(i-1, j)], cost[(i-1, j-1)])\\n                 \\n         return min([cost[(len(triangle)-1, j)] for j in range(len(triangle[-1]))])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         layer_costs = triangle[-1]\\n         \\n         for layer in range(len(triangle)-2, -1, -1): # range(start, end +- 1, step)\\n             for pos in range(len(triangle[layer])):\\n                 min_cost = min(layer_costs[pos], layer_costs[pos+1]) + triangle[layer][pos]\\n                 layer_costs[pos] = min_cost\\n         \\n         return(layer_costs[0])\", \"class Solution:\\n         \\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         memory = []\\n         for i in range(len(triangle)):\\n             memory.append({})\\n         \\n         return self.helper(memory, triangle, 0, 0) \\n \\n     def helper(self, memory, triangle, i, j):\\n         if i >= len(triangle):\\n             return 0 \\n         \\n         if memory[i].get(j) is None:\\n             memory[i][j] = triangle[i][j] + \\\\\\n                            min(self.helper(memory, triangle, i+1, j), self.helper(memory, triangle, i+1, j+1))\\n             \\n         return memory[i][j]\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\\"\\\"\\\"\\n         For each position (column) C within a row R,\\n         the minimum path sum that leads to (r, c) is defined as:\\n         min_path_sum[r][c] = min(min_path_sum[r-1][adj] for adj in adjacent columns of (r,c)) + triangle[r][c]\\n         \\\"\\\"\\\"\\n         \\n         if not triangle or not triangle[0]:\\n             return 0\\n         \\n         min_path_sum = [[0] * len(triangle) for row in range(len(triangle))]\\n         min_path_sum[0][0] = triangle[0][0]\\n         \\n         ROWS = len(triangle)\\n         \\n         \\n         def get_valid_adjacents_of(row, col):\\n             if col == 0:\\n                 return [(row - 1, col)]\\n             if col == row:\\n                 return [(row - 1, col - 1)]\\n             return [(row - 1, col - 1), (row - 1, col)]\\n         \\n         for row in range(1, ROWS):\\n             for col in range(0, row + 1):\\n                 adjacents = get_valid_adjacents_of(row, col)\\n                 min_path_sum[row][col] = triangle[row][col] + min(min_path_sum[r][c] for r, c in adjacents)\\n         \\n         return min(min_path_sum[-1])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not triangle:\\n             return 0\\n         \\n         dp = triangle.pop()\\n         \\n         while triangle:\\n             level = triangle.pop()\\n             \\n             for i in range(len(level)):\\n                 dp[i] = min(dp[i], dp[i + 1]) + level[i]\\n                     \\n         return dp[0]\", \"class Solution:\\n \\tdef minimumTotal(self, triangle):\\n \\t\\tself._min_cache = {} \\n \\t\\tfor i in reversed(list(range(len(triangle)))):\\n \\t\\t\\tfor j in reversed(list(range(len(triangle[i])))):\\n \\t\\t\\t\\tself.get_minimum_sum(triangle, i, j)\\n \\t\\treturn self.get_minimum_sum(triangle, 0, 0)\\n \\n \\tdef get_minimum_sum(self, triangle, level_i, item_j):\\n \\t\\tif ((level_i, item_j) in self._min_cache):\\n \\t\\t\\treturn self._min_cache[(level_i, item_j)]\\n \\t\\tif level_i < 0 or level_i >= len(triangle):\\n \\t\\t\\treturn 0\\n \\t\\tlevel = triangle[level_i]\\n \\n \\t\\tif item_j<0 or item_j>len(level):\\n \\t\\t\\treturn 0\\n \\n \\t\\titem = level[item_j]\\n \\n \\t\\tmin_v = item+min(self.get_minimum_sum(triangle,level_i+1,item_j),\\n \\t\\t\\t\\t\\t    self.get_minimum_sum(triangle,level_i+1,item_j+1))\\n \\t\\tself._min_cache[(level_i, item_j)] = min_v\\n \\n \\t\\treturn min_v\\n \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    2
                ],
                [
                    3,
                    4
                ],
                [
                    6,
                    5,
                    7
                ],
                [
                    4,
                    1,
                    8,
                    3
                ]
            ]
        ],
        "output": 11,
        "halu_type": "Identification Hallucination",
        "fn_name": "minimumTotal",
        "starter_code": "\nclass Solution:\n    def minimumTotal(self, triangle: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/triangle/"
    },
    {
        "id": 1286,
        "task_id": 2638,
        "test_case_id": 2,
        "question": "Given a triangle, find the minimum path sum from top to bottom. Each step you may move to adjacent numbers on the row below.\n\nFor example, given the following triangle\n\n\n[\n     [2],\n    [3,4],\n   [6,5,7],\n  [4,1,8,3]\n]\n\n\nThe minimum path sum from top to bottom is 11 (i.e., 2 + 3 + 5 + 1 = 11).\n\nNote:\n\nBonus point if you are able to do this using only O(n) extra space, where n is the total number of rows in the triangle.",
        "solutions": "[\"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         length = len(triangle)\\n         for i in range(length - 1, 0, -1):\\n             for j in range(1, len(triangle[i])):\\n                 if triangle[i][j] < triangle[i][j-1]:\\n                     triangle[i-1][j-1] += triangle[i][j]\\n                 else:\\n                     triangle[i-1][j-1] += triangle[i][j - 1]\\n         return triangle[0][0]\", \"class Solution:\\n     \\\"\\\"\\\"\\n     def minmum(self, triangle, i, j):\\n         if (i + 1) == len(triangle):\\n             return triangle[i][j]\\n         return min(self.minmum(triangle, i+1, j), self.minmum(triangle, i+1, j+1)) + triangle[i][j]\\n     \\\"\\\"\\\"\\n     \\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         #return self.minmum(triangle, 0, 0)\\n         \\n         rows = len(triangle)\\n         minPath = triangle[rows-1]\\n         for i in range(rows-1)[::-1]:\\n             for j in range(i+1):\\n                 minPath[j] = min(minPath[j], minPath[j+1]) + triangle[i][j]\\n         return minPath[0]\\n         \\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         for i in range(1,len(triangle)):\\n             for j in range(i+1):\\n                 if j == 0:\\n                     triangle[i][j] += triangle[i-1][0]\\n                 elif j == i:\\n                     triangle[i][j] += triangle[i-1][-1]\\n                 else:\\n                     triangle[i][j] += min(triangle[i-1][j-1], triangle[i-1][j])\\n         return min(triangle[-1])\\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\n         for i in range(1,len(triangle)):\\n             triangle[i][0] = triangle[i][0] + triangle[i-1][0]\\n             triangle[i][-1] = triangle[i][-1] + triangle[i-1][-1]\\n             for j in range(1,len(triangle[i])-1):\\n                 triangle[i][j] = min(triangle[i-1][j-1], triangle[i-1][j]) + triangle[i][j]\\n                 \\n         return min(triangle[-1])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         length = len(triangle)\\n         dp = []\\n         for i in range(length):\\n             dp.append([])\\n             for j in range(len(triangle[i])):\\n                 dp[i].append(float('inf'))\\n         dp[0][0] = triangle[0][0]\\n         for i in range(1, length):\\n             dp[i][0] = dp[i-1][0] + triangle[i][0]\\n             dp[i][-1] = dp[i-1][-1] + triangle[i][-1]\\n         \\n         for i in range(1, length):\\n             for j in range(1, len(triangle[i]) - 1):\\n                 dp[i][j] = min(dp[i][j], dp[i-1][j] + triangle[i][j], dp[i-1][j-1] + triangle[i][j])\\n         return min(dp[length-1])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         mini = 100000000\\n         level =0 \\n         while level < (len(triangle)-1):\\n             for i in range(len(triangle[level+1])):\\n                 if i==0 :\\n                     triangle[level+1][i] += triangle[level][i]\\n                 elif i == len(triangle[level]):\\n                     triangle[level+1][i] += triangle[level][i-1]\\n                 else :\\n                     triangle[level+1][i] += min(triangle[level][i-1], triangle[level][i])\\n                     \\n             level +=1\\n         \\n         return  min(triangle[len(triangle)-1])\\n             \\n         \\n \\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     '''\\n     def minimumTotal(self, triangle):\\n         cost = [[0 for j in range(len(i))] for i in triangle]\\n         for row in range(len(triangle)):\\n             for col in range(len(triangle[row])):\\n                 if row == 0:\\n                     cost[row][col] = triangle[row][col]\\n                 else:\\n                     if col-1 >= 0 and col < len(triangle[row-1]):\\n                         cost[row][col] = min(cost[row-1][col-1]+triangle[row][col],cost[row-1][col]+triangle[row][col])\\n                     elif col-1 >= 0:\\n                         cost[row][col] = cost[row-1][col-1]+triangle[row][col]\\n                     elif col < len(triangle[row-1]):\\n                         cost[row][col] = cost[row-1][col]+triangle[row][col]\\n         return min(cost[len(triangle)-1])\\n     '''\\n     #O(n) extra space:\\n     def minimumTotal(self, triangle):\\n         cost = [[0 for j in range(len(triangle))] for i in range(2)]\\n         for row in range(len(triangle)):\\n             for col in range(len(triangle[row])):\\n                 if row == 0:\\n                     cost[row][col] = triangle[row][col]\\n                 else:\\n                     if col-1 >= 0 and col < len(triangle[row-1]):\\n                         cost[row%2][col] = min(cost[(row-1)%2][col-1]+triangle[row][col],cost[(row-1)%2][col]+triangle[row][col])\\n                     elif col-1 >= 0:\\n                         cost[row%2][col] = cost[(row-1)%2][col-1]+triangle[row][col]\\n                     elif col < len(triangle[row-1]):\\n                         cost[row%2][col] = cost[(row-1)%2][col]+triangle[row][col]\\n         return min(cost[(len(triangle)-1)%2])\", \"class Solution:\\n     def nodeSum(self, L, U):\\n         ans = [0]*len(U)\\n         for i,v in enumerate(U):\\n             ans[i] = (v+min(L[i],L[i+1]))\\n         return ans\\n     \\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         L = triangle[-1]\\n         for i in range(len(triangle)-1, 0, -1):\\n             L=self.nodeSum(L, triangle[i-1])\\n         return L[0]\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         row = [triangle[0][0]]\\n         for i in range(1, len(triangle)):\\n             new_row = []\\n             for j in range(len(row)+1):\\n                 new_row.append(triangle[i][j]+min(row[j-1] if j-1>=0 else sys.maxsize, row[j] if j<len(row) else sys.maxsize))\\n             row = new_row\\n         return min(row)\\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         res = [0]\\n         for row in triangle:\\n             res = [row[i]+min([res[j] for j in (i-1, i) if 0<=j<len(res)]) for i in range(len(row))]\\n         return min(res)\\n\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         import math\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         cost = {(-1, -1): math.inf, (-1,0):0, (-1, 1):math.inf}\\n         for i in range(len(triangle)):\\n             cost[(i, -1)] = math.inf\\n             cost[i, len(triangle[i])] = math.inf\\n \\n         for i in range(len(triangle)):\\n             for j in range(len(triangle[i])):\\n                 cost[(i, j)] = triangle[i][j] + min(cost[(i-1, j)], cost[(i-1, j-1)])\\n                 \\n         return min([cost[(len(triangle)-1, j)] for j in range(len(triangle[-1]))])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         layer_costs = triangle[-1]\\n         \\n         for layer in range(len(triangle)-2, -1, -1): # range(start, end +- 1, step)\\n             for pos in range(len(triangle[layer])):\\n                 min_cost = min(layer_costs[pos], layer_costs[pos+1]) + triangle[layer][pos]\\n                 layer_costs[pos] = min_cost\\n         \\n         return(layer_costs[0])\", \"class Solution:\\n         \\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         memory = []\\n         for i in range(len(triangle)):\\n             memory.append({})\\n         \\n         return self.helper(memory, triangle, 0, 0) \\n \\n     def helper(self, memory, triangle, i, j):\\n         if i >= len(triangle):\\n             return 0 \\n         \\n         if memory[i].get(j) is None:\\n             memory[i][j] = triangle[i][j] + \\\\\\n                            min(self.helper(memory, triangle, i+1, j), self.helper(memory, triangle, i+1, j+1))\\n             \\n         return memory[i][j]\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         \\\"\\\"\\\"\\n         For each position (column) C within a row R,\\n         the minimum path sum that leads to (r, c) is defined as:\\n         min_path_sum[r][c] = min(min_path_sum[r-1][adj] for adj in adjacent columns of (r,c)) + triangle[r][c]\\n         \\\"\\\"\\\"\\n         \\n         if not triangle or not triangle[0]:\\n             return 0\\n         \\n         min_path_sum = [[0] * len(triangle) for row in range(len(triangle))]\\n         min_path_sum[0][0] = triangle[0][0]\\n         \\n         ROWS = len(triangle)\\n         \\n         \\n         def get_valid_adjacents_of(row, col):\\n             if col == 0:\\n                 return [(row - 1, col)]\\n             if col == row:\\n                 return [(row - 1, col - 1)]\\n             return [(row - 1, col - 1), (row - 1, col)]\\n         \\n         for row in range(1, ROWS):\\n             for col in range(0, row + 1):\\n                 adjacents = get_valid_adjacents_of(row, col)\\n                 min_path_sum[row][col] = triangle[row][col] + min(min_path_sum[r][c] for r, c in adjacents)\\n         \\n         return min(min_path_sum[-1])\", \"class Solution:\\n     def minimumTotal(self, triangle):\\n         \\\"\\\"\\\"\\n         :type triangle: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not triangle:\\n             return 0\\n         \\n         dp = triangle.pop()\\n         \\n         while triangle:\\n             level = triangle.pop()\\n             \\n             for i in range(len(level)):\\n                 dp[i] = min(dp[i], dp[i + 1]) + level[i]\\n                     \\n         return dp[0]\", \"class Solution:\\n \\tdef minimumTotal(self, triangle):\\n \\t\\tself._min_cache = {} \\n \\t\\tfor i in reversed(list(range(len(triangle)))):\\n \\t\\t\\tfor j in reversed(list(range(len(triangle[i])))):\\n \\t\\t\\t\\tself.get_minimum_sum(triangle, i, j)\\n \\t\\treturn self.get_minimum_sum(triangle, 0, 0)\\n \\n \\tdef get_minimum_sum(self, triangle, level_i, item_j):\\n \\t\\tif ((level_i, item_j) in self._min_cache):\\n \\t\\t\\treturn self._min_cache[(level_i, item_j)]\\n \\t\\tif level_i < 0 or level_i >= len(triangle):\\n \\t\\t\\treturn 0\\n \\t\\tlevel = triangle[level_i]\\n \\n \\t\\tif item_j<0 or item_j>len(level):\\n \\t\\t\\treturn 0\\n \\n \\t\\titem = level[item_j]\\n \\n \\t\\tmin_v = item+min(self.get_minimum_sum(triangle,level_i+1,item_j),\\n \\t\\t\\t\\t\\t    self.get_minimum_sum(triangle,level_i+1,item_j+1))\\n \\t\\tself._min_cache[(level_i, item_j)] = min_v\\n \\n \\t\\treturn min_v\\n \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    -10
                ]
            ]
        ],
        "output": -10,
        "halu_type": "Identification Hallucination",
        "fn_name": "minimumTotal",
        "starter_code": "\nclass Solution:\n    def minimumTotal(self, triangle: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/triangle/"
    },
    {
        "id": 1287,
        "task_id": 2639,
        "test_case_id": 1,
        "question": "Given a collection of integers that might contain duplicates, nums, return all possible subsets (the power set).\n\nNote: The solution set must not contain duplicate subsets.\n\nExample:\n\n\nInput: [1,2,2]\nOutput:\n[\n  [2],\n  [1],\n  [1,2,2],\n  [2,2],\n  [1,2],\n  []\n]",
        "solutions": "[\"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(idx, path):\\n             subsets.append(path)\\n             \\n             for i in range(idx, len(nums)):\\n                 if i > idx and nums[i] == nums[i-1]:\\n                     continue\\n                 dfs(i + 1, path + [nums[i]])     \\n         nums.sort()\\n         subsets = []\\n         dfs(0, [])\\n         \\n         return subsets\\n         \\n         \\n         \\n         \\n             \\n             \\n                 \\n         \\n             \\n                 \\n\", \"class Solution:\\n     def subsets(self, nums):\\n         if len(nums) == 0:\\n             return [[]]\\n         ret = []\\n         for i, n in enumerate(nums):\\n             if i > 0 and n == nums[i - 1]:\\n                 continue\\n             for s in self.subsets(nums[i + 1:]):\\n                 ret.append([n] + s)\\n         return [[]] + ret\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         nums.sort()\\n         return self.subsets(nums)\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = []\\n         res = []\\n         \\n         self.df(nums, 0, result, res)\\n         return res\\n     \\n     def df(self, nums, idx, result, res):\\n         if idx > len(nums):\\n             return\\n         if result not in res:\\n             res.append(result)\\n         \\n         for i in range(idx, len(nums)):\\n             self.df(nums, i+1, sorted(result + [nums[i]]), res)\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(depth, start, cur):\\n             if cur not in res:\\n                 res.append(cur)\\n             if depth == len(nums):\\n                 return \\n             for i in range(start, len(nums)):\\n                 dfs(depth + 1, i + 1, cur + [nums[i]])\\n         \\n         nums.sort()\\n         res = []\\n         dfs(0, 0, [])\\n         return list(res)\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         def backtrack(nums, start, tmp, res):\\n             res.append(tmp[:])\\n             \\n             for i in range(start, len(nums)):\\n                 if i>start and nums[i] == nums[i-1]:\\n                     continue\\n                 else:\\n                     tmp.append(nums[i])\\n                     backtrack(nums, i+1, tmp, res)\\n                     del tmp[-1]\\n                     \\n         res = list()\\n         backtrack(sorted(nums), 0, [], res)\\n         return res \\n\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         nums = sorted(nums)\\n         res = [[]]\\n         self._helpFun(nums,  res, [])\\n         return(res)\\n     \\n     def _helpFun(self, nums, res, curr):\\n         if not nums:\\n             if curr not in res:\\n                 res.append(curr)\\n             return\\n         \\n         for i in range(len(nums)):\\n             newCurr = curr + [nums[i]]\\n             if newCurr not in res:\\n                 res.append(newCurr)\\n             \\n             self._helpFun(nums[i+1:], res, newCurr )\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                2,
                2
            ]
        ],
        "output": [
            [],
            [
                1
            ],
            [
                1,
                2
            ],
            [
                1,
                2,
                2
            ],
            [
                2
            ],
            [
                2,
                2
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "subsetsWithDup",
        "starter_code": "\nclass Solution:\n    def subsetsWithDup(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/subsets-ii/"
    },
    {
        "id": 1288,
        "task_id": 2639,
        "test_case_id": 2,
        "question": "Given a collection of integers that might contain duplicates, nums, return all possible subsets (the power set).\n\nNote: The solution set must not contain duplicate subsets.\n\nExample:\n\n\nInput: [1,2,2]\nOutput:\n[\n  [2],\n  [1],\n  [1,2,2],\n  [2,2],\n  [1,2],\n  []\n]",
        "solutions": "[\"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(idx, path):\\n             subsets.append(path)\\n             \\n             for i in range(idx, len(nums)):\\n                 if i > idx and nums[i] == nums[i-1]:\\n                     continue\\n                 dfs(i + 1, path + [nums[i]])     \\n         nums.sort()\\n         subsets = []\\n         dfs(0, [])\\n         \\n         return subsets\\n         \\n         \\n         \\n         \\n             \\n             \\n                 \\n         \\n             \\n                 \\n\", \"class Solution:\\n     def subsets(self, nums):\\n         if len(nums) == 0:\\n             return [[]]\\n         ret = []\\n         for i, n in enumerate(nums):\\n             if i > 0 and n == nums[i - 1]:\\n                 continue\\n             for s in self.subsets(nums[i + 1:]):\\n                 ret.append([n] + s)\\n         return [[]] + ret\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         nums.sort()\\n         return self.subsets(nums)\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = []\\n         res = []\\n         \\n         self.df(nums, 0, result, res)\\n         return res\\n     \\n     def df(self, nums, idx, result, res):\\n         if idx > len(nums):\\n             return\\n         if result not in res:\\n             res.append(result)\\n         \\n         for i in range(idx, len(nums)):\\n             self.df(nums, i+1, sorted(result + [nums[i]]), res)\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(depth, start, cur):\\n             if cur not in res:\\n                 res.append(cur)\\n             if depth == len(nums):\\n                 return \\n             for i in range(start, len(nums)):\\n                 dfs(depth + 1, i + 1, cur + [nums[i]])\\n         \\n         nums.sort()\\n         res = []\\n         dfs(0, 0, [])\\n         return list(res)\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         def backtrack(nums, start, tmp, res):\\n             res.append(tmp[:])\\n             \\n             for i in range(start, len(nums)):\\n                 if i>start and nums[i] == nums[i-1]:\\n                     continue\\n                 else:\\n                     tmp.append(nums[i])\\n                     backtrack(nums, i+1, tmp, res)\\n                     del tmp[-1]\\n                     \\n         res = list()\\n         backtrack(sorted(nums), 0, [], res)\\n         return res \\n\", \"class Solution:\\n     def subsetsWithDup(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         nums = sorted(nums)\\n         res = [[]]\\n         self._helpFun(nums,  res, [])\\n         return(res)\\n     \\n     def _helpFun(self, nums, res, curr):\\n         if not nums:\\n             if curr not in res:\\n                 res.append(curr)\\n             return\\n         \\n         for i in range(len(nums)):\\n             newCurr = curr + [nums[i]]\\n             if newCurr not in res:\\n                 res.append(newCurr)\\n             \\n             self._helpFun(nums[i+1:], res, newCurr )\"]",
        "difficulty": "interview",
        "input": [
            [
                0
            ]
        ],
        "output": [
            [],
            [
                0
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "subsetsWithDup",
        "starter_code": "\nclass Solution:\n    def subsetsWithDup(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/subsets-ii/"
    },
    {
        "id": 1289,
        "task_id": 2883,
        "test_case_id": 1,
        "question": "Given a collection of intervals, merge all overlapping intervals.\n\nExample 1:\n\n\nInput: [[1,3],[2,6],[8,10],[15,18]]\nOutput: [[1,6],[8,10],[15,18]]\nExplanation: Since intervals [1,3] and [2,6] overlaps, merge them into [1,6].\n\n\nExample 2:\n\n\nInput: [[1,4],[4,5]]\nOutput: [[1,5]]\nExplanation: Intervals [1,4] and [4,5] are considerred overlapping.",
        "solutions": "[\"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n       new_intervals = []\\n       for interval in sorted(intervals, key=lambda i: i.start):\\n         if new_intervals and interval.start <= new_intervals[-1].end:\\n           new_intervals[-1].end = max(new_intervals[-1].end, interval.end)\\n         else:\\n           new_intervals.append(interval)\\n       return new_intervals\\n\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if len(intervals) < 2:\\n             return intervals\\n         intervals = sorted(intervals, key=lambda s: s.start)\\n         rs = []\\n         cur = intervals[0]\\n         for r in intervals[1:]:\\n             if r.start <= cur.end:\\n                 cur.end = max(r.end, cur.end)\\n             else:\\n                 rs.append(cur)\\n                 cur = r\\n         rs.append(cur)\\n         return rs\\n         \", \"class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         inters = sorted(intervals, key=lambda intval: intval.start)\\n         ret = []\\n         n = len(intervals)\\n         if n == 0:\\n             return ret\\n         s = inters[0].start\\n         e = inters[0].end\\n         for i in range(1, n):\\n             if inters[i].start <= e:\\n                 e = max(inters[i].end, e)\\n             else:\\n                 ret.append(Interval(s, e))\\n                 s = inters[i].start\\n                 e = inters[i].end\\n         ret.append(Interval(s,e))\\n         return ret\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         start = sorted([interval.start for interval in intervals])\\n         end = sorted([interval.end for interval in intervals])\\n         if not start:\\n             return end\\n         if not end:\\n             return start\\n         result = list()\\n         make_interval = 1\\n         i, j = 1, 0\\n         first_start = 0\\n         while i < len(start) and j < len(end):\\n             if start[i] <= end [j]:\\n                 make_interval += 1\\n                 i += 1\\n             else:\\n                 make_interval -= 1\\n                 if make_interval == 0:\\n                     result.append([start[first_start], end[j]])\\n                     first_start = i\\n                 j += 1\\n                     \\n         if j < len(end) :\\n             result.append([start[first_start], end[-1]])\\n         return result\\n                 \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         length=len(intervals)\\n         if length==1 or length==0: return intervals\\n         intervals.sort(key=lambda l:l.start)\\n         result=[]\\n         curr=intervals.pop(0)\\n         while intervals:\\n             next_int=intervals.pop(0)\\n             if curr.end>=next_int.start:\\n                 curr.end=max(next_int.end,curr.end)\\n             else:\\n                 result.append(curr)\\n                 curr=next_int\\n         result.append(curr)\\n         return result\\n     \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         length=len(intervals)\\n         if length==1 or length==0: return intervals\\n         intervals.sort(key=lambda l:l.start)\\n         result=[]\\n         curr=intervals.pop(0)\\n         while intervals:\\n             next_int=intervals.pop(0)\\n             if curr.end>=next_int.start:\\n                 if curr.end<next_int.end:\\n                     curr.end=next_int.end\\n             else:\\n                 result.append(curr)\\n                 curr=next_int\\n         result.append(curr)\\n         return result\\n     \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if not intervals: return []\\n         intervals = sorted(intervals, key=lambda x: (x.start, x.end))\\n         res = [Interval(intervals[0].start, intervals[0].end), ]\\n         for i in range(1, len(intervals)):\\n             if intervals[i].start <= res[len(res) - 1].end:\\n                 res[len(res) - 1].start = min(res[len(res) - 1].start, intervals[i].start)\\n                 res[len(res) - 1].end = max(res[len(res) - 1].end, intervals[i].end)\\n             else:\\n                 res.append(Interval(intervals[i].start, intervals[i].end))\\n         return res\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if not intervals:\\n             return []\\n         intervals.sort(key=lambda a: (a.start, a.end))\\n         ptr = 0\\n         while ptr < len(intervals) - 1:\\n             if intervals[ptr].end >= intervals[ptr+1].start:\\n                 p = intervals.pop(ptr + 1)\\n                 intervals[ptr].end = max(intervals[ptr].end, p.end)\\n             else:\\n                 ptr += 1\\n         \\n         return intervals\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         si = sorted(intervals, key=lambda x: (x.start, x.end))\\n         cur = 1  # index of current tuple\\n         while cur < len(si):\\n             # check intersection of current and previous tuple\\n             # if end of previous is less than begin of current\\n             # there are the intersection, we should merge them\\n             # into the bigger interval\\n             if si[cur - 1].end >= si[cur].start:\\n                 # change end of the interval(tuple)\\n                 si[cur - 1] = Interval(si[cur - 1].start, max(si[cur - 1].end, si[cur].end))\\n                 # delete unnecessary interval(tuple)\\n                 del si[cur]\\n             # there are no intersection, go to the next tuple\\n             else:\\n                 cur += 1\\n         return si\\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         '''\\n         \\u5148\\u6309\\u7167\\u6bcf\\u4e2a\\u4f1a\\u8bae\\u7684\\u5f00\\u59cb\\u65f6\\u95f4\\u6392\\u5e8f\\uff0c\\u7528\\u4e00\\u4e2a\\u6570\\u5217\\u6765\\u4fdd\\u5b58\\u4f1a\\u8bae\\uff0c\\u6761\\u4ef6\\u662f\\n         \\u5982\\u679c\\u5f53\\u524d\\u4f1a\\u8bae\\u7684\\u5f00\\u59cb\\u65f6\\u95f4\\u6bd4\\u6570\\u5217\\u4e2d\\u6700\\u540e\\u4e00\\u4e2a\\u4f1a\\u8bae\\u7684\\u7ed3\\u675f\\u65f6\\u95f4\\u8fd8\\u665a\\uff0c\\u53e6\\u8d77\\u7089\\u7076\\u3002\\n         \\u5982\\u679c\\u5f00\\u59cb\\u65f6\\u95f4\\u6bd4\\u7ed3\\u675f\\u65f6\\u95f4\\u8fd8\\u65e9\\uff0c\\u6324\\u8fdb\\u53bb\\uff01\\u6bd4\\u8f83\\u4f1a\\u8bae\\u7684\\u7ed3\\u675f\\u65f6\\u95f4\\uff0c\\u66f4\\u65b0\\u3002\\n         '''\\n         intervals.sort(key=lambda x : x.start)\\n         res = []\\n         for interval in intervals:\\n             # \\u5982\\u679cres \\u662f\\u7a7a\\u7684\\uff0c\\u521d\\u59cb\\u5316\\u7684\\u60c5\\u51b5\\uff0c\\u8981\\u628a\\u7b2c\\u4e00\\u4e2a\\u4f1a\\u8bae\\u52a0\\u8fdb\\u53bb\\uff01\\n             if not res or interval.start > res[-1].end:\\n                 res.append(interval)\\n             else:\\n                 res[-1].end = max(interval.end, res[-1].end)\\n         return res\\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         length=len(intervals)\\n         if length==1 or length==0: return intervals\\n         intervals.sort(key=lambda l:l.start)\\n         index=0\\n         while index<len(intervals)-1:\\n             curr=intervals[index]\\n             if curr.end>=intervals[index+1].start:\\n                 intervals.pop(index)\\n                 next_int=intervals.pop(index)\\n                 if curr.end<next_int.end:\\n                     curr.end=next_int.end\\n                 intervals.insert(index,curr)\\n             else:\\n                 index+=1\\n         return intervals\\n     \"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    3
                ],
                [
                    2,
                    6
                ],
                [
                    8,
                    10
                ],
                [
                    15,
                    18
                ]
            ]
        ],
        "output": [
            [
                1,
                6
            ],
            [
                8,
                10
            ],
            [
                15,
                18
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "merge",
        "starter_code": "\nclass Solution:\n    def merge(self, intervals: List[List[int]]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/merge-intervals/"
    },
    {
        "id": 1290,
        "task_id": 2883,
        "test_case_id": 2,
        "question": "Given a collection of intervals, merge all overlapping intervals.\n\nExample 1:\n\n\nInput: [[1,3],[2,6],[8,10],[15,18]]\nOutput: [[1,6],[8,10],[15,18]]\nExplanation: Since intervals [1,3] and [2,6] overlaps, merge them into [1,6].\n\n\nExample 2:\n\n\nInput: [[1,4],[4,5]]\nOutput: [[1,5]]\nExplanation: Intervals [1,4] and [4,5] are considerred overlapping.",
        "solutions": "[\"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n       new_intervals = []\\n       for interval in sorted(intervals, key=lambda i: i.start):\\n         if new_intervals and interval.start <= new_intervals[-1].end:\\n           new_intervals[-1].end = max(new_intervals[-1].end, interval.end)\\n         else:\\n           new_intervals.append(interval)\\n       return new_intervals\\n\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if len(intervals) < 2:\\n             return intervals\\n         intervals = sorted(intervals, key=lambda s: s.start)\\n         rs = []\\n         cur = intervals[0]\\n         for r in intervals[1:]:\\n             if r.start <= cur.end:\\n                 cur.end = max(r.end, cur.end)\\n             else:\\n                 rs.append(cur)\\n                 cur = r\\n         rs.append(cur)\\n         return rs\\n         \", \"class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         inters = sorted(intervals, key=lambda intval: intval.start)\\n         ret = []\\n         n = len(intervals)\\n         if n == 0:\\n             return ret\\n         s = inters[0].start\\n         e = inters[0].end\\n         for i in range(1, n):\\n             if inters[i].start <= e:\\n                 e = max(inters[i].end, e)\\n             else:\\n                 ret.append(Interval(s, e))\\n                 s = inters[i].start\\n                 e = inters[i].end\\n         ret.append(Interval(s,e))\\n         return ret\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         start = sorted([interval.start for interval in intervals])\\n         end = sorted([interval.end for interval in intervals])\\n         if not start:\\n             return end\\n         if not end:\\n             return start\\n         result = list()\\n         make_interval = 1\\n         i, j = 1, 0\\n         first_start = 0\\n         while i < len(start) and j < len(end):\\n             if start[i] <= end [j]:\\n                 make_interval += 1\\n                 i += 1\\n             else:\\n                 make_interval -= 1\\n                 if make_interval == 0:\\n                     result.append([start[first_start], end[j]])\\n                     first_start = i\\n                 j += 1\\n                     \\n         if j < len(end) :\\n             result.append([start[first_start], end[-1]])\\n         return result\\n                 \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         length=len(intervals)\\n         if length==1 or length==0: return intervals\\n         intervals.sort(key=lambda l:l.start)\\n         result=[]\\n         curr=intervals.pop(0)\\n         while intervals:\\n             next_int=intervals.pop(0)\\n             if curr.end>=next_int.start:\\n                 curr.end=max(next_int.end,curr.end)\\n             else:\\n                 result.append(curr)\\n                 curr=next_int\\n         result.append(curr)\\n         return result\\n     \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         length=len(intervals)\\n         if length==1 or length==0: return intervals\\n         intervals.sort(key=lambda l:l.start)\\n         result=[]\\n         curr=intervals.pop(0)\\n         while intervals:\\n             next_int=intervals.pop(0)\\n             if curr.end>=next_int.start:\\n                 if curr.end<next_int.end:\\n                     curr.end=next_int.end\\n             else:\\n                 result.append(curr)\\n                 curr=next_int\\n         result.append(curr)\\n         return result\\n     \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if not intervals: return []\\n         intervals = sorted(intervals, key=lambda x: (x.start, x.end))\\n         res = [Interval(intervals[0].start, intervals[0].end), ]\\n         for i in range(1, len(intervals)):\\n             if intervals[i].start <= res[len(res) - 1].end:\\n                 res[len(res) - 1].start = min(res[len(res) - 1].start, intervals[i].start)\\n                 res[len(res) - 1].end = max(res[len(res) - 1].end, intervals[i].end)\\n             else:\\n                 res.append(Interval(intervals[i].start, intervals[i].end))\\n         return res\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         if not intervals:\\n             return []\\n         intervals.sort(key=lambda a: (a.start, a.end))\\n         ptr = 0\\n         while ptr < len(intervals) - 1:\\n             if intervals[ptr].end >= intervals[ptr+1].start:\\n                 p = intervals.pop(ptr + 1)\\n                 intervals[ptr].end = max(intervals[ptr].end, p.end)\\n             else:\\n                 ptr += 1\\n         \\n         return intervals\", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         si = sorted(intervals, key=lambda x: (x.start, x.end))\\n         cur = 1  # index of current tuple\\n         while cur < len(si):\\n             # check intersection of current and previous tuple\\n             # if end of previous is less than begin of current\\n             # there are the intersection, we should merge them\\n             # into the bigger interval\\n             if si[cur - 1].end >= si[cur].start:\\n                 # change end of the interval(tuple)\\n                 si[cur - 1] = Interval(si[cur - 1].start, max(si[cur - 1].end, si[cur].end))\\n                 # delete unnecessary interval(tuple)\\n                 del si[cur]\\n             # there are no intersection, go to the next tuple\\n             else:\\n                 cur += 1\\n         return si\\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         '''\\n         \\u5148\\u6309\\u7167\\u6bcf\\u4e2a\\u4f1a\\u8bae\\u7684\\u5f00\\u59cb\\u65f6\\u95f4\\u6392\\u5e8f\\uff0c\\u7528\\u4e00\\u4e2a\\u6570\\u5217\\u6765\\u4fdd\\u5b58\\u4f1a\\u8bae\\uff0c\\u6761\\u4ef6\\u662f\\n         \\u5982\\u679c\\u5f53\\u524d\\u4f1a\\u8bae\\u7684\\u5f00\\u59cb\\u65f6\\u95f4\\u6bd4\\u6570\\u5217\\u4e2d\\u6700\\u540e\\u4e00\\u4e2a\\u4f1a\\u8bae\\u7684\\u7ed3\\u675f\\u65f6\\u95f4\\u8fd8\\u665a\\uff0c\\u53e6\\u8d77\\u7089\\u7076\\u3002\\n         \\u5982\\u679c\\u5f00\\u59cb\\u65f6\\u95f4\\u6bd4\\u7ed3\\u675f\\u65f6\\u95f4\\u8fd8\\u65e9\\uff0c\\u6324\\u8fdb\\u53bb\\uff01\\u6bd4\\u8f83\\u4f1a\\u8bae\\u7684\\u7ed3\\u675f\\u65f6\\u95f4\\uff0c\\u66f4\\u65b0\\u3002\\n         '''\\n         intervals.sort(key=lambda x : x.start)\\n         res = []\\n         for interval in intervals:\\n             # \\u5982\\u679cres \\u662f\\u7a7a\\u7684\\uff0c\\u521d\\u59cb\\u5316\\u7684\\u60c5\\u51b5\\uff0c\\u8981\\u628a\\u7b2c\\u4e00\\u4e2a\\u4f1a\\u8bae\\u52a0\\u8fdb\\u53bb\\uff01\\n             if not res or interval.start > res[-1].end:\\n                 res.append(interval)\\n             else:\\n                 res[-1].end = max(interval.end, res[-1].end)\\n         return res\\n         \", \"# Definition for an interval.\\n # class Interval:\\n #     def __init__(self, s=0, e=0):\\n #         self.start = s\\n #         self.end = e\\n \\n class Solution:\\n     def merge(self, intervals):\\n         \\\"\\\"\\\"\\n         :type intervals: List[Interval]\\n         :rtype: List[Interval]\\n         \\\"\\\"\\\"\\n         length=len(intervals)\\n         if length==1 or length==0: return intervals\\n         intervals.sort(key=lambda l:l.start)\\n         index=0\\n         while index<len(intervals)-1:\\n             curr=intervals[index]\\n             if curr.end>=intervals[index+1].start:\\n                 intervals.pop(index)\\n                 next_int=intervals.pop(index)\\n                 if curr.end<next_int.end:\\n                     curr.end=next_int.end\\n                 intervals.insert(index,curr)\\n             else:\\n                 index+=1\\n         return intervals\\n     \"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    1,
                    4
                ],
                [
                    4,
                    5
                ]
            ]
        ],
        "output": [
            [
                1,
                5
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "merge",
        "starter_code": "\nclass Solution:\n    def merge(self, intervals: List[List[int]]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/merge-intervals/"
    },
    {
        "id": 1291,
        "task_id": 2884,
        "test_case_id": 1,
        "question": "Given a collection of candidate numbers (candidates) and a target number (target), find all unique combinations in candidates where the candidate numbers sums to target.\n\nEach number in candidates may only be used once in the combination.\n\nNote:\n\n\n       All numbers (including target) will be positive integers.\n       The solution set must not contain duplicate combinations.\n\n\nExample 1:\n\n\nInput: candidates = [10,1,2,7,6,1,5], target = 8,\nA solution set is:\n[\n  [1, 7],\n  [1, 2, 5],\n  [2, 6],\n  [1, 1, 6]\n]\n\n\nExample 2:\n\n\nInput: candidates = [2,5,2,1,2], target = 5,\nA solution set is:\n[\n  [1,2,2],\n  [5]\n]",
        "solutions": "[\"class Solution:\\n     def combinationSum2(self, candidates, target):\\n                 \\n         def dfs(i, val, path):\\n             while i < len(candidates):\\n                 num = candidates[i]\\n                 val_ = val + num\\n                 path_ = path + [num]\\n                 if val_ > target:\\n                     return\\n                 elif val_ == target:\\n                     ans.append(path_)\\n                     return                  \\n                 dfs(i+1, val_, path_)\\n                 while i<len(candidates)-1 and candidates[i]==candidates[i+1]:\\n                     i += 1\\n                 i += 1\\n                \\n         candidates = sorted(candidates)\\n         ans = []\\n         dfs(0, 0, [])\\n         return ans\", \"\\n class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if len(candidates)==0:\\n             return []\\n \\n \\n         candidates = sorted(candidates)\\n         print(candidates)\\n \\n         results = []\\n \\n \\n         def find(curr_comb, curr_sum, candidates): #\\u9760\\u526f\\u4f5c\\u7528\\u586b\\u5145results\\n             #curr_comb = curr_comb[:]\\n             for i,cand in enumerate(candidates):\\n \\n                 if (cand+curr_sum) < target:\\n                     new_curr_comb = curr_comb[:]\\n                     new_curr_comb.append(cand)\\n                     find(new_curr_comb, curr_sum+cand, candidates[i+1:])\\n                 elif (cand+curr_sum) == target:\\n                     new_curr_comb = curr_comb[:]\\n                     new_curr_comb.append(cand)\\n                     if new_curr_comb not in results:\\n                         results.append(new_curr_comb)\\n                     #print(curr_comb, curr_sum, cand, 'target:',target)\\n                 else:\\n                     break\\n \\n \\n         find([], 0, candidates)\\n \\n         return results\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not candidates:\\n             return []\\n         result = []\\n         candidates.sort()\\n         self.dfs(result, [], 0, target, candidates)\\n         return result\\n     \\n     def dfs(self, result, combin, start, target, candidates):\\n         if target == 0:\\n             result.append([i for i in combin])\\n         else:\\n             for index in range(start, len(candidates)):\\n                 if candidates[index] > target:\\n                     break\\n                 if (index > start and candidates[index] == candidates[index - 1]):\\n                     continue\\n                 combin.append(candidates[index])\\n                 self.dfs(result, combin, index + 1, target - candidates[index], candidates)\\n                 combin.pop()\\n\", \"class Solution:\\n     def combinationSum2(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(a, i, target):\\n             for j in range(i, len(nums)):\\n                 if target <= nums[j]:\\n                     if target == nums[j]:\\n                         ans.append(a+[nums[j]])\\n                     break\\n                 if i == j or nums[j] != nums[j-1]:\\n                     dfs(a+[nums[j]], j+1, target-nums[j])\\n \\n         ans = []\\n         nums.sort()\\n         dfs([], 0, target)\\n         return ans\\n\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def helper(L, target, path, ans):\\n             if target == 0:\\n                 ans.append(path)\\n                 return\\n             for i in range(len(L)):\\n                 if L[i] > target:\\n                     return\\n                 if i > 0 and L[i] == L[i-1]:\\n                     continue\\n                 helper(L[i+1:], target-L[i], path+[L[i]], ans)\\n         \\n         ans = []\\n         helper(sorted(candidates), target, [], ans)\\n         return ans\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         candidates.sort()\\n         self.dfs(candidates,target,0,[],res)\\n         return res\\n #60%\\n     \\n     def dfs(self,nums,target,index,path,res):\\n         if target<0:\\n             return \\n         if target == 0:\\n             res.append(path)\\n             return \\n         for i in range(index,len(nums)):\\n             if i>index and nums[i] == nums[i-1]:\\n                 continue\\n             if nums[i] > target:    #   \\u63d0\\u5347\\u523093%\\n                 break\\n             self.dfs(nums,target-nums[i],i+1,path+[nums[i]],res)\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result=[]\\n         def combinationSumRecu(self,candidates,target,start,intermediate,result):\\n             if target==0:\\n                 result.append(list(intermediate))\\n             while start<len(candidates) and candidates[start]<=target:\\n                 intermediate.append(candidates[start])\\n                 combinationSumRecu(self,candidates,target-candidates[start],start+1,intermediate,result)\\n                 start+=1\\n                 intermediate.pop()\\n                 while start>0 and start<len(candidates) and candidates[start]==candidates[start-1]:\\n                     start+=1\\n                     \\n                 \\n         \\n         combinationSumRecu(self,sorted(candidates),target,0,[],result)\\n         return result\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         dp = [None] + [set() for i in range(target)]\\n         candidates.sort()\\n         for i in candidates:\\n             if i > target:\\n                 break\\n             for j in range(target-i,0,-1):\\n                 dp[i+j] |= {_ + (i,) for _ in dp[j]}\\n             dp[i].add((i,))\\n         return list(dp[-1])\", \"class Solution:\\n     def find(self,candidates,result,current,target,pos,num,current_num):\\n         if target==0:\\n             temp = current[:]\\n             result.append(temp)\\n         else:\\n             for i in range(pos,len(candidates)):\\n                 n = str(candidates[i])\\n                 if candidates[i]>target or num[n]<current_num[n]:\\n                     break\\n                 if num[n]==current_num[n]:\\n                     continue\\n                 current.append(candidates[i])\\n                 current_num[n] +=1\\n                 self.find(candidates,result,current,target-candidates[i],i,num,current_num)\\n                 value = current.pop()\\n                 current_num[str(value)] -=1\\n \\n     def combinationSum2(self,candidates, target):\\n         result = []\\n         current =[]\\n         candidates.sort()\\n         can = candidates[:]\\n         count = 0;\\n         num = {}\\n         current_num={}\\n         for i in range(len(candidates)):\\n             if i==0 or candidates[i]!=candidates[i-1]:\\n                 num[str(candidates[i])]=1\\n                 current_num[str(candidates[i])]=0\\n                 continue\\n             else:\\n                 num[str(candidates[i])]+=1\\n                 del can[i-count]\\n                 count +=1\\n         self.find(can,result,current,target,0,num,current_num)\\n         return result\\n\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def produce(nums, target, current, output):\\n             if target == 0:\\n                 output.append(list(current))\\n                 return\\n             for i, n in enumerate(nums):\\n                 if n > target: continue\\n                 # avoid duplicate solutions\\n                 if i > 0 and nums[i] == nums[i-1]: continue\\n                 current.append(n)\\n                 # each num can be used only once: nums[i+1:]\\n                 produce(nums[i+1:], target - n, current, output)\\n                 current.pop()\\n \\n         output = []\\n         candidates.sort()\\n         produce(candidates, target, [], output)\\n         return output\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         sortedCandidates = sorted(candidates)\\n         if len(candidates) == 0 or target < sortedCandidates[0]:\\n             return []\\n         if target == sortedCandidates[0]:\\n             return [[sortedCandidates[0]]]\\n         resultWith0 = self.combinationSum2(sortedCandidates[1:], target - sortedCandidates[0])\\n         nextIndex = 0\\n         while nextIndex < len(candidates) and sortedCandidates[nextIndex] == sortedCandidates[0]:\\n             nextIndex += 1\\n         \\n         resultNo0 = self.combinationSum2(sortedCandidates[nextIndex:], target)\\n         \\n         return [[sortedCandidates[0]] + item for item in resultWith0] + resultNo0\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def backtracing(nums,target,idx,path,ret):\\n             if not target :\\n                 if path not in ret :\\n                     ret.append(path)\\n             for i in range (idx,len(nums)):\\n                 if nums[i] > target :\\n                     break\\n                 backtracing(nums,target-nums[i],i+1,path+[nums[i]],ret)\\n         candidates.sort()\\n         ret = []\\n         backtracing(candidates,target,0,[],ret)\\n         return ret\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def combination(candidates, k, target, res, result):\\n             for i in range(k,len(candidates)):\\n                 if target == candidates[i]:\\n                     temp = [j for j in res]\\n                     temp.append(candidates[i])\\n                     temp.sort()\\n                     if temp not in result:\\n                         result.append(temp)\\n                     return\\n                 elif target < candidates[i]:\\n                     return\\n                 else:\\n                     res.append(candidates[i])\\n                     combination(candidates, i+1, target - candidates[i], res, result)\\n                     res.pop()\\n \\n         candidates.sort()\\n         res = []\\n         result = []\\n         combination(candidates, 0, target, res, result)\\n         return(result)\", \"class Solution(object):\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = []\\n         temp = []\\n         candidates.sort(reverse=True)\\n \\n         self.util(candidates, target, result, temp)\\n         return result       \\n \\n     def util(self, nums, target, result, temp):\\n \\n         for i in range(len(nums)): \\n             if nums[i] == target and (temp + [nums[i]] not in result):\\n                 result.append(temp + [nums[i]])\\n             elif nums[i] < target:\\n                 self.util(nums[i + 1:], target - nums[i], result, temp + [nums[i]])\\n \\n         return \"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    10,
                    1,
                    2,
                    7,
                    6,
                    1,
                    5
                ]
            ],
            [
                8
            ]
        ],
        "output": [
            [
                1,
                1,
                6
            ],
            [
                1,
                2,
                5
            ],
            [
                1,
                7
            ],
            [
                2,
                6
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "combinationSum2",
        "starter_code": "\nclass Solution:\n    def combinationSum2(self, candidates: List[int], target: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/combination-sum-ii/"
    },
    {
        "id": 1292,
        "task_id": 2884,
        "test_case_id": 2,
        "question": "Given a collection of candidate numbers (candidates) and a target number (target), find all unique combinations in candidates where the candidate numbers sums to target.\n\nEach number in candidates may only be used once in the combination.\n\nNote:\n\n\n       All numbers (including target) will be positive integers.\n       The solution set must not contain duplicate combinations.\n\n\nExample 1:\n\n\nInput: candidates = [10,1,2,7,6,1,5], target = 8,\nA solution set is:\n[\n  [1, 7],\n  [1, 2, 5],\n  [2, 6],\n  [1, 1, 6]\n]\n\n\nExample 2:\n\n\nInput: candidates = [2,5,2,1,2], target = 5,\nA solution set is:\n[\n  [1,2,2],\n  [5]\n]",
        "solutions": "[\"class Solution:\\n     def combinationSum2(self, candidates, target):\\n                 \\n         def dfs(i, val, path):\\n             while i < len(candidates):\\n                 num = candidates[i]\\n                 val_ = val + num\\n                 path_ = path + [num]\\n                 if val_ > target:\\n                     return\\n                 elif val_ == target:\\n                     ans.append(path_)\\n                     return                  \\n                 dfs(i+1, val_, path_)\\n                 while i<len(candidates)-1 and candidates[i]==candidates[i+1]:\\n                     i += 1\\n                 i += 1\\n                \\n         candidates = sorted(candidates)\\n         ans = []\\n         dfs(0, 0, [])\\n         return ans\", \"\\n class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if len(candidates)==0:\\n             return []\\n \\n \\n         candidates = sorted(candidates)\\n         print(candidates)\\n \\n         results = []\\n \\n \\n         def find(curr_comb, curr_sum, candidates): #\\u9760\\u526f\\u4f5c\\u7528\\u586b\\u5145results\\n             #curr_comb = curr_comb[:]\\n             for i,cand in enumerate(candidates):\\n \\n                 if (cand+curr_sum) < target:\\n                     new_curr_comb = curr_comb[:]\\n                     new_curr_comb.append(cand)\\n                     find(new_curr_comb, curr_sum+cand, candidates[i+1:])\\n                 elif (cand+curr_sum) == target:\\n                     new_curr_comb = curr_comb[:]\\n                     new_curr_comb.append(cand)\\n                     if new_curr_comb not in results:\\n                         results.append(new_curr_comb)\\n                     #print(curr_comb, curr_sum, cand, 'target:',target)\\n                 else:\\n                     break\\n \\n \\n         find([], 0, candidates)\\n \\n         return results\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not candidates:\\n             return []\\n         result = []\\n         candidates.sort()\\n         self.dfs(result, [], 0, target, candidates)\\n         return result\\n     \\n     def dfs(self, result, combin, start, target, candidates):\\n         if target == 0:\\n             result.append([i for i in combin])\\n         else:\\n             for index in range(start, len(candidates)):\\n                 if candidates[index] > target:\\n                     break\\n                 if (index > start and candidates[index] == candidates[index - 1]):\\n                     continue\\n                 combin.append(candidates[index])\\n                 self.dfs(result, combin, index + 1, target - candidates[index], candidates)\\n                 combin.pop()\\n\", \"class Solution:\\n     def combinationSum2(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def dfs(a, i, target):\\n             for j in range(i, len(nums)):\\n                 if target <= nums[j]:\\n                     if target == nums[j]:\\n                         ans.append(a+[nums[j]])\\n                     break\\n                 if i == j or nums[j] != nums[j-1]:\\n                     dfs(a+[nums[j]], j+1, target-nums[j])\\n \\n         ans = []\\n         nums.sort()\\n         dfs([], 0, target)\\n         return ans\\n\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def helper(L, target, path, ans):\\n             if target == 0:\\n                 ans.append(path)\\n                 return\\n             for i in range(len(L)):\\n                 if L[i] > target:\\n                     return\\n                 if i > 0 and L[i] == L[i-1]:\\n                     continue\\n                 helper(L[i+1:], target-L[i], path+[L[i]], ans)\\n         \\n         ans = []\\n         helper(sorted(candidates), target, [], ans)\\n         return ans\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         candidates.sort()\\n         self.dfs(candidates,target,0,[],res)\\n         return res\\n #60%\\n     \\n     def dfs(self,nums,target,index,path,res):\\n         if target<0:\\n             return \\n         if target == 0:\\n             res.append(path)\\n             return \\n         for i in range(index,len(nums)):\\n             if i>index and nums[i] == nums[i-1]:\\n                 continue\\n             if nums[i] > target:    #   \\u63d0\\u5347\\u523093%\\n                 break\\n             self.dfs(nums,target-nums[i],i+1,path+[nums[i]],res)\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result=[]\\n         def combinationSumRecu(self,candidates,target,start,intermediate,result):\\n             if target==0:\\n                 result.append(list(intermediate))\\n             while start<len(candidates) and candidates[start]<=target:\\n                 intermediate.append(candidates[start])\\n                 combinationSumRecu(self,candidates,target-candidates[start],start+1,intermediate,result)\\n                 start+=1\\n                 intermediate.pop()\\n                 while start>0 and start<len(candidates) and candidates[start]==candidates[start-1]:\\n                     start+=1\\n                     \\n                 \\n         \\n         combinationSumRecu(self,sorted(candidates),target,0,[],result)\\n         return result\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         dp = [None] + [set() for i in range(target)]\\n         candidates.sort()\\n         for i in candidates:\\n             if i > target:\\n                 break\\n             for j in range(target-i,0,-1):\\n                 dp[i+j] |= {_ + (i,) for _ in dp[j]}\\n             dp[i].add((i,))\\n         return list(dp[-1])\", \"class Solution:\\n     def find(self,candidates,result,current,target,pos,num,current_num):\\n         if target==0:\\n             temp = current[:]\\n             result.append(temp)\\n         else:\\n             for i in range(pos,len(candidates)):\\n                 n = str(candidates[i])\\n                 if candidates[i]>target or num[n]<current_num[n]:\\n                     break\\n                 if num[n]==current_num[n]:\\n                     continue\\n                 current.append(candidates[i])\\n                 current_num[n] +=1\\n                 self.find(candidates,result,current,target-candidates[i],i,num,current_num)\\n                 value = current.pop()\\n                 current_num[str(value)] -=1\\n \\n     def combinationSum2(self,candidates, target):\\n         result = []\\n         current =[]\\n         candidates.sort()\\n         can = candidates[:]\\n         count = 0;\\n         num = {}\\n         current_num={}\\n         for i in range(len(candidates)):\\n             if i==0 or candidates[i]!=candidates[i-1]:\\n                 num[str(candidates[i])]=1\\n                 current_num[str(candidates[i])]=0\\n                 continue\\n             else:\\n                 num[str(candidates[i])]+=1\\n                 del can[i-count]\\n                 count +=1\\n         self.find(can,result,current,target,0,num,current_num)\\n         return result\\n\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def produce(nums, target, current, output):\\n             if target == 0:\\n                 output.append(list(current))\\n                 return\\n             for i, n in enumerate(nums):\\n                 if n > target: continue\\n                 # avoid duplicate solutions\\n                 if i > 0 and nums[i] == nums[i-1]: continue\\n                 current.append(n)\\n                 # each num can be used only once: nums[i+1:]\\n                 produce(nums[i+1:], target - n, current, output)\\n                 current.pop()\\n \\n         output = []\\n         candidates.sort()\\n         produce(candidates, target, [], output)\\n         return output\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         sortedCandidates = sorted(candidates)\\n         if len(candidates) == 0 or target < sortedCandidates[0]:\\n             return []\\n         if target == sortedCandidates[0]:\\n             return [[sortedCandidates[0]]]\\n         resultWith0 = self.combinationSum2(sortedCandidates[1:], target - sortedCandidates[0])\\n         nextIndex = 0\\n         while nextIndex < len(candidates) and sortedCandidates[nextIndex] == sortedCandidates[0]:\\n             nextIndex += 1\\n         \\n         resultNo0 = self.combinationSum2(sortedCandidates[nextIndex:], target)\\n         \\n         return [[sortedCandidates[0]] + item for item in resultWith0] + resultNo0\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def backtracing(nums,target,idx,path,ret):\\n             if not target :\\n                 if path not in ret :\\n                     ret.append(path)\\n             for i in range (idx,len(nums)):\\n                 if nums[i] > target :\\n                     break\\n                 backtracing(nums,target-nums[i],i+1,path+[nums[i]],ret)\\n         candidates.sort()\\n         ret = []\\n         backtracing(candidates,target,0,[],ret)\\n         return ret\", \"class Solution:\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         def combination(candidates, k, target, res, result):\\n             for i in range(k,len(candidates)):\\n                 if target == candidates[i]:\\n                     temp = [j for j in res]\\n                     temp.append(candidates[i])\\n                     temp.sort()\\n                     if temp not in result:\\n                         result.append(temp)\\n                     return\\n                 elif target < candidates[i]:\\n                     return\\n                 else:\\n                     res.append(candidates[i])\\n                     combination(candidates, i+1, target - candidates[i], res, result)\\n                     res.pop()\\n \\n         candidates.sort()\\n         res = []\\n         result = []\\n         combination(candidates, 0, target, res, result)\\n         return(result)\", \"class Solution(object):\\n     def combinationSum2(self, candidates, target):\\n         \\\"\\\"\\\"\\n         :type candidates: List[int]\\n         :type target: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         result = []\\n         temp = []\\n         candidates.sort(reverse=True)\\n \\n         self.util(candidates, target, result, temp)\\n         return result       \\n \\n     def util(self, nums, target, result, temp):\\n \\n         for i in range(len(nums)): \\n             if nums[i] == target and (temp + [nums[i]] not in result):\\n                 result.append(temp + [nums[i]])\\n             elif nums[i] < target:\\n                 self.util(nums[i + 1:], target - nums[i], result, temp + [nums[i]])\\n \\n         return \"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    2,
                    5,
                    2,
                    1,
                    2
                ]
            ],
            [
                5
            ]
        ],
        "output": [
            [
                1,
                2,
                2
            ],
            [
                5
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "combinationSum2",
        "starter_code": "\nclass Solution:\n    def combinationSum2(self, candidates: List[int], target: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/combination-sum-ii/"
    },
    {
        "id": 1293,
        "task_id": 4480,
        "test_case_id": 1,
        "question": "Given an array A of integers, return true if and only if we can partition the array into three non-empty parts with equal sums.\nFormally, we can partition the array if we can find indexes i+1 < j with (A[0] + A[1] + ... + A[i] == A[i+1] + A[i+2] + ... + A[j-1] == A[j] + A[j-1] + ... + A[A.length - 1])\n \nExample 1:\nInput: A = [0,2,1,-6,6,-7,9,1,2,0,1]\nOutput: true\nExplanation: 0 + 2 + 1 = -6 + 6 - 7 + 9 + 1 = 2 + 0 + 1\n\nExample 2:\nInput: A = [0,2,1,-6,6,7,9,-1,2,0,1]\nOutput: false\n\nExample 3:\nInput: A = [3,3,6,5,-2,2,5,1,-9,4]\nOutput: true\nExplanation: 3 + 3 = 6 = 5 - 2 + 2 + 5 + 1 - 9 + 4\n\n \nConstraints:\n\n3 <= A.length <= 50000\n-10^4 <= A[i] <= 10^4",
        "solutions": "[\"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n#         if len(A)<3:\\n#             return\\n        \\n#         i = 1\\n#         j = len(A)-2\\n        \\n#         tgtsum = sum(A)/3\\n#         sum1 = sum(A[:i])\\n#         while i<(len(A)-2) and sum1!=tgtsum:\\n#             sum1 = sum1 + A[i]\\n#             i+=1\\n        \\n#         sum3=sum(A[j+1:])\\n#         while j>1 and sum3!=tgtsum:\\n#             # print (tgtsum, sum1, sum3, A[j])\\n#             sum3 = sum3 + A[j]\\n#             j-=1\\n#         # print (i,j)\\n#         if j>=i and sum1==tgtsum and sum3==tgtsum:\\n#             return True\\n#         else:\\n#             return False\\n\\n        if len(A)<3:\\n            return False\\n        suma = sum(A)\\n        if suma%3!=0:\\n            return False\\n        \\n        runsum,target, count = 0,suma/3,0\\n        \\n        for val in A[:-1]:\\n            runsum += val\\n            if runsum==target:\\n                count+=1\\n                runsum=0\\n                if count==2:\\n                    return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for i in range(len(A)-1): \\n            Sum += A[i]\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        print(A)\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=0\\n        res=0\\n        s1=sum(A)\\n        if s1%3 != 0:\\n            return False\\n        target=s1//3\\n        for a in A:\\n            s+=a\\n            if s == target:\\n                s=0\\n                res+=1\\n        return res >2\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for i in A:\\n            sum += i\\n        s = 0\\n        c = 0\\n        for i in A:\\n            s += i\\n            if s == int(sum/3):\\n                s = 0\\n                c += 1\\n        return c >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp:\\n                Sum = 0\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        \\n        # second solution\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cusum = 0\\n        cusum_list = list()\\n        for i in range(len(A)):\\n            cusum += A[i]\\n            cusum_list.append(cusum)\\n        if sum(A)/3 in cusum_list:\\n            if sum(A)*2/3 in cusum_list[cusum_list.index(sum(A)/3) + 1:-1]:\\n                return True\\n            else:\\n                return False\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        required_sum = sum(A)//3\\n        count_partition = 0\\n        curr_sum = A[0]\\n        i = 1\\n        while i < len(A):\\n            while i < len(A) and curr_sum != required_sum:\\n                curr_sum += A[i]\\n                i += 1\\n            if curr_sum == required_sum:\\n                count_partition += 1\\n                if i < len(A):\\n                    curr_sum = A[i]\\n                    i += 1\\n                    if i == len(A) and curr_sum == required_sum:\\n                        count_partition += 1\\n        if count_partition > 3 and required_sum == 0:\\n            count_partition = 3\\n        return count_partition == 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        avg = 0\\n        for i in A:\\n            avg += i\\n        if(avg%3 != 0): return False\\n        count = 0\\n        sum = 0\\n        for i in range(0,len(A)):\\n            if(sum == avg//3):\\n                if(i!=0):\\n                    count += 1\\n                    sum = A[i]\\n            elif(sum != avg//3):\\n                sum += A[i]\\n        if(sum == avg//3): count += 1\\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        \\n        count, tmp_sum, target = 0, 0, sum(A)//3\\n        \\n        for num in A:\\n            tmp_sum += num\\n            if tmp_sum == target:\\n                count += 1 \\n                tmp_sum = 0\\n        \\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3 : return False\\n        prefixSumMap = defaultdict(list)\\n        sumUpto = {}\\n        total = 0\\n        for i in range(len(A)) :\\n            prefixSumMap[total].append(i)\\n            total += A[i]\\n            sumUpto[i] = total\\n        s = 0\\n        for i in range(len(A)-1, 1, -1) :\\n            s += A[i]\\n            target = total - 2*s\\n            if prefixSumMap.get(target) != None :\\n                for j in prefixSumMap[target] :\\n                    if j < i and j > 0 and sumUpto[j-1] == s :\\n                        return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        sumA = sum(A)\\n        if sumA % 3 != 0:\\n            return False\\n        tmpSum, isFound, eachPart = 0, 0 ,sumA//3\\n        for i in A:\\n            tmpSum += i\\n            if tmpSum == eachPart:\\n                tmpSum = 0 \\n                isFound += 1\\n        if isFound >=3 :\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: list) -> bool:\\n        ''' returns True if indices i and j can be found such that A[0]+...+A[i]==A[i+1]+...+A[j]==A[j+1]+...+A[-1]\\n\\n        Algo: form [A[0], A[0]+A[1], A[0]+A[1]+A[2], ..., sum(A)]\\n        if A can be partitioned, then n=sum(A) is a multiple of 3,\\n        2*n//3 must appear at some index j, and n//3 must appear at some index i<j'''\\n        if len(A)<3:\\n            return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i-1]+A[i]\\n        end_value=A[-1]\\n        if end_value%3!=0:\\n            return False\\n        index=len(A)-2\\n        while A[index] != 2*end_value//3:\\n            index-=1\\n            if index==0:\\n                return False\\n        index-=1\\n        while A[index] != end_value//3:\\n            index-=1\\n            if index<0:\\n                return False\\n        return True\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        d = defaultdict(list)\\n        sumn = 0\\n        for i,num in enumerate(A) :\\n            sumn += num\\n            d[sumn].append(i)\\n\\n        if sumn % 3 != 0 :\\n            return False\\n        div = sumn // 3\\n\\n        return div in d and div*2 in d and d[div][0] < (d[div*2][-1] if d[div*2][-1] != len(A)-1 else d[div*2][-2])\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n                if count > 2:\\n                    return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=sum(A)\\n        if s%3!=0:\\n            return 0\\n        n=len(A)\\n        cnt=[0]*n\\n        s//=3\\n        ss=0\\n        for i in range(n-1,-1,-1):\\n            ss+=A[i]\\n            if i==n-1:\\n                cnt[i]= 1 if ss==s else 0\\n            else:\\n                cnt[i]=cnt[i+1]+ (1 if ss==s else 0)\\n        ss=0\\n        ans=0\\n        print(cnt)\\n        for i in range(0,n-2):\\n            ss+=A[i];\\n            if ss==s:\\n                ans+=cnt[i+2]\\n        print(ans)\\n        return True if ans>0 else False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        x = s // 3\\n        c = 0\\n        d = 0\\n        for a in A[:-1]:\\n            c += a\\n            if d == 0 and c == x:\\n                d = 1\\n            elif d == 1 and c == 2 * x:\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A) % 3 != 0:\\n            return False\\n        targetSum = sum(A) // 3\\n        a = 1\\n        runningSum = A[0]\\n        while a < len(A) and runningSum != targetSum:\\n            runningSum += A[a]\\n            a += 1\\n        b = len(A) - 2\\n        runningSum = A[-1]\\n        while b > -1 and runningSum != targetSum:\\n            runningSum += A[b]\\n            b -= 1\\n        return not a > b\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3: return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i]+A[i-1]\\n        \\n        sumTarg = A[-1]/3\\n        \\n        first = False\\n        \\n        for i in range(len(A)):\\n            if A[i] == sumTarg and first == False:\\n                first = True\\n            elif (first == True and A[i] == 2*sumTarg and i != len(A)-1):\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cummulative_sum = 0\\n        counter = 0\\n        target = sum(A) / 3\\n        if target != int(target):\\n            return False\\n        print(target)\\n        for idx in range(len(A)):\\n            cummulative_sum += A[idx]\\n            if cummulative_sum == target:\\n                cummulative_sum = 0\\n                counter += 1\\n        \\n        if counter == 4 and target == 0:\\n            return True\\n        if counter == 3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)-(sum(A)//3)*3!=0:\\n            return 0\\n        else:\\n            s=sum(A)//3\\n            c=0\\n            ts=0\\n            j=0\\n            for i in range(len(A)):\\n                j=i\\n                ts+=A[i]\\n                if ts==s:\\n                    c+=1\\n                    ts=0\\n                if c==2 and j+1<len(A):\\n                    for k in range(j+1,len(A)):\\n                        ts+=A[k]\\n                    if ts==s:\\n                        return 1\\n                    else:\\n                        return 0\\n            return 0\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        partSum = sum(A)/3\\n        curSum = 0\\n        partNum = 0\\n        for i  in A:\\n            curSum +=i\\n            if(curSum == partSum):\\n                curSum = 0\\n                partNum +=1\\n        if(partSum == 0):\\n            return partNum>=3\\n        else:\\n            return partNum == 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        third = total // 3\\n        cumsum, count = 0, 0\\n        for i in range(0, len(A)):\\n            cumsum += A[i]\\n            if cumsum == third:\\n                count += 1\\n                cumsum = 0\\n        return count == 3 or count > 3 and total == 0\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        x= 0\\n        for i in A:\\n            x += i\\n        if x%3 !=0 :\\n            return False\\n        x = x/3\\n        count = 0\\n        sum1 = 0\\n        for i in A :\\n            sum1 += i\\n            if sum1 == x :\\n                count += 1\\n                sum1 = 0\\n        if count >= 3 :\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3!=0:\\n            return False\\n        \\n        req=sum(A)//3\\n        \\n        s=0\\n        c=0\\n        print(sum(A))\\n        print(req)\\n        for i in A:\\n            s+=i\\n            if s==req:\\n                s=0\\n                c+=1\\n        \\n        return c>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        i = 0\\n        count = A[i]\\n        while (i < len(A)-1) and (count != total/3):\\n            i += 1\\n            count += A[i]\\n            \\n            \\n        if (count != total/3) or (i+1 > len(A)-2):\\n            return False\\n        \\n        j = i +1\\n        count = A[j]\\n        while (j < len(A)-1) and (count != total/3):\\n            j += 1\\n            count += A[j]\\n           \\n            \\n        if (count != total/3) or (j+1 == len(A)):\\n            return False\\n        \\n        if sum(A[j+1:]) == total/3:\\n            return True\\n        else:\\n            return False\\n        \\n        \\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        #a + ....+ b = num \\n        # if 3 equal amount of indices have the same sum  return true\\n        # how do we find the sum? \\n        # use the sum and then divide it by three thats the equal part sum\\n        # use len(A) - 1  to find the number of indexes and \\n        equalSum = sum(A) // 3\\n        part = 0\\n        numparts = 0\\n        r = sum(A) % 3\\n        for i in A:\\n            part += i \\n            if part == equalSum:\\n                part = 0\\n                numparts += 1\\n        return not r and numparts >= 3\\n\\n\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum1=sum(A)\\n        if sum1%3!=0:\\n            return False\\n        \\n        else:\\n            div=sum1//3\\n            print(div)\\n            i1,sum2=0,0\\n            for i in A:\\n                sum2=sum2+i\\n                print(sum2,i)\\n                if sum2==div:\\n                    i1=i1+1\\n                    print('sum2',sum2)\\n                    sum2=0\\n            if i1>=3 and div==0:\\n                return True\\n            if i1==3:\\n                return True\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n        return count >= 3\\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        isum = 0\\n        \\n        for i in range(0,len(A)-2):\\n            isum += A[i]\\n            if isum == total/3:\\n                jsum = 0\\n                for j in range(i+1,len(A)-1):\\n                    jsum += A[j]\\n                    if isum == jsum == total - isum - jsum:\\n                        return True\\n        return False\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        target = sum(A)//3\\n        \\n        temp = 0\\n        count = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if(temp == target):\\n                temp = 0\\n                count += 1\\n                if(count == 3):\\n                    return True\\n        return False\\n                \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= sum(A)//3\\n        \\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        sum3 = sum(A[i:])\\n        # for i in range(i, j):\\n        #     sum3 = sum3 + A[i]\\n        #     i = i +1\\n        #     if sum3 == num:\\n        #         break\\n        # print(\\\\\\\"sum=\\\\\\\",sum3)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= 0\\n        for i in range(0, j):\\n            num = num + A[i]\\n        num = floor(num /3);\\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        for i in range(i, j):\\n            sum3 = sum3 + A[i]\\n            i = i +1\\n            if sum3 == num:\\n                break\\n        print(sum1)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        flag = 0\\n        temp = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if flag == 0 and temp == s//3:\\n                flag += 1\\n            elif flag == 1 and temp == s*2/3 and i != len(A)-1:\\n                return True\\n        return False\\n                \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"\\nclass Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        curr_sum = 0\\n        first_found_flag = False\\n        \\n        for i in range(len(A) - 1): #will go until the last element.\\n            curr_sum += A[i]\\n            if not first_found_flag: #looking for the first group\\n                if curr_sum == total / 3: #we found the first group.\\n                    first_found_flag = True\\n            else: #looking for the second group\\n                if curr_sum == total * 2 / 3:\\n                    return True\\n                \\n        return False\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        l,r,s=1, len(A)-2, sum(A)\\n        ls, rs, avgs = A[0], A[-1], s//3\\n        while l<r:\\n            if l < r and ls != avgs:\\n                ls+=A[l]\\n                l+=1\\n            if l<r and rs!=avgs:\\n                rs+=A[r]\\n                r-=1\\n            if ls == rs == avgs and s%3==0:\\n                return True\\n            \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        t=sum(A)\\n        if t%3!=0:\\n            return False\\n        s=0\\n        p=0\\n        for i in range(len(A)):\\n            s+=A[i]\\n            if s==t//3:\\n                s=0\\n                p+=1\\n        if p>=3:\\n            return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, part, cnt = sum(A) // 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return cnt >= 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        s = s / 3\\n        t = 0\\n        count = 0\\n        for x in A:\\n            t += x\\n            if t == s:\\n                t = 0\\n                count += 1\\n        if t == 0 and count >= 3:\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        target = s//3\\n        current = 0\\n        count = 0\\n        for v in A:\\n            current += v\\n            if current == target:\\n                count += 1\\n                current = 0\\n                if count >= 3:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        totalSum = sum(A)\\n        if totalSum % 3 != 0:\\n            return False\\n        \\n        target = totalSum // 3\\n        numPartitions = 0\\n        currSum = 0\\n        \\n        for num in A:\\n            currSum += num\\n            if currSum == target:\\n                numPartitions += 1\\n                currSum = 0\\n                if numPartitions == 3:\\n                    return True\\n                \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, li: List[int]) -> bool:\\n        s = sum(li)\\n        if s % 3: return False\\n        ts = s // 3\\n        ss = 0\\n        chunk = 0\\n        for n in li[:-1]:\\n            ss += n\\n            if ss == ts:\\n                chunk += 1\\n                if chunk == 2:\\n                    return True\\n                ss = 0\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        curr, target = 0, s // 3\\n        count = 0\\n        for ix in range(len(A)):\\n            curr += A[ix]\\n            if curr == target:\\n                curr = 0\\n                count += 1\\n                \\n            if count == 2 and ix < len(A) - 1:\\n                return True\\n        \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        partition_sum = total/3\\n        numberofpartition=0\\n        tempsum=0\\n        for i in range(len(A)):\\n            tempsum+=A[i]\\n            if tempsum == partition_sum:\\n                numberofpartition+=1\\n                tempsum=0\\n            \\n            if numberofpartition == 3:\\n                return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        S = sum(A)\\n        if S % 3 != 0:\\n            return False\\n        \\n        S_1 = S / 3\\n        S_2 = S_1 + S_1\\n        cur = A[0]\\n        for i in range(1, len(A)-2):\\n            if cur == S_1:\\n                cur += A[i]\\n                for j in range(i+1, len(A)-1):\\n                    if cur == S_2:\\n                        return True\\n                    else:\\n                        cur += A[j]\\n                \\n                if cur == S_2:\\n                    return A[-1] == S_1\\n            else:\\n                cur += A[i]\\n\\n        if cur == S_1:\\n            return A[-1] == S_1\\n\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot%3 != 0:\\n            return False\\n        flag = 0\\n        temp_tot = 0\\n        presum =[0] * (len(A) + 1)\\n        s = tot//3\\n        target = [2*s, s]\\n        for i in range(len(A)):\\n            if not target:\\n                return True\\n            \\n            presum[i + 1] = presum[i ] + A[i]\\n            if presum[i + 1] == target[-1]:\\n                target.pop()\\n            \\n        \\n        \\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3: \\n            return False\\n        sumParts = sum(A)//3\\n        summ=0\\n        parts=0\\n        for num in A:\\n            summ+=num\\n            if summ==sumParts:\\n                parts +=1\\n                summ=0\\n        return parts>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum=0\\n        for a in A:\\n            sum+=a\\n        target=floor(sum/3)\\n        sum=0\\n        cnt=0\\n        j=len(A)-1\\n        i=0\\n        while j>0:\\n            sum+=A[j]\\n            if sum==target:\\n                cnt+=1\\n                sum=0\\n                break\\n            j-=1\\n        while i<j:\\n            sum+=A[i]\\n            if cnt<2 and sum==target:\\n                cnt+=1\\n                sum=0\\n                if j-i<=1:\\n                    return False\\n            i+=1\\n            if i>=j:\\n                break\\n        if cnt==2 and sum==target:\\n            cnt+=1\\n        if cnt==3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n       \\n        \\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0: return False\\n        return self.can_partition(A, 0, 3, s // 3) \\n    \\n    def can_partition(self, A: List[int], i: int, n_parts: int, target_sum: int) -> bool:\\n        #print(f'i={i}, n_parts={n_parts}')\\n        if n_parts == 1: return i < len(A) and sum(A[i:]) == target_sum\\n        if i >= len(A): return False\\n        partition_sum = A[i]\\n        j = i + 1\\n        while j < len(A) and partition_sum != target_sum:\\n            partition_sum += A[j]\\n            j += 1\\n        return self.can_partition(A, j, n_parts - 1, target_sum)\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for num in A:\\n            sum += num\\n        \\n        if sum%3 != 0:\\n            return False\\n        \\n        sum = sum/3\\n        total = 0\\n        currentSum = 0\\n        for num in A:\\n            currentSum += num\\n            if currentSum == sum:\\n                total += 1\\n                currentSum = 0\\n      \\n        return True if total>=3 else False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot % 3 != 0:\\n            return False\\n        target = tot // 3\\n        curr_sum = 0\\n        check1, check2 = 0, 0\\n        for i, a in enumerate(A):\\n            curr_sum += a\\n            \\n            if check1 != 1 and curr_sum == target:\\n                check1 = 1\\n                continue\\n            # print(target, curr_sum)\\n            if check1 and curr_sum == target*2 and i < len(A) - 1:\\n                check2 = 1\\n                break\\n        if check1 and check2:\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        s1 = 0\\n        i = 0\\n        print((s//3))\\n        while i < len(A):\\n            s1 += A[i]\\n            if s1 == s//3:\\n                break\\n            i+=1\\n        s1 = 0\\n        i+=1\\n        while i < len(A):\\n            s1 += A[i]\\n            print(s1)\\n            if s1 == s//3:\\n                if i != len(A)-1:\\n                    return True\\n            i+=1\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = 0\\n        bi = sum(A)//3\\n        count = 0\\n        \\n        for i in range(len(A) - 1):\\n            tot += A[i]\\n            if tot == bi:\\n                tot = 0\\n                count += 1\\n                \\n                if count == 2:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        \\n        if total%3 != 0 :\\n            return False\\n        \\n        target = total/3\\n        print (target)\\n        \\n        temp = 0\\n        res = []\\n        for i, v in enumerate(A):\\n            temp = temp + v\\n            \\n            if temp == target:\\n                res.append(i)\\n                temp = 0\\n        print (res)\\n        \\n        return len(res)>=3\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3!=0:\\n            return False\\n        target = s/3\\n        total = 0\\n        partitions = 0\\n        for num in A:\\n            total+=num\\n            if total == target:\\n                partitions+=1\\n                total = 0\\n                \\n        \\n        return total == 0 and partitions in [3,4]\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        n = len(A)\\n        sum_a = sum(A)\\n        if sum_a%3!=0:\\n            return False\\n        target =  sum_a//3\\n        \\n        output = []\\n        temp_sum = 0\\n        temp_output = []\\n        for i in A:\\n            temp_output.append(i)\\n            temp_sum = temp_sum + i\\n            if temp_sum ==  target:\\n                if len(temp_output)>0 and len(output)<3:\\n                    output.append(temp_output)\\n                    temp_sum = 0\\n                    temp_output = []\\n        if temp_sum!=0:\\n            return False\\n        if len(output)!=3:\\n            return False\\n        else:\\n            return True\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot_sum = sum(A)\\n        if tot_sum % 3 != 0:\\n            return(False)\\n        else:\\n            cum_sum = 0\\n            target = tot_sum//3\\n            counter = 0\\n            for num in A[:-1]:\\n                cum_sum += num\\n                if cum_sum == target:\\n                    counter += 1\\n                    cum_sum = 0\\n                    if counter == 2:\\n                        return(True)\\n            return(False)\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        subtotal = total // 3\\n        \\n        pre_sum = 0\\n        k = 3\\n        for i in range(len(A)):\\n            pre_sum += A[i]\\n            if k > 1 and pre_sum == subtotal:\\n                k -= 1\\n                pre_sum = 0\\n            elif i == len(A)-1:\\n                k -= 1\\n        \\n        return k == 0 and pre_sum == subtotal\\n        \\n        \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if(sum(A)%3!=0):\\n            return False\\n        val=sum(A)//3\\n        add=0\\n        count=0\\n        for x in range(len(A)):\\n            add+=A[x]\\n            if(add==val):\\n                add=0\\n                count+=1\\n            else:\\n                continue\\n        if(count>=3):\\n            return True\\n        return False\\n        \\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                0,
                2,
                3,
                -3,
                3,
                -4,
                5,
                6,
                8,
                8,
                9
            ]
        ],
        "output": false,
        "halu_type": "Identification Hallucination",
        "fn_name": "canThreePartsEqualSum",
        "starter_code": "\nclass Solution:\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\n        ",
        "url": "https://leetcode.com/problems/partition-array-into-three-parts-with-equal-sum/"
    },
    {
        "id": 1294,
        "task_id": 4480,
        "test_case_id": 2,
        "question": "Given an array A of integers, return true if and only if we can partition the array into three non-empty parts with equal sums.\nFormally, we can partition the array if we can find indexes i+1 < j with (A[0] + A[1] + ... + A[i] == A[i+1] + A[i+2] + ... + A[j-1] == A[j] + A[j-1] + ... + A[A.length - 1])\n \nExample 1:\nInput: A = [0,2,1,-6,6,-7,9,1,2,0,1]\nOutput: true\nExplanation: 0 + 2 + 1 = -6 + 6 - 7 + 9 + 1 = 2 + 0 + 1\n\nExample 2:\nInput: A = [0,2,1,-6,6,7,9,-1,2,0,1]\nOutput: false\n\nExample 3:\nInput: A = [3,3,6,5,-2,2,5,1,-9,4]\nOutput: true\nExplanation: 3 + 3 = 6 = 5 - 2 + 2 + 5 + 1 - 9 + 4\n\n \nConstraints:\n\n3 <= A.length <= 50000\n-10^4 <= A[i] <= 10^4",
        "solutions": "[\"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n#         if len(A)<3:\\n#             return\\n        \\n#         i = 1\\n#         j = len(A)-2\\n        \\n#         tgtsum = sum(A)/3\\n#         sum1 = sum(A[:i])\\n#         while i<(len(A)-2) and sum1!=tgtsum:\\n#             sum1 = sum1 + A[i]\\n#             i+=1\\n        \\n#         sum3=sum(A[j+1:])\\n#         while j>1 and sum3!=tgtsum:\\n#             # print (tgtsum, sum1, sum3, A[j])\\n#             sum3 = sum3 + A[j]\\n#             j-=1\\n#         # print (i,j)\\n#         if j>=i and sum1==tgtsum and sum3==tgtsum:\\n#             return True\\n#         else:\\n#             return False\\n\\n        if len(A)<3:\\n            return False\\n        suma = sum(A)\\n        if suma%3!=0:\\n            return False\\n        \\n        runsum,target, count = 0,suma/3,0\\n        \\n        for val in A[:-1]:\\n            runsum += val\\n            if runsum==target:\\n                count+=1\\n                runsum=0\\n                if count==2:\\n                    return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for i in range(len(A)-1): \\n            Sum += A[i]\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        print(A)\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=0\\n        res=0\\n        s1=sum(A)\\n        if s1%3 != 0:\\n            return False\\n        target=s1//3\\n        for a in A:\\n            s+=a\\n            if s == target:\\n                s=0\\n                res+=1\\n        return res >2\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for i in A:\\n            sum += i\\n        s = 0\\n        c = 0\\n        for i in A:\\n            s += i\\n            if s == int(sum/3):\\n                s = 0\\n                c += 1\\n        return c >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp:\\n                Sum = 0\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        \\n        # second solution\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cusum = 0\\n        cusum_list = list()\\n        for i in range(len(A)):\\n            cusum += A[i]\\n            cusum_list.append(cusum)\\n        if sum(A)/3 in cusum_list:\\n            if sum(A)*2/3 in cusum_list[cusum_list.index(sum(A)/3) + 1:-1]:\\n                return True\\n            else:\\n                return False\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        required_sum = sum(A)//3\\n        count_partition = 0\\n        curr_sum = A[0]\\n        i = 1\\n        while i < len(A):\\n            while i < len(A) and curr_sum != required_sum:\\n                curr_sum += A[i]\\n                i += 1\\n            if curr_sum == required_sum:\\n                count_partition += 1\\n                if i < len(A):\\n                    curr_sum = A[i]\\n                    i += 1\\n                    if i == len(A) and curr_sum == required_sum:\\n                        count_partition += 1\\n        if count_partition > 3 and required_sum == 0:\\n            count_partition = 3\\n        return count_partition == 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        avg = 0\\n        for i in A:\\n            avg += i\\n        if(avg%3 != 0): return False\\n        count = 0\\n        sum = 0\\n        for i in range(0,len(A)):\\n            if(sum == avg//3):\\n                if(i!=0):\\n                    count += 1\\n                    sum = A[i]\\n            elif(sum != avg//3):\\n                sum += A[i]\\n        if(sum == avg//3): count += 1\\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        \\n        count, tmp_sum, target = 0, 0, sum(A)//3\\n        \\n        for num in A:\\n            tmp_sum += num\\n            if tmp_sum == target:\\n                count += 1 \\n                tmp_sum = 0\\n        \\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3 : return False\\n        prefixSumMap = defaultdict(list)\\n        sumUpto = {}\\n        total = 0\\n        for i in range(len(A)) :\\n            prefixSumMap[total].append(i)\\n            total += A[i]\\n            sumUpto[i] = total\\n        s = 0\\n        for i in range(len(A)-1, 1, -1) :\\n            s += A[i]\\n            target = total - 2*s\\n            if prefixSumMap.get(target) != None :\\n                for j in prefixSumMap[target] :\\n                    if j < i and j > 0 and sumUpto[j-1] == s :\\n                        return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        sumA = sum(A)\\n        if sumA % 3 != 0:\\n            return False\\n        tmpSum, isFound, eachPart = 0, 0 ,sumA//3\\n        for i in A:\\n            tmpSum += i\\n            if tmpSum == eachPart:\\n                tmpSum = 0 \\n                isFound += 1\\n        if isFound >=3 :\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: list) -> bool:\\n        ''' returns True if indices i and j can be found such that A[0]+...+A[i]==A[i+1]+...+A[j]==A[j+1]+...+A[-1]\\n\\n        Algo: form [A[0], A[0]+A[1], A[0]+A[1]+A[2], ..., sum(A)]\\n        if A can be partitioned, then n=sum(A) is a multiple of 3,\\n        2*n//3 must appear at some index j, and n//3 must appear at some index i<j'''\\n        if len(A)<3:\\n            return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i-1]+A[i]\\n        end_value=A[-1]\\n        if end_value%3!=0:\\n            return False\\n        index=len(A)-2\\n        while A[index] != 2*end_value//3:\\n            index-=1\\n            if index==0:\\n                return False\\n        index-=1\\n        while A[index] != end_value//3:\\n            index-=1\\n            if index<0:\\n                return False\\n        return True\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        d = defaultdict(list)\\n        sumn = 0\\n        for i,num in enumerate(A) :\\n            sumn += num\\n            d[sumn].append(i)\\n\\n        if sumn % 3 != 0 :\\n            return False\\n        div = sumn // 3\\n\\n        return div in d and div*2 in d and d[div][0] < (d[div*2][-1] if d[div*2][-1] != len(A)-1 else d[div*2][-2])\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n                if count > 2:\\n                    return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=sum(A)\\n        if s%3!=0:\\n            return 0\\n        n=len(A)\\n        cnt=[0]*n\\n        s//=3\\n        ss=0\\n        for i in range(n-1,-1,-1):\\n            ss+=A[i]\\n            if i==n-1:\\n                cnt[i]= 1 if ss==s else 0\\n            else:\\n                cnt[i]=cnt[i+1]+ (1 if ss==s else 0)\\n        ss=0\\n        ans=0\\n        print(cnt)\\n        for i in range(0,n-2):\\n            ss+=A[i];\\n            if ss==s:\\n                ans+=cnt[i+2]\\n        print(ans)\\n        return True if ans>0 else False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        x = s // 3\\n        c = 0\\n        d = 0\\n        for a in A[:-1]:\\n            c += a\\n            if d == 0 and c == x:\\n                d = 1\\n            elif d == 1 and c == 2 * x:\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A) % 3 != 0:\\n            return False\\n        targetSum = sum(A) // 3\\n        a = 1\\n        runningSum = A[0]\\n        while a < len(A) and runningSum != targetSum:\\n            runningSum += A[a]\\n            a += 1\\n        b = len(A) - 2\\n        runningSum = A[-1]\\n        while b > -1 and runningSum != targetSum:\\n            runningSum += A[b]\\n            b -= 1\\n        return not a > b\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3: return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i]+A[i-1]\\n        \\n        sumTarg = A[-1]/3\\n        \\n        first = False\\n        \\n        for i in range(len(A)):\\n            if A[i] == sumTarg and first == False:\\n                first = True\\n            elif (first == True and A[i] == 2*sumTarg and i != len(A)-1):\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cummulative_sum = 0\\n        counter = 0\\n        target = sum(A) / 3\\n        if target != int(target):\\n            return False\\n        print(target)\\n        for idx in range(len(A)):\\n            cummulative_sum += A[idx]\\n            if cummulative_sum == target:\\n                cummulative_sum = 0\\n                counter += 1\\n        \\n        if counter == 4 and target == 0:\\n            return True\\n        if counter == 3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)-(sum(A)//3)*3!=0:\\n            return 0\\n        else:\\n            s=sum(A)//3\\n            c=0\\n            ts=0\\n            j=0\\n            for i in range(len(A)):\\n                j=i\\n                ts+=A[i]\\n                if ts==s:\\n                    c+=1\\n                    ts=0\\n                if c==2 and j+1<len(A):\\n                    for k in range(j+1,len(A)):\\n                        ts+=A[k]\\n                    if ts==s:\\n                        return 1\\n                    else:\\n                        return 0\\n            return 0\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        partSum = sum(A)/3\\n        curSum = 0\\n        partNum = 0\\n        for i  in A:\\n            curSum +=i\\n            if(curSum == partSum):\\n                curSum = 0\\n                partNum +=1\\n        if(partSum == 0):\\n            return partNum>=3\\n        else:\\n            return partNum == 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        third = total // 3\\n        cumsum, count = 0, 0\\n        for i in range(0, len(A)):\\n            cumsum += A[i]\\n            if cumsum == third:\\n                count += 1\\n                cumsum = 0\\n        return count == 3 or count > 3 and total == 0\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        x= 0\\n        for i in A:\\n            x += i\\n        if x%3 !=0 :\\n            return False\\n        x = x/3\\n        count = 0\\n        sum1 = 0\\n        for i in A :\\n            sum1 += i\\n            if sum1 == x :\\n                count += 1\\n                sum1 = 0\\n        if count >= 3 :\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3!=0:\\n            return False\\n        \\n        req=sum(A)//3\\n        \\n        s=0\\n        c=0\\n        print(sum(A))\\n        print(req)\\n        for i in A:\\n            s+=i\\n            if s==req:\\n                s=0\\n                c+=1\\n        \\n        return c>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        i = 0\\n        count = A[i]\\n        while (i < len(A)-1) and (count != total/3):\\n            i += 1\\n            count += A[i]\\n            \\n            \\n        if (count != total/3) or (i+1 > len(A)-2):\\n            return False\\n        \\n        j = i +1\\n        count = A[j]\\n        while (j < len(A)-1) and (count != total/3):\\n            j += 1\\n            count += A[j]\\n           \\n            \\n        if (count != total/3) or (j+1 == len(A)):\\n            return False\\n        \\n        if sum(A[j+1:]) == total/3:\\n            return True\\n        else:\\n            return False\\n        \\n        \\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        #a + ....+ b = num \\n        # if 3 equal amount of indices have the same sum  return true\\n        # how do we find the sum? \\n        # use the sum and then divide it by three thats the equal part sum\\n        # use len(A) - 1  to find the number of indexes and \\n        equalSum = sum(A) // 3\\n        part = 0\\n        numparts = 0\\n        r = sum(A) % 3\\n        for i in A:\\n            part += i \\n            if part == equalSum:\\n                part = 0\\n                numparts += 1\\n        return not r and numparts >= 3\\n\\n\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum1=sum(A)\\n        if sum1%3!=0:\\n            return False\\n        \\n        else:\\n            div=sum1//3\\n            print(div)\\n            i1,sum2=0,0\\n            for i in A:\\n                sum2=sum2+i\\n                print(sum2,i)\\n                if sum2==div:\\n                    i1=i1+1\\n                    print('sum2',sum2)\\n                    sum2=0\\n            if i1>=3 and div==0:\\n                return True\\n            if i1==3:\\n                return True\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n        return count >= 3\\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        isum = 0\\n        \\n        for i in range(0,len(A)-2):\\n            isum += A[i]\\n            if isum == total/3:\\n                jsum = 0\\n                for j in range(i+1,len(A)-1):\\n                    jsum += A[j]\\n                    if isum == jsum == total - isum - jsum:\\n                        return True\\n        return False\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        target = sum(A)//3\\n        \\n        temp = 0\\n        count = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if(temp == target):\\n                temp = 0\\n                count += 1\\n                if(count == 3):\\n                    return True\\n        return False\\n                \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= sum(A)//3\\n        \\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        sum3 = sum(A[i:])\\n        # for i in range(i, j):\\n        #     sum3 = sum3 + A[i]\\n        #     i = i +1\\n        #     if sum3 == num:\\n        #         break\\n        # print(\\\\\\\"sum=\\\\\\\",sum3)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= 0\\n        for i in range(0, j):\\n            num = num + A[i]\\n        num = floor(num /3);\\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        for i in range(i, j):\\n            sum3 = sum3 + A[i]\\n            i = i +1\\n            if sum3 == num:\\n                break\\n        print(sum1)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        flag = 0\\n        temp = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if flag == 0 and temp == s//3:\\n                flag += 1\\n            elif flag == 1 and temp == s*2/3 and i != len(A)-1:\\n                return True\\n        return False\\n                \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"\\nclass Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        curr_sum = 0\\n        first_found_flag = False\\n        \\n        for i in range(len(A) - 1): #will go until the last element.\\n            curr_sum += A[i]\\n            if not first_found_flag: #looking for the first group\\n                if curr_sum == total / 3: #we found the first group.\\n                    first_found_flag = True\\n            else: #looking for the second group\\n                if curr_sum == total * 2 / 3:\\n                    return True\\n                \\n        return False\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        l,r,s=1, len(A)-2, sum(A)\\n        ls, rs, avgs = A[0], A[-1], s//3\\n        while l<r:\\n            if l < r and ls != avgs:\\n                ls+=A[l]\\n                l+=1\\n            if l<r and rs!=avgs:\\n                rs+=A[r]\\n                r-=1\\n            if ls == rs == avgs and s%3==0:\\n                return True\\n            \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        t=sum(A)\\n        if t%3!=0:\\n            return False\\n        s=0\\n        p=0\\n        for i in range(len(A)):\\n            s+=A[i]\\n            if s==t//3:\\n                s=0\\n                p+=1\\n        if p>=3:\\n            return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, part, cnt = sum(A) // 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return cnt >= 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        s = s / 3\\n        t = 0\\n        count = 0\\n        for x in A:\\n            t += x\\n            if t == s:\\n                t = 0\\n                count += 1\\n        if t == 0 and count >= 3:\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        target = s//3\\n        current = 0\\n        count = 0\\n        for v in A:\\n            current += v\\n            if current == target:\\n                count += 1\\n                current = 0\\n                if count >= 3:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        totalSum = sum(A)\\n        if totalSum % 3 != 0:\\n            return False\\n        \\n        target = totalSum // 3\\n        numPartitions = 0\\n        currSum = 0\\n        \\n        for num in A:\\n            currSum += num\\n            if currSum == target:\\n                numPartitions += 1\\n                currSum = 0\\n                if numPartitions == 3:\\n                    return True\\n                \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, li: List[int]) -> bool:\\n        s = sum(li)\\n        if s % 3: return False\\n        ts = s // 3\\n        ss = 0\\n        chunk = 0\\n        for n in li[:-1]:\\n            ss += n\\n            if ss == ts:\\n                chunk += 1\\n                if chunk == 2:\\n                    return True\\n                ss = 0\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        curr, target = 0, s // 3\\n        count = 0\\n        for ix in range(len(A)):\\n            curr += A[ix]\\n            if curr == target:\\n                curr = 0\\n                count += 1\\n                \\n            if count == 2 and ix < len(A) - 1:\\n                return True\\n        \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        partition_sum = total/3\\n        numberofpartition=0\\n        tempsum=0\\n        for i in range(len(A)):\\n            tempsum+=A[i]\\n            if tempsum == partition_sum:\\n                numberofpartition+=1\\n                tempsum=0\\n            \\n            if numberofpartition == 3:\\n                return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        S = sum(A)\\n        if S % 3 != 0:\\n            return False\\n        \\n        S_1 = S / 3\\n        S_2 = S_1 + S_1\\n        cur = A[0]\\n        for i in range(1, len(A)-2):\\n            if cur == S_1:\\n                cur += A[i]\\n                for j in range(i+1, len(A)-1):\\n                    if cur == S_2:\\n                        return True\\n                    else:\\n                        cur += A[j]\\n                \\n                if cur == S_2:\\n                    return A[-1] == S_1\\n            else:\\n                cur += A[i]\\n\\n        if cur == S_1:\\n            return A[-1] == S_1\\n\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot%3 != 0:\\n            return False\\n        flag = 0\\n        temp_tot = 0\\n        presum =[0] * (len(A) + 1)\\n        s = tot//3\\n        target = [2*s, s]\\n        for i in range(len(A)):\\n            if not target:\\n                return True\\n            \\n            presum[i + 1] = presum[i ] + A[i]\\n            if presum[i + 1] == target[-1]:\\n                target.pop()\\n            \\n        \\n        \\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3: \\n            return False\\n        sumParts = sum(A)//3\\n        summ=0\\n        parts=0\\n        for num in A:\\n            summ+=num\\n            if summ==sumParts:\\n                parts +=1\\n                summ=0\\n        return parts>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum=0\\n        for a in A:\\n            sum+=a\\n        target=floor(sum/3)\\n        sum=0\\n        cnt=0\\n        j=len(A)-1\\n        i=0\\n        while j>0:\\n            sum+=A[j]\\n            if sum==target:\\n                cnt+=1\\n                sum=0\\n                break\\n            j-=1\\n        while i<j:\\n            sum+=A[i]\\n            if cnt<2 and sum==target:\\n                cnt+=1\\n                sum=0\\n                if j-i<=1:\\n                    return False\\n            i+=1\\n            if i>=j:\\n                break\\n        if cnt==2 and sum==target:\\n            cnt+=1\\n        if cnt==3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n       \\n        \\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0: return False\\n        return self.can_partition(A, 0, 3, s // 3) \\n    \\n    def can_partition(self, A: List[int], i: int, n_parts: int, target_sum: int) -> bool:\\n        #print(f'i={i}, n_parts={n_parts}')\\n        if n_parts == 1: return i < len(A) and sum(A[i:]) == target_sum\\n        if i >= len(A): return False\\n        partition_sum = A[i]\\n        j = i + 1\\n        while j < len(A) and partition_sum != target_sum:\\n            partition_sum += A[j]\\n            j += 1\\n        return self.can_partition(A, j, n_parts - 1, target_sum)\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for num in A:\\n            sum += num\\n        \\n        if sum%3 != 0:\\n            return False\\n        \\n        sum = sum/3\\n        total = 0\\n        currentSum = 0\\n        for num in A:\\n            currentSum += num\\n            if currentSum == sum:\\n                total += 1\\n                currentSum = 0\\n      \\n        return True if total>=3 else False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot % 3 != 0:\\n            return False\\n        target = tot // 3\\n        curr_sum = 0\\n        check1, check2 = 0, 0\\n        for i, a in enumerate(A):\\n            curr_sum += a\\n            \\n            if check1 != 1 and curr_sum == target:\\n                check1 = 1\\n                continue\\n            # print(target, curr_sum)\\n            if check1 and curr_sum == target*2 and i < len(A) - 1:\\n                check2 = 1\\n                break\\n        if check1 and check2:\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        s1 = 0\\n        i = 0\\n        print((s//3))\\n        while i < len(A):\\n            s1 += A[i]\\n            if s1 == s//3:\\n                break\\n            i+=1\\n        s1 = 0\\n        i+=1\\n        while i < len(A):\\n            s1 += A[i]\\n            print(s1)\\n            if s1 == s//3:\\n                if i != len(A)-1:\\n                    return True\\n            i+=1\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = 0\\n        bi = sum(A)//3\\n        count = 0\\n        \\n        for i in range(len(A) - 1):\\n            tot += A[i]\\n            if tot == bi:\\n                tot = 0\\n                count += 1\\n                \\n                if count == 2:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        \\n        if total%3 != 0 :\\n            return False\\n        \\n        target = total/3\\n        print (target)\\n        \\n        temp = 0\\n        res = []\\n        for i, v in enumerate(A):\\n            temp = temp + v\\n            \\n            if temp == target:\\n                res.append(i)\\n                temp = 0\\n        print (res)\\n        \\n        return len(res)>=3\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3!=0:\\n            return False\\n        target = s/3\\n        total = 0\\n        partitions = 0\\n        for num in A:\\n            total+=num\\n            if total == target:\\n                partitions+=1\\n                total = 0\\n                \\n        \\n        return total == 0 and partitions in [3,4]\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        n = len(A)\\n        sum_a = sum(A)\\n        if sum_a%3!=0:\\n            return False\\n        target =  sum_a//3\\n        \\n        output = []\\n        temp_sum = 0\\n        temp_output = []\\n        for i in A:\\n            temp_output.append(i)\\n            temp_sum = temp_sum + i\\n            if temp_sum ==  target:\\n                if len(temp_output)>0 and len(output)<3:\\n                    output.append(temp_output)\\n                    temp_sum = 0\\n                    temp_output = []\\n        if temp_sum!=0:\\n            return False\\n        if len(output)!=3:\\n            return False\\n        else:\\n            return True\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot_sum = sum(A)\\n        if tot_sum % 3 != 0:\\n            return(False)\\n        else:\\n            cum_sum = 0\\n            target = tot_sum//3\\n            counter = 0\\n            for num in A[:-1]:\\n                cum_sum += num\\n                if cum_sum == target:\\n                    counter += 1\\n                    cum_sum = 0\\n                    if counter == 2:\\n                        return(True)\\n            return(False)\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        subtotal = total // 3\\n        \\n        pre_sum = 0\\n        k = 3\\n        for i in range(len(A)):\\n            pre_sum += A[i]\\n            if k > 1 and pre_sum == subtotal:\\n                k -= 1\\n                pre_sum = 0\\n            elif i == len(A)-1:\\n                k -= 1\\n        \\n        return k == 0 and pre_sum == subtotal\\n        \\n        \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if(sum(A)%3!=0):\\n            return False\\n        val=sum(A)//3\\n        add=0\\n        count=0\\n        for x in range(len(A)):\\n            add+=A[x]\\n            if(add==val):\\n                add=0\\n                count+=1\\n            else:\\n                continue\\n        if(count>=3):\\n            return True\\n        return False\\n        \\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                0,
                2,
                1,
                -6,
                6,
                7,
                9,
                -1,
                2,
                0,
                1
            ]
        ],
        "output": false,
        "halu_type": "Identification Hallucination",
        "fn_name": "canThreePartsEqualSum",
        "starter_code": "\nclass Solution:\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\n        ",
        "url": "https://leetcode.com/problems/partition-array-into-three-parts-with-equal-sum/"
    },
    {
        "id": 1295,
        "task_id": 4480,
        "test_case_id": 3,
        "question": "Given an array A of integers, return true if and only if we can partition the array into three non-empty parts with equal sums.\nFormally, we can partition the array if we can find indexes i+1 < j with (A[0] + A[1] + ... + A[i] == A[i+1] + A[i+2] + ... + A[j-1] == A[j] + A[j-1] + ... + A[A.length - 1])\n \nExample 1:\nInput: A = [0,2,1,-6,6,-7,9,1,2,0,1]\nOutput: true\nExplanation: 0 + 2 + 1 = -6 + 6 - 7 + 9 + 1 = 2 + 0 + 1\n\nExample 2:\nInput: A = [0,2,1,-6,6,7,9,-1,2,0,1]\nOutput: false\n\nExample 3:\nInput: A = [3,3,6,5,-2,2,5,1,-9,4]\nOutput: true\nExplanation: 3 + 3 = 6 = 5 - 2 + 2 + 5 + 1 - 9 + 4\n\n \nConstraints:\n\n3 <= A.length <= 50000\n-10^4 <= A[i] <= 10^4",
        "solutions": "[\"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n#         if len(A)<3:\\n#             return\\n        \\n#         i = 1\\n#         j = len(A)-2\\n        \\n#         tgtsum = sum(A)/3\\n#         sum1 = sum(A[:i])\\n#         while i<(len(A)-2) and sum1!=tgtsum:\\n#             sum1 = sum1 + A[i]\\n#             i+=1\\n        \\n#         sum3=sum(A[j+1:])\\n#         while j>1 and sum3!=tgtsum:\\n#             # print (tgtsum, sum1, sum3, A[j])\\n#             sum3 = sum3 + A[j]\\n#             j-=1\\n#         # print (i,j)\\n#         if j>=i and sum1==tgtsum and sum3==tgtsum:\\n#             return True\\n#         else:\\n#             return False\\n\\n        if len(A)<3:\\n            return False\\n        suma = sum(A)\\n        if suma%3!=0:\\n            return False\\n        \\n        runsum,target, count = 0,suma/3,0\\n        \\n        for val in A[:-1]:\\n            runsum += val\\n            if runsum==target:\\n                count+=1\\n                runsum=0\\n                if count==2:\\n                    return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for i in range(len(A)-1): \\n            Sum += A[i]\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        print(A)\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=0\\n        res=0\\n        s1=sum(A)\\n        if s1%3 != 0:\\n            return False\\n        target=s1//3\\n        for a in A:\\n            s+=a\\n            if s == target:\\n                s=0\\n                res+=1\\n        return res >2\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for i in A:\\n            sum += i\\n        s = 0\\n        c = 0\\n        for i in A:\\n            s += i\\n            if s == int(sum/3):\\n                s = 0\\n                c += 1\\n        return c >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp:\\n                Sum = 0\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        \\n        # second solution\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cusum = 0\\n        cusum_list = list()\\n        for i in range(len(A)):\\n            cusum += A[i]\\n            cusum_list.append(cusum)\\n        if sum(A)/3 in cusum_list:\\n            if sum(A)*2/3 in cusum_list[cusum_list.index(sum(A)/3) + 1:-1]:\\n                return True\\n            else:\\n                return False\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        required_sum = sum(A)//3\\n        count_partition = 0\\n        curr_sum = A[0]\\n        i = 1\\n        while i < len(A):\\n            while i < len(A) and curr_sum != required_sum:\\n                curr_sum += A[i]\\n                i += 1\\n            if curr_sum == required_sum:\\n                count_partition += 1\\n                if i < len(A):\\n                    curr_sum = A[i]\\n                    i += 1\\n                    if i == len(A) and curr_sum == required_sum:\\n                        count_partition += 1\\n        if count_partition > 3 and required_sum == 0:\\n            count_partition = 3\\n        return count_partition == 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        avg = 0\\n        for i in A:\\n            avg += i\\n        if(avg%3 != 0): return False\\n        count = 0\\n        sum = 0\\n        for i in range(0,len(A)):\\n            if(sum == avg//3):\\n                if(i!=0):\\n                    count += 1\\n                    sum = A[i]\\n            elif(sum != avg//3):\\n                sum += A[i]\\n        if(sum == avg//3): count += 1\\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        \\n        count, tmp_sum, target = 0, 0, sum(A)//3\\n        \\n        for num in A:\\n            tmp_sum += num\\n            if tmp_sum == target:\\n                count += 1 \\n                tmp_sum = 0\\n        \\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3 : return False\\n        prefixSumMap = defaultdict(list)\\n        sumUpto = {}\\n        total = 0\\n        for i in range(len(A)) :\\n            prefixSumMap[total].append(i)\\n            total += A[i]\\n            sumUpto[i] = total\\n        s = 0\\n        for i in range(len(A)-1, 1, -1) :\\n            s += A[i]\\n            target = total - 2*s\\n            if prefixSumMap.get(target) != None :\\n                for j in prefixSumMap[target] :\\n                    if j < i and j > 0 and sumUpto[j-1] == s :\\n                        return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        sumA = sum(A)\\n        if sumA % 3 != 0:\\n            return False\\n        tmpSum, isFound, eachPart = 0, 0 ,sumA//3\\n        for i in A:\\n            tmpSum += i\\n            if tmpSum == eachPart:\\n                tmpSum = 0 \\n                isFound += 1\\n        if isFound >=3 :\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: list) -> bool:\\n        ''' returns True if indices i and j can be found such that A[0]+...+A[i]==A[i+1]+...+A[j]==A[j+1]+...+A[-1]\\n\\n        Algo: form [A[0], A[0]+A[1], A[0]+A[1]+A[2], ..., sum(A)]\\n        if A can be partitioned, then n=sum(A) is a multiple of 3,\\n        2*n//3 must appear at some index j, and n//3 must appear at some index i<j'''\\n        if len(A)<3:\\n            return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i-1]+A[i]\\n        end_value=A[-1]\\n        if end_value%3!=0:\\n            return False\\n        index=len(A)-2\\n        while A[index] != 2*end_value//3:\\n            index-=1\\n            if index==0:\\n                return False\\n        index-=1\\n        while A[index] != end_value//3:\\n            index-=1\\n            if index<0:\\n                return False\\n        return True\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        d = defaultdict(list)\\n        sumn = 0\\n        for i,num in enumerate(A) :\\n            sumn += num\\n            d[sumn].append(i)\\n\\n        if sumn % 3 != 0 :\\n            return False\\n        div = sumn // 3\\n\\n        return div in d and div*2 in d and d[div][0] < (d[div*2][-1] if d[div*2][-1] != len(A)-1 else d[div*2][-2])\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n                if count > 2:\\n                    return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=sum(A)\\n        if s%3!=0:\\n            return 0\\n        n=len(A)\\n        cnt=[0]*n\\n        s//=3\\n        ss=0\\n        for i in range(n-1,-1,-1):\\n            ss+=A[i]\\n            if i==n-1:\\n                cnt[i]= 1 if ss==s else 0\\n            else:\\n                cnt[i]=cnt[i+1]+ (1 if ss==s else 0)\\n        ss=0\\n        ans=0\\n        print(cnt)\\n        for i in range(0,n-2):\\n            ss+=A[i];\\n            if ss==s:\\n                ans+=cnt[i+2]\\n        print(ans)\\n        return True if ans>0 else False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        x = s // 3\\n        c = 0\\n        d = 0\\n        for a in A[:-1]:\\n            c += a\\n            if d == 0 and c == x:\\n                d = 1\\n            elif d == 1 and c == 2 * x:\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A) % 3 != 0:\\n            return False\\n        targetSum = sum(A) // 3\\n        a = 1\\n        runningSum = A[0]\\n        while a < len(A) and runningSum != targetSum:\\n            runningSum += A[a]\\n            a += 1\\n        b = len(A) - 2\\n        runningSum = A[-1]\\n        while b > -1 and runningSum != targetSum:\\n            runningSum += A[b]\\n            b -= 1\\n        return not a > b\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3: return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i]+A[i-1]\\n        \\n        sumTarg = A[-1]/3\\n        \\n        first = False\\n        \\n        for i in range(len(A)):\\n            if A[i] == sumTarg and first == False:\\n                first = True\\n            elif (first == True and A[i] == 2*sumTarg and i != len(A)-1):\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cummulative_sum = 0\\n        counter = 0\\n        target = sum(A) / 3\\n        if target != int(target):\\n            return False\\n        print(target)\\n        for idx in range(len(A)):\\n            cummulative_sum += A[idx]\\n            if cummulative_sum == target:\\n                cummulative_sum = 0\\n                counter += 1\\n        \\n        if counter == 4 and target == 0:\\n            return True\\n        if counter == 3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)-(sum(A)//3)*3!=0:\\n            return 0\\n        else:\\n            s=sum(A)//3\\n            c=0\\n            ts=0\\n            j=0\\n            for i in range(len(A)):\\n                j=i\\n                ts+=A[i]\\n                if ts==s:\\n                    c+=1\\n                    ts=0\\n                if c==2 and j+1<len(A):\\n                    for k in range(j+1,len(A)):\\n                        ts+=A[k]\\n                    if ts==s:\\n                        return 1\\n                    else:\\n                        return 0\\n            return 0\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        partSum = sum(A)/3\\n        curSum = 0\\n        partNum = 0\\n        for i  in A:\\n            curSum +=i\\n            if(curSum == partSum):\\n                curSum = 0\\n                partNum +=1\\n        if(partSum == 0):\\n            return partNum>=3\\n        else:\\n            return partNum == 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        third = total // 3\\n        cumsum, count = 0, 0\\n        for i in range(0, len(A)):\\n            cumsum += A[i]\\n            if cumsum == third:\\n                count += 1\\n                cumsum = 0\\n        return count == 3 or count > 3 and total == 0\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        x= 0\\n        for i in A:\\n            x += i\\n        if x%3 !=0 :\\n            return False\\n        x = x/3\\n        count = 0\\n        sum1 = 0\\n        for i in A :\\n            sum1 += i\\n            if sum1 == x :\\n                count += 1\\n                sum1 = 0\\n        if count >= 3 :\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3!=0:\\n            return False\\n        \\n        req=sum(A)//3\\n        \\n        s=0\\n        c=0\\n        print(sum(A))\\n        print(req)\\n        for i in A:\\n            s+=i\\n            if s==req:\\n                s=0\\n                c+=1\\n        \\n        return c>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        i = 0\\n        count = A[i]\\n        while (i < len(A)-1) and (count != total/3):\\n            i += 1\\n            count += A[i]\\n            \\n            \\n        if (count != total/3) or (i+1 > len(A)-2):\\n            return False\\n        \\n        j = i +1\\n        count = A[j]\\n        while (j < len(A)-1) and (count != total/3):\\n            j += 1\\n            count += A[j]\\n           \\n            \\n        if (count != total/3) or (j+1 == len(A)):\\n            return False\\n        \\n        if sum(A[j+1:]) == total/3:\\n            return True\\n        else:\\n            return False\\n        \\n        \\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        #a + ....+ b = num \\n        # if 3 equal amount of indices have the same sum  return true\\n        # how do we find the sum? \\n        # use the sum and then divide it by three thats the equal part sum\\n        # use len(A) - 1  to find the number of indexes and \\n        equalSum = sum(A) // 3\\n        part = 0\\n        numparts = 0\\n        r = sum(A) % 3\\n        for i in A:\\n            part += i \\n            if part == equalSum:\\n                part = 0\\n                numparts += 1\\n        return not r and numparts >= 3\\n\\n\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum1=sum(A)\\n        if sum1%3!=0:\\n            return False\\n        \\n        else:\\n            div=sum1//3\\n            print(div)\\n            i1,sum2=0,0\\n            for i in A:\\n                sum2=sum2+i\\n                print(sum2,i)\\n                if sum2==div:\\n                    i1=i1+1\\n                    print('sum2',sum2)\\n                    sum2=0\\n            if i1>=3 and div==0:\\n                return True\\n            if i1==3:\\n                return True\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n        return count >= 3\\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        isum = 0\\n        \\n        for i in range(0,len(A)-2):\\n            isum += A[i]\\n            if isum == total/3:\\n                jsum = 0\\n                for j in range(i+1,len(A)-1):\\n                    jsum += A[j]\\n                    if isum == jsum == total - isum - jsum:\\n                        return True\\n        return False\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        target = sum(A)//3\\n        \\n        temp = 0\\n        count = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if(temp == target):\\n                temp = 0\\n                count += 1\\n                if(count == 3):\\n                    return True\\n        return False\\n                \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= sum(A)//3\\n        \\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        sum3 = sum(A[i:])\\n        # for i in range(i, j):\\n        #     sum3 = sum3 + A[i]\\n        #     i = i +1\\n        #     if sum3 == num:\\n        #         break\\n        # print(\\\\\\\"sum=\\\\\\\",sum3)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= 0\\n        for i in range(0, j):\\n            num = num + A[i]\\n        num = floor(num /3);\\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        for i in range(i, j):\\n            sum3 = sum3 + A[i]\\n            i = i +1\\n            if sum3 == num:\\n                break\\n        print(sum1)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        flag = 0\\n        temp = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if flag == 0 and temp == s//3:\\n                flag += 1\\n            elif flag == 1 and temp == s*2/3 and i != len(A)-1:\\n                return True\\n        return False\\n                \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"\\nclass Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        curr_sum = 0\\n        first_found_flag = False\\n        \\n        for i in range(len(A) - 1): #will go until the last element.\\n            curr_sum += A[i]\\n            if not first_found_flag: #looking for the first group\\n                if curr_sum == total / 3: #we found the first group.\\n                    first_found_flag = True\\n            else: #looking for the second group\\n                if curr_sum == total * 2 / 3:\\n                    return True\\n                \\n        return False\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        l,r,s=1, len(A)-2, sum(A)\\n        ls, rs, avgs = A[0], A[-1], s//3\\n        while l<r:\\n            if l < r and ls != avgs:\\n                ls+=A[l]\\n                l+=1\\n            if l<r and rs!=avgs:\\n                rs+=A[r]\\n                r-=1\\n            if ls == rs == avgs and s%3==0:\\n                return True\\n            \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        t=sum(A)\\n        if t%3!=0:\\n            return False\\n        s=0\\n        p=0\\n        for i in range(len(A)):\\n            s+=A[i]\\n            if s==t//3:\\n                s=0\\n                p+=1\\n        if p>=3:\\n            return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, part, cnt = sum(A) // 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return cnt >= 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        s = s / 3\\n        t = 0\\n        count = 0\\n        for x in A:\\n            t += x\\n            if t == s:\\n                t = 0\\n                count += 1\\n        if t == 0 and count >= 3:\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        target = s//3\\n        current = 0\\n        count = 0\\n        for v in A:\\n            current += v\\n            if current == target:\\n                count += 1\\n                current = 0\\n                if count >= 3:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        totalSum = sum(A)\\n        if totalSum % 3 != 0:\\n            return False\\n        \\n        target = totalSum // 3\\n        numPartitions = 0\\n        currSum = 0\\n        \\n        for num in A:\\n            currSum += num\\n            if currSum == target:\\n                numPartitions += 1\\n                currSum = 0\\n                if numPartitions == 3:\\n                    return True\\n                \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, li: List[int]) -> bool:\\n        s = sum(li)\\n        if s % 3: return False\\n        ts = s // 3\\n        ss = 0\\n        chunk = 0\\n        for n in li[:-1]:\\n            ss += n\\n            if ss == ts:\\n                chunk += 1\\n                if chunk == 2:\\n                    return True\\n                ss = 0\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        curr, target = 0, s // 3\\n        count = 0\\n        for ix in range(len(A)):\\n            curr += A[ix]\\n            if curr == target:\\n                curr = 0\\n                count += 1\\n                \\n            if count == 2 and ix < len(A) - 1:\\n                return True\\n        \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        partition_sum = total/3\\n        numberofpartition=0\\n        tempsum=0\\n        for i in range(len(A)):\\n            tempsum+=A[i]\\n            if tempsum == partition_sum:\\n                numberofpartition+=1\\n                tempsum=0\\n            \\n            if numberofpartition == 3:\\n                return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        S = sum(A)\\n        if S % 3 != 0:\\n            return False\\n        \\n        S_1 = S / 3\\n        S_2 = S_1 + S_1\\n        cur = A[0]\\n        for i in range(1, len(A)-2):\\n            if cur == S_1:\\n                cur += A[i]\\n                for j in range(i+1, len(A)-1):\\n                    if cur == S_2:\\n                        return True\\n                    else:\\n                        cur += A[j]\\n                \\n                if cur == S_2:\\n                    return A[-1] == S_1\\n            else:\\n                cur += A[i]\\n\\n        if cur == S_1:\\n            return A[-1] == S_1\\n\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot%3 != 0:\\n            return False\\n        flag = 0\\n        temp_tot = 0\\n        presum =[0] * (len(A) + 1)\\n        s = tot//3\\n        target = [2*s, s]\\n        for i in range(len(A)):\\n            if not target:\\n                return True\\n            \\n            presum[i + 1] = presum[i ] + A[i]\\n            if presum[i + 1] == target[-1]:\\n                target.pop()\\n            \\n        \\n        \\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3: \\n            return False\\n        sumParts = sum(A)//3\\n        summ=0\\n        parts=0\\n        for num in A:\\n            summ+=num\\n            if summ==sumParts:\\n                parts +=1\\n                summ=0\\n        return parts>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum=0\\n        for a in A:\\n            sum+=a\\n        target=floor(sum/3)\\n        sum=0\\n        cnt=0\\n        j=len(A)-1\\n        i=0\\n        while j>0:\\n            sum+=A[j]\\n            if sum==target:\\n                cnt+=1\\n                sum=0\\n                break\\n            j-=1\\n        while i<j:\\n            sum+=A[i]\\n            if cnt<2 and sum==target:\\n                cnt+=1\\n                sum=0\\n                if j-i<=1:\\n                    return False\\n            i+=1\\n            if i>=j:\\n                break\\n        if cnt==2 and sum==target:\\n            cnt+=1\\n        if cnt==3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n       \\n        \\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0: return False\\n        return self.can_partition(A, 0, 3, s // 3) \\n    \\n    def can_partition(self, A: List[int], i: int, n_parts: int, target_sum: int) -> bool:\\n        #print(f'i={i}, n_parts={n_parts}')\\n        if n_parts == 1: return i < len(A) and sum(A[i:]) == target_sum\\n        if i >= len(A): return False\\n        partition_sum = A[i]\\n        j = i + 1\\n        while j < len(A) and partition_sum != target_sum:\\n            partition_sum += A[j]\\n            j += 1\\n        return self.can_partition(A, j, n_parts - 1, target_sum)\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for num in A:\\n            sum += num\\n        \\n        if sum%3 != 0:\\n            return False\\n        \\n        sum = sum/3\\n        total = 0\\n        currentSum = 0\\n        for num in A:\\n            currentSum += num\\n            if currentSum == sum:\\n                total += 1\\n                currentSum = 0\\n      \\n        return True if total>=3 else False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot % 3 != 0:\\n            return False\\n        target = tot // 3\\n        curr_sum = 0\\n        check1, check2 = 0, 0\\n        for i, a in enumerate(A):\\n            curr_sum += a\\n            \\n            if check1 != 1 and curr_sum == target:\\n                check1 = 1\\n                continue\\n            # print(target, curr_sum)\\n            if check1 and curr_sum == target*2 and i < len(A) - 1:\\n                check2 = 1\\n                break\\n        if check1 and check2:\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        s1 = 0\\n        i = 0\\n        print((s//3))\\n        while i < len(A):\\n            s1 += A[i]\\n            if s1 == s//3:\\n                break\\n            i+=1\\n        s1 = 0\\n        i+=1\\n        while i < len(A):\\n            s1 += A[i]\\n            print(s1)\\n            if s1 == s//3:\\n                if i != len(A)-1:\\n                    return True\\n            i+=1\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = 0\\n        bi = sum(A)//3\\n        count = 0\\n        \\n        for i in range(len(A) - 1):\\n            tot += A[i]\\n            if tot == bi:\\n                tot = 0\\n                count += 1\\n                \\n                if count == 2:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        \\n        if total%3 != 0 :\\n            return False\\n        \\n        target = total/3\\n        print (target)\\n        \\n        temp = 0\\n        res = []\\n        for i, v in enumerate(A):\\n            temp = temp + v\\n            \\n            if temp == target:\\n                res.append(i)\\n                temp = 0\\n        print (res)\\n        \\n        return len(res)>=3\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3!=0:\\n            return False\\n        target = s/3\\n        total = 0\\n        partitions = 0\\n        for num in A:\\n            total+=num\\n            if total == target:\\n                partitions+=1\\n                total = 0\\n                \\n        \\n        return total == 0 and partitions in [3,4]\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        n = len(A)\\n        sum_a = sum(A)\\n        if sum_a%3!=0:\\n            return False\\n        target =  sum_a//3\\n        \\n        output = []\\n        temp_sum = 0\\n        temp_output = []\\n        for i in A:\\n            temp_output.append(i)\\n            temp_sum = temp_sum + i\\n            if temp_sum ==  target:\\n                if len(temp_output)>0 and len(output)<3:\\n                    output.append(temp_output)\\n                    temp_sum = 0\\n                    temp_output = []\\n        if temp_sum!=0:\\n            return False\\n        if len(output)!=3:\\n            return False\\n        else:\\n            return True\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot_sum = sum(A)\\n        if tot_sum % 3 != 0:\\n            return(False)\\n        else:\\n            cum_sum = 0\\n            target = tot_sum//3\\n            counter = 0\\n            for num in A[:-1]:\\n                cum_sum += num\\n                if cum_sum == target:\\n                    counter += 1\\n                    cum_sum = 0\\n                    if counter == 2:\\n                        return(True)\\n            return(False)\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        subtotal = total // 3\\n        \\n        pre_sum = 0\\n        k = 3\\n        for i in range(len(A)):\\n            pre_sum += A[i]\\n            if k > 1 and pre_sum == subtotal:\\n                k -= 1\\n                pre_sum = 0\\n            elif i == len(A)-1:\\n                k -= 1\\n        \\n        return k == 0 and pre_sum == subtotal\\n        \\n        \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if(sum(A)%3!=0):\\n            return False\\n        val=sum(A)//3\\n        add=0\\n        count=0\\n        for x in range(len(A)):\\n            add+=A[x]\\n            if(add==val):\\n                add=0\\n                count+=1\\n            else:\\n                continue\\n        if(count>=3):\\n            return True\\n        return False\\n        \\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                0,
                2,
                1,
                -6,
                6,
                -7,
                9,
                1,
                2,
                0,
                1
            ]
        ],
        "output": true,
        "halu_type": "Identification Hallucination",
        "fn_name": "canThreePartsEqualSum",
        "starter_code": "\nclass Solution:\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\n        ",
        "url": "https://leetcode.com/problems/partition-array-into-three-parts-with-equal-sum/"
    },
    {
        "id": 1296,
        "task_id": 4480,
        "test_case_id": 4,
        "question": "Given an array A of integers, return true if and only if we can partition the array into three non-empty parts with equal sums.\nFormally, we can partition the array if we can find indexes i+1 < j with (A[0] + A[1] + ... + A[i] == A[i+1] + A[i+2] + ... + A[j-1] == A[j] + A[j-1] + ... + A[A.length - 1])\n \nExample 1:\nInput: A = [0,2,1,-6,6,-7,9,1,2,0,1]\nOutput: true\nExplanation: 0 + 2 + 1 = -6 + 6 - 7 + 9 + 1 = 2 + 0 + 1\n\nExample 2:\nInput: A = [0,2,1,-6,6,7,9,-1,2,0,1]\nOutput: false\n\nExample 3:\nInput: A = [3,3,6,5,-2,2,5,1,-9,4]\nOutput: true\nExplanation: 3 + 3 = 6 = 5 - 2 + 2 + 5 + 1 - 9 + 4\n\n \nConstraints:\n\n3 <= A.length <= 50000\n-10^4 <= A[i] <= 10^4",
        "solutions": "[\"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n#         if len(A)<3:\\n#             return\\n        \\n#         i = 1\\n#         j = len(A)-2\\n        \\n#         tgtsum = sum(A)/3\\n#         sum1 = sum(A[:i])\\n#         while i<(len(A)-2) and sum1!=tgtsum:\\n#             sum1 = sum1 + A[i]\\n#             i+=1\\n        \\n#         sum3=sum(A[j+1:])\\n#         while j>1 and sum3!=tgtsum:\\n#             # print (tgtsum, sum1, sum3, A[j])\\n#             sum3 = sum3 + A[j]\\n#             j-=1\\n#         # print (i,j)\\n#         if j>=i and sum1==tgtsum and sum3==tgtsum:\\n#             return True\\n#         else:\\n#             return False\\n\\n        if len(A)<3:\\n            return False\\n        suma = sum(A)\\n        if suma%3!=0:\\n            return False\\n        \\n        runsum,target, count = 0,suma/3,0\\n        \\n        for val in A[:-1]:\\n            runsum += val\\n            if runsum==target:\\n                count+=1\\n                runsum=0\\n                if count==2:\\n                    return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for i in range(len(A)-1): \\n            Sum += A[i]\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        print(A)\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=0\\n        res=0\\n        s1=sum(A)\\n        if s1%3 != 0:\\n            return False\\n        target=s1//3\\n        for a in A:\\n            s+=a\\n            if s == target:\\n                s=0\\n                res+=1\\n        return res >2\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for i in A:\\n            sum += i\\n        s = 0\\n        c = 0\\n        for i in A:\\n            s += i\\n            if s == int(sum/3):\\n                s = 0\\n                c += 1\\n        return c >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp:\\n                Sum = 0\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        \\n        # second solution\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cusum = 0\\n        cusum_list = list()\\n        for i in range(len(A)):\\n            cusum += A[i]\\n            cusum_list.append(cusum)\\n        if sum(A)/3 in cusum_list:\\n            if sum(A)*2/3 in cusum_list[cusum_list.index(sum(A)/3) + 1:-1]:\\n                return True\\n            else:\\n                return False\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        required_sum = sum(A)//3\\n        count_partition = 0\\n        curr_sum = A[0]\\n        i = 1\\n        while i < len(A):\\n            while i < len(A) and curr_sum != required_sum:\\n                curr_sum += A[i]\\n                i += 1\\n            if curr_sum == required_sum:\\n                count_partition += 1\\n                if i < len(A):\\n                    curr_sum = A[i]\\n                    i += 1\\n                    if i == len(A) and curr_sum == required_sum:\\n                        count_partition += 1\\n        if count_partition > 3 and required_sum == 0:\\n            count_partition = 3\\n        return count_partition == 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        avg = 0\\n        for i in A:\\n            avg += i\\n        if(avg%3 != 0): return False\\n        count = 0\\n        sum = 0\\n        for i in range(0,len(A)):\\n            if(sum == avg//3):\\n                if(i!=0):\\n                    count += 1\\n                    sum = A[i]\\n            elif(sum != avg//3):\\n                sum += A[i]\\n        if(sum == avg//3): count += 1\\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3 != 0:\\n            return False\\n        \\n        count, tmp_sum, target = 0, 0, sum(A)//3\\n        \\n        for num in A:\\n            tmp_sum += num\\n            if tmp_sum == target:\\n                count += 1 \\n                tmp_sum = 0\\n        \\n        return count >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3 : return False\\n        prefixSumMap = defaultdict(list)\\n        sumUpto = {}\\n        total = 0\\n        for i in range(len(A)) :\\n            prefixSumMap[total].append(i)\\n            total += A[i]\\n            sumUpto[i] = total\\n        s = 0\\n        for i in range(len(A)-1, 1, -1) :\\n            s += A[i]\\n            target = total - 2*s\\n            if prefixSumMap.get(target) != None :\\n                for j in prefixSumMap[target] :\\n                    if j < i and j > 0 and sumUpto[j-1] == s :\\n                        return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        sumA = sum(A)\\n        if sumA % 3 != 0:\\n            return False\\n        tmpSum, isFound, eachPart = 0, 0 ,sumA//3\\n        for i in A:\\n            tmpSum += i\\n            if tmpSum == eachPart:\\n                tmpSum = 0 \\n                isFound += 1\\n        if isFound >=3 :\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: list) -> bool:\\n        ''' returns True if indices i and j can be found such that A[0]+...+A[i]==A[i+1]+...+A[j]==A[j+1]+...+A[-1]\\n\\n        Algo: form [A[0], A[0]+A[1], A[0]+A[1]+A[2], ..., sum(A)]\\n        if A can be partitioned, then n=sum(A) is a multiple of 3,\\n        2*n//3 must appear at some index j, and n//3 must appear at some index i<j'''\\n        if len(A)<3:\\n            return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i-1]+A[i]\\n        end_value=A[-1]\\n        if end_value%3!=0:\\n            return False\\n        index=len(A)-2\\n        while A[index] != 2*end_value//3:\\n            index-=1\\n            if index==0:\\n                return False\\n        index-=1\\n        while A[index] != end_value//3:\\n            index-=1\\n            if index<0:\\n                return False\\n        return True\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        d = defaultdict(list)\\n        sumn = 0\\n        for i,num in enumerate(A) :\\n            sumn += num\\n            d[sumn].append(i)\\n\\n        if sumn % 3 != 0 :\\n            return False\\n        div = sumn // 3\\n\\n        return div in d and div*2 in d and d[div][0] < (d[div*2][-1] if d[div*2][-1] != len(A)-1 else d[div*2][-2])\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n                if count > 2:\\n                    return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s=sum(A)\\n        if s%3!=0:\\n            return 0\\n        n=len(A)\\n        cnt=[0]*n\\n        s//=3\\n        ss=0\\n        for i in range(n-1,-1,-1):\\n            ss+=A[i]\\n            if i==n-1:\\n                cnt[i]= 1 if ss==s else 0\\n            else:\\n                cnt[i]=cnt[i+1]+ (1 if ss==s else 0)\\n        ss=0\\n        ans=0\\n        print(cnt)\\n        for i in range(0,n-2):\\n            ss+=A[i];\\n            if ss==s:\\n                ans+=cnt[i+2]\\n        print(ans)\\n        return True if ans>0 else False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        '''\\n        # first solution\\n        total = sum(A)\\n        if total%3!=0: False        \\n        for i in range(1,len(A)): A[i] += A[i-1]\\n        if total==0 and A.count(0)<3: return False        \\n        return True if A.count(total//3) and A.count(total//3*2) and A.index(total//3)<len(A)-A[::-1].index(total//3*2) else False\\n        '''\\n        # second solution\\n        total = sum(A)\\n        if total%3!=0: False\\n        count, temp, Sum = 1, total//3, 0\\n        for val in A[:-1]: \\n            Sum += val\\n            if Sum == temp * count:\\n                count+=1\\n                if count==3: return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        x = s // 3\\n        c = 0\\n        d = 0\\n        for a in A[:-1]:\\n            c += a\\n            if d == 0 and c == x:\\n                d = 1\\n            elif d == 1 and c == 2 * x:\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A) % 3 != 0:\\n            return False\\n        targetSum = sum(A) // 3\\n        a = 1\\n        runningSum = A[0]\\n        while a < len(A) and runningSum != targetSum:\\n            runningSum += A[a]\\n            a += 1\\n        b = len(A) - 2\\n        runningSum = A[-1]\\n        while b > -1 and runningSum != targetSum:\\n            runningSum += A[b]\\n            b -= 1\\n        return not a > b\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if len(A) < 3: return False\\n        for i in range(1,len(A)):\\n            A[i] = A[i]+A[i-1]\\n        \\n        sumTarg = A[-1]/3\\n        \\n        first = False\\n        \\n        for i in range(len(A)):\\n            if A[i] == sumTarg and first == False:\\n                first = True\\n            elif (first == True and A[i] == 2*sumTarg and i != len(A)-1):\\n                return True\\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        cummulative_sum = 0\\n        counter = 0\\n        target = sum(A) / 3\\n        if target != int(target):\\n            return False\\n        print(target)\\n        for idx in range(len(A)):\\n            cummulative_sum += A[idx]\\n            if cummulative_sum == target:\\n                cummulative_sum = 0\\n                counter += 1\\n        \\n        if counter == 4 and target == 0:\\n            return True\\n        if counter == 3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)-(sum(A)//3)*3!=0:\\n            return 0\\n        else:\\n            s=sum(A)//3\\n            c=0\\n            ts=0\\n            j=0\\n            for i in range(len(A)):\\n                j=i\\n                ts+=A[i]\\n                if ts==s:\\n                    c+=1\\n                    ts=0\\n                if c==2 and j+1<len(A):\\n                    for k in range(j+1,len(A)):\\n                        ts+=A[k]\\n                    if ts==s:\\n                        return 1\\n                    else:\\n                        return 0\\n            return 0\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        partSum = sum(A)/3\\n        curSum = 0\\n        partNum = 0\\n        for i  in A:\\n            curSum +=i\\n            if(curSum == partSum):\\n                curSum = 0\\n                partNum +=1\\n        if(partSum == 0):\\n            return partNum>=3\\n        else:\\n            return partNum == 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        third = total // 3\\n        cumsum, count = 0, 0\\n        for i in range(0, len(A)):\\n            cumsum += A[i]\\n            if cumsum == third:\\n                count += 1\\n                cumsum = 0\\n        return count == 3 or count > 3 and total == 0\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        x= 0\\n        for i in A:\\n            x += i\\n        if x%3 !=0 :\\n            return False\\n        x = x/3\\n        count = 0\\n        sum1 = 0\\n        for i in A :\\n            sum1 += i\\n            if sum1 == x :\\n                count += 1\\n                sum1 = 0\\n        if count >= 3 :\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3!=0:\\n            return False\\n        \\n        req=sum(A)//3\\n        \\n        s=0\\n        c=0\\n        print(sum(A))\\n        print(req)\\n        for i in A:\\n            s+=i\\n            if s==req:\\n                s=0\\n                c+=1\\n        \\n        return c>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        i = 0\\n        count = A[i]\\n        while (i < len(A)-1) and (count != total/3):\\n            i += 1\\n            count += A[i]\\n            \\n            \\n        if (count != total/3) or (i+1 > len(A)-2):\\n            return False\\n        \\n        j = i +1\\n        count = A[j]\\n        while (j < len(A)-1) and (count != total/3):\\n            j += 1\\n            count += A[j]\\n           \\n            \\n        if (count != total/3) or (j+1 == len(A)):\\n            return False\\n        \\n        if sum(A[j+1:]) == total/3:\\n            return True\\n        else:\\n            return False\\n        \\n        \\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        #a + ....+ b = num \\n        # if 3 equal amount of indices have the same sum  return true\\n        # how do we find the sum? \\n        # use the sum and then divide it by three thats the equal part sum\\n        # use len(A) - 1  to find the number of indexes and \\n        equalSum = sum(A) // 3\\n        part = 0\\n        numparts = 0\\n        r = sum(A) % 3\\n        for i in A:\\n            part += i \\n            if part == equalSum:\\n                part = 0\\n                numparts += 1\\n        return not r and numparts >= 3\\n\\n\\n\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum1=sum(A)\\n        if sum1%3!=0:\\n            return False\\n        \\n        else:\\n            div=sum1//3\\n            print(div)\\n            i1,sum2=0,0\\n            for i in A:\\n                sum2=sum2+i\\n                print(sum2,i)\\n                if sum2==div:\\n                    i1=i1+1\\n                    print('sum2',sum2)\\n                    sum2=0\\n            if i1>=3 and div==0:\\n                return True\\n            if i1==3:\\n                return True\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3 != 0: \\n            return False\\n        ps = s//3\\n        count = 0\\n        for x in A:\\n            ps -= x\\n            if not ps:\\n                ps = s//3\\n                count+=1\\n        return count >= 3\\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        isum = 0\\n        \\n        for i in range(0,len(A)-2):\\n            isum += A[i]\\n            if isum == total/3:\\n                jsum = 0\\n                for j in range(i+1,len(A)-1):\\n                    jsum += A[j]\\n                    if isum == jsum == total - isum - jsum:\\n                        return True\\n        return False\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        target = sum(A)//3\\n        \\n        temp = 0\\n        count = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if(temp == target):\\n                temp = 0\\n                count += 1\\n                if(count == 3):\\n                    return True\\n        return False\\n                \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= sum(A)//3\\n        \\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        sum3 = sum(A[i:])\\n        # for i in range(i, j):\\n        #     sum3 = sum3 + A[i]\\n        #     i = i +1\\n        #     if sum3 == num:\\n        #         break\\n        # print(\\\\\\\"sum=\\\\\\\",sum3)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        j = len(A)\\n        i = 0\\n        num= 0\\n        for i in range(0, j):\\n            num = num + A[i]\\n        num = floor(num /3);\\n        print(num)\\n        sum1 = 0\\n        sum2 = 0\\n        sum3 = 0\\n        for i in range(0, j):\\n            sum1 = sum1 + A[i]\\n            i = i +1\\n            if sum1 == num:\\n                break\\n        for i in range(i, j):\\n            sum2 = sum2 + A[i]\\n            i = i +1\\n            if sum2 == num:\\n                break\\n        \\n        if i == j:\\n            return False\\n        for i in range(i, j):\\n            sum3 = sum3 + A[i]\\n            i = i +1\\n            if sum3 == num:\\n                break\\n        print(sum1)\\n        return sum1==sum2 and sum1 == sum3\\n    \\n        # return sum3\\n#         for i in range(0, floor(j/3)):\\n#             sum1 = sum1 + A[i]\\n#             i = i+1\\n\\n#         sum2 =0 \\n#         print(\\\\\\\"sum1 = \\\\\\\" , sum1)\\n#         while sum2 is not sum1:\\n#             sum2 = sum2 + A[i]\\n#             print(sum2)\\n#             i= i+1\\n        \\n#         sum3 =0\\n#         for i in range(i,j):\\n#             sum3 = sum3 +A[i]\\n#             # print(sum3)\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        flag = 0\\n        temp = 0\\n        for i in range(len(A)):\\n            temp += A[i]\\n            if flag == 0 and temp == s//3:\\n                flag += 1\\n            elif flag == 1 and temp == s*2/3 and i != len(A)-1:\\n                return True\\n        return False\\n                \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"\\nclass Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        curr_sum = 0\\n        first_found_flag = False\\n        \\n        for i in range(len(A) - 1): #will go until the last element.\\n            curr_sum += A[i]\\n            if not first_found_flag: #looking for the first group\\n                if curr_sum == total / 3: #we found the first group.\\n                    first_found_flag = True\\n            else: #looking for the second group\\n                if curr_sum == total * 2 / 3:\\n                    return True\\n                \\n        return False\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        l,r,s=1, len(A)-2, sum(A)\\n        ls, rs, avgs = A[0], A[-1], s//3\\n        while l<r:\\n            if l < r and ls != avgs:\\n                ls+=A[l]\\n                l+=1\\n            if l<r and rs!=avgs:\\n                rs+=A[r]\\n                r-=1\\n            if ls == rs == avgs and s%3==0:\\n                return True\\n            \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        t=sum(A)\\n        if t%3!=0:\\n            return False\\n        s=0\\n        p=0\\n        for i in range(len(A)):\\n            s+=A[i]\\n            if s==t//3:\\n                s=0\\n                p+=1\\n        if p>=3:\\n            return True\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        average, part, cnt = sum(A) // 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return cnt >= 3\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        s = s / 3\\n        t = 0\\n        count = 0\\n        for x in A:\\n            t += x\\n            if t == s:\\n                t = 0\\n                count += 1\\n        if t == 0 and count >= 3:\\n            return True\\n        else:\\n            return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        target = s//3\\n        current = 0\\n        count = 0\\n        for v in A:\\n            current += v\\n            if current == target:\\n                count += 1\\n                current = 0\\n                if count >= 3:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        totalSum = sum(A)\\n        if totalSum % 3 != 0:\\n            return False\\n        \\n        target = totalSum // 3\\n        numPartitions = 0\\n        currSum = 0\\n        \\n        for num in A:\\n            currSum += num\\n            if currSum == target:\\n                numPartitions += 1\\n                currSum = 0\\n                if numPartitions == 3:\\n                    return True\\n                \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, li: List[int]) -> bool:\\n        s = sum(li)\\n        if s % 3: return False\\n        ts = s // 3\\n        ss = 0\\n        chunk = 0\\n        for n in li[:-1]:\\n            ss += n\\n            if ss == ts:\\n                chunk += 1\\n                if chunk == 2:\\n                    return True\\n                ss = 0\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0:\\n            return False\\n        curr, target = 0, s // 3\\n        count = 0\\n        for ix in range(len(A)):\\n            curr += A[ix]\\n            if curr == target:\\n                curr = 0\\n                count += 1\\n                \\n            if count == 2 and ix < len(A) - 1:\\n                return True\\n        \\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        partition_sum = total/3\\n        numberofpartition=0\\n        tempsum=0\\n        for i in range(len(A)):\\n            tempsum+=A[i]\\n            if tempsum == partition_sum:\\n                numberofpartition+=1\\n                tempsum=0\\n            \\n            if numberofpartition == 3:\\n                return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        S = sum(A)\\n        if S % 3 != 0:\\n            return False\\n        \\n        S_1 = S / 3\\n        S_2 = S_1 + S_1\\n        cur = A[0]\\n        for i in range(1, len(A)-2):\\n            if cur == S_1:\\n                cur += A[i]\\n                for j in range(i+1, len(A)-1):\\n                    if cur == S_2:\\n                        return True\\n                    else:\\n                        cur += A[j]\\n                \\n                if cur == S_2:\\n                    return A[-1] == S_1\\n            else:\\n                cur += A[i]\\n\\n        if cur == S_1:\\n            return A[-1] == S_1\\n\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot%3 != 0:\\n            return False\\n        flag = 0\\n        temp_tot = 0\\n        presum =[0] * (len(A) + 1)\\n        s = tot//3\\n        target = [2*s, s]\\n        for i in range(len(A)):\\n            if not target:\\n                return True\\n            \\n            presum[i + 1] = presum[i ] + A[i]\\n            if presum[i + 1] == target[-1]:\\n                target.pop()\\n            \\n        \\n        \\n        \\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if sum(A)%3: \\n            return False\\n        sumParts = sum(A)//3\\n        summ=0\\n        parts=0\\n        for num in A:\\n            summ+=num\\n            if summ==sumParts:\\n                parts +=1\\n                summ=0\\n        return parts>=3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum=0\\n        for a in A:\\n            sum+=a\\n        target=floor(sum/3)\\n        sum=0\\n        cnt=0\\n        j=len(A)-1\\n        i=0\\n        while j>0:\\n            sum+=A[j]\\n            if sum==target:\\n                cnt+=1\\n                sum=0\\n                break\\n            j-=1\\n        while i<j:\\n            sum+=A[i]\\n            if cnt<2 and sum==target:\\n                cnt+=1\\n                sum=0\\n                if j-i<=1:\\n                    return False\\n            i+=1\\n            if i>=j:\\n                break\\n        if cnt==2 and sum==target:\\n            cnt+=1\\n        if cnt==3:\\n            return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n       \\n        \\n        average, remainder, part, cnt = sum(A) // 3, sum(A) % 3, 0, 0\\n        for a in A:\\n            part += a\\n            if part == average:\\n                cnt += 1\\n                part = 0\\n        return not remainder and cnt >= 3\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s % 3 != 0: return False\\n        return self.can_partition(A, 0, 3, s // 3) \\n    \\n    def can_partition(self, A: List[int], i: int, n_parts: int, target_sum: int) -> bool:\\n        #print(f'i={i}, n_parts={n_parts}')\\n        if n_parts == 1: return i < len(A) and sum(A[i:]) == target_sum\\n        if i >= len(A): return False\\n        partition_sum = A[i]\\n        j = i + 1\\n        while j < len(A) and partition_sum != target_sum:\\n            partition_sum += A[j]\\n            j += 1\\n        return self.can_partition(A, j, n_parts - 1, target_sum)\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        sum = 0\\n        for num in A:\\n            sum += num\\n        \\n        if sum%3 != 0:\\n            return False\\n        \\n        sum = sum/3\\n        total = 0\\n        currentSum = 0\\n        for num in A:\\n            currentSum += num\\n            if currentSum == sum:\\n                total += 1\\n                currentSum = 0\\n      \\n        return True if total>=3 else False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = sum(A)\\n        if tot % 3 != 0:\\n            return False\\n        target = tot // 3\\n        curr_sum = 0\\n        check1, check2 = 0, 0\\n        for i, a in enumerate(A):\\n            curr_sum += a\\n            \\n            if check1 != 1 and curr_sum == target:\\n                check1 = 1\\n                continue\\n            # print(target, curr_sum)\\n            if check1 and curr_sum == target*2 and i < len(A) - 1:\\n                check2 = 1\\n                break\\n        if check1 and check2:\\n            return True\\n        else:\\n            return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        s1 = 0\\n        i = 0\\n        print((s//3))\\n        while i < len(A):\\n            s1 += A[i]\\n            if s1 == s//3:\\n                break\\n            i+=1\\n        s1 = 0\\n        i+=1\\n        while i < len(A):\\n            s1 += A[i]\\n            print(s1)\\n            if s1 == s//3:\\n                if i != len(A)-1:\\n                    return True\\n            i+=1\\n        return False\\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot = 0\\n        bi = sum(A)//3\\n        count = 0\\n        \\n        for i in range(len(A) - 1):\\n            tot += A[i]\\n            if tot == bi:\\n                tot = 0\\n                count += 1\\n                \\n                if count == 2:\\n                    return True\\n        return False\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        \\n        total = sum(A)\\n        \\n        if total%3 != 0 :\\n            return False\\n        \\n        target = total/3\\n        print (target)\\n        \\n        temp = 0\\n        res = []\\n        for i, v in enumerate(A):\\n            temp = temp + v\\n            \\n            if temp == target:\\n                res.append(i)\\n                temp = 0\\n        print (res)\\n        \\n        return len(res)>=3\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        s = sum(A)\\n        if s%3!=0:\\n            return False\\n        target = s/3\\n        total = 0\\n        partitions = 0\\n        for num in A:\\n            total+=num\\n            if total == target:\\n                partitions+=1\\n                total = 0\\n                \\n        \\n        return total == 0 and partitions in [3,4]\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        n = len(A)\\n        sum_a = sum(A)\\n        if sum_a%3!=0:\\n            return False\\n        target =  sum_a//3\\n        \\n        output = []\\n        temp_sum = 0\\n        temp_output = []\\n        for i in A:\\n            temp_output.append(i)\\n            temp_sum = temp_sum + i\\n            if temp_sum ==  target:\\n                if len(temp_output)>0 and len(output)<3:\\n                    output.append(temp_output)\\n                    temp_sum = 0\\n                    temp_output = []\\n        if temp_sum!=0:\\n            return False\\n        if len(output)!=3:\\n            return False\\n        else:\\n            return True\\n            \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        tot_sum = sum(A)\\n        if tot_sum % 3 != 0:\\n            return(False)\\n        else:\\n            cum_sum = 0\\n            target = tot_sum//3\\n            counter = 0\\n            for num in A[:-1]:\\n                cum_sum += num\\n                if cum_sum == target:\\n                    counter += 1\\n                    cum_sum = 0\\n                    if counter == 2:\\n                        return(True)\\n            return(False)\\n            \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        total = sum(A)\\n        if total % 3 != 0:\\n            return False\\n        \\n        subtotal = total // 3\\n        \\n        pre_sum = 0\\n        k = 3\\n        for i in range(len(A)):\\n            pre_sum += A[i]\\n            if k > 1 and pre_sum == subtotal:\\n                k -= 1\\n                pre_sum = 0\\n            elif i == len(A)-1:\\n                k -= 1\\n        \\n        return k == 0 and pre_sum == subtotal\\n        \\n        \\n        \\n\", \"class Solution:\\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\\n        if(sum(A)%3!=0):\\n            return False\\n        val=sum(A)//3\\n        add=0\\n        count=0\\n        for x in range(len(A)):\\n            add+=A[x]\\n            if(add==val):\\n                add=0\\n                count+=1\\n            else:\\n                continue\\n        if(count>=3):\\n            return True\\n        return False\\n        \\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                3,
                3,
                6,
                5,
                -2,
                2,
                5,
                1,
                -9,
                4
            ]
        ],
        "output": true,
        "halu_type": "Identification Hallucination",
        "fn_name": "canThreePartsEqualSum",
        "starter_code": "\nclass Solution:\n    def canThreePartsEqualSum(self, A: List[int]) -> bool:\n        ",
        "url": "https://leetcode.com/problems/partition-array-into-three-parts-with-equal-sum/"
    },
    {
        "id": 1297,
        "task_id": 4534,
        "test_case_id": 1,
        "question": "Given a non-negative index k where k ≤ 33, return the kth index row of the Pascal's triangle.\n\nNote that the row index starts from 0.\n\n\nIn Pascal's triangle, each number is the sum of the two numbers directly above it.\n\nExample:\n\n\nInput: 3\nOutput: [1,3,3,1]\n\n\nFollow up:\n\nCould you optimize your algorithm to use only O(k) extra space?",
        "solutions": "[\"class Solution:\\n     def getRow(self, k):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = [1]\\n         cur = k\\n         for i in range(k//2):\\n             res += res[-1] * cur // (i+1),\\n             cur -= 1\\n         if k % 2 == 0:\\n             res = res + res[:-1][::-1]\\n         else:\\n             res = res + res[::-1]\\n         return res\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         a=[1]\\n         i=0\\n         b=[1]\\n         while i<rowIndex:\\n             a.append(0)\\n             b=a[:]\\n             for j in  range(i+2):\\n                 b[j]=a[j]+a[i-j+1]\\n             a=b\\n             i+=1\\n         return b\\n         \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = [1]\\n         for i in range(1, rowIndex + 1):\\n             res += [1]\\n             for j in range(len(res) - 2, -1, -1):\\n                 if j > 0:\\n                     res[j] = res[j] + res[j -1]\\n                 else:\\n                     res[j] = 1\\n                     \\n         return res\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         out = [1]\\n         for i in range(1,rowIndex+1):\\n             temp = []\\n             for j in range(0, i-1):\\n                 temp.append(out[j]+out[j+1])\\n             out = [1]+temp+[1]\\n         return out\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         l = [1]\\n         for i in range(rowIndex):\\n             l = [j+k for j, k in zip([0]+l, l+[0])]\\n         return l\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result=[]\\n         for n in range(rowIndex+1):\\n             num=1\\n             for i in range(n):\\n                 num=int(num*(rowIndex-i)/(i+1))\\n             result=result+[num]\\n         return result\\n     \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if rowIndex < 0:\\n             return list()\\n         if rowIndex == 0:\\n             return list([1])\\n         l = list([1])\\n         for i in range(1,rowIndex+1):\\n             pre_value = l[0] #1\\n             for j in range(1,i):#j = 3\\n                 temp = l[j] #1\\n                 l[j] = pre_value+l[j] #4\\n                 pre_value = temp #1\\n             l.append(1)\\n         return l\\n                 \\n             \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         rows = [1]\\n         for i in range(rowIndex):\\n             rows = [x+y for x, y in zip([0]+rows, rows+[0])]\\n         return rows\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         row = [1]\\n         for _ in range(rowIndex):\\n             row = [x + y for x, y in zip([0]+row, row+[0])]\\n         return row\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if rowIndex < 0: return []\\n         result = [0 for _ in range(rowIndex+1)]\\n         result[0] = 1\\n         for i in range(1, rowIndex+1):\\n             result[i] = 1\\n             for j in range(i-1, 0, -1):\\n                 result[j] = result[j] + result[j-1]\\n         return result\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         i=1\\n         res = [1]\\n         while rowIndex >=1:\\n             res.append(int(res[-1]*rowIndex/i))\\n             rowIndex,i = rowIndex-1,i+1\\n         return(res)\"]",
        "difficulty": "introductory",
        "input": [
            3
        ],
        "output": [
            1,
            3,
            3,
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "getRow",
        "starter_code": "\nclass Solution:\n    def getRow(self, rowIndex: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/pascals-triangle-ii/"
    },
    {
        "id": 1298,
        "task_id": 4534,
        "test_case_id": 2,
        "question": "Given a non-negative index k where k ≤ 33, return the kth index row of the Pascal's triangle.\n\nNote that the row index starts from 0.\n\n\nIn Pascal's triangle, each number is the sum of the two numbers directly above it.\n\nExample:\n\n\nInput: 3\nOutput: [1,3,3,1]\n\n\nFollow up:\n\nCould you optimize your algorithm to use only O(k) extra space?",
        "solutions": "[\"class Solution:\\n     def getRow(self, k):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = [1]\\n         cur = k\\n         for i in range(k//2):\\n             res += res[-1] * cur // (i+1),\\n             cur -= 1\\n         if k % 2 == 0:\\n             res = res + res[:-1][::-1]\\n         else:\\n             res = res + res[::-1]\\n         return res\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         a=[1]\\n         i=0\\n         b=[1]\\n         while i<rowIndex:\\n             a.append(0)\\n             b=a[:]\\n             for j in  range(i+2):\\n                 b[j]=a[j]+a[i-j+1]\\n             a=b\\n             i+=1\\n         return b\\n         \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = [1]\\n         for i in range(1, rowIndex + 1):\\n             res += [1]\\n             for j in range(len(res) - 2, -1, -1):\\n                 if j > 0:\\n                     res[j] = res[j] + res[j -1]\\n                 else:\\n                     res[j] = 1\\n                     \\n         return res\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         out = [1]\\n         for i in range(1,rowIndex+1):\\n             temp = []\\n             for j in range(0, i-1):\\n                 temp.append(out[j]+out[j+1])\\n             out = [1]+temp+[1]\\n         return out\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         l = [1]\\n         for i in range(rowIndex):\\n             l = [j+k for j, k in zip([0]+l, l+[0])]\\n         return l\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result=[]\\n         for n in range(rowIndex+1):\\n             num=1\\n             for i in range(n):\\n                 num=int(num*(rowIndex-i)/(i+1))\\n             result=result+[num]\\n         return result\\n     \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if rowIndex < 0:\\n             return list()\\n         if rowIndex == 0:\\n             return list([1])\\n         l = list([1])\\n         for i in range(1,rowIndex+1):\\n             pre_value = l[0] #1\\n             for j in range(1,i):#j = 3\\n                 temp = l[j] #1\\n                 l[j] = pre_value+l[j] #4\\n                 pre_value = temp #1\\n             l.append(1)\\n         return l\\n                 \\n             \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         rows = [1]\\n         for i in range(rowIndex):\\n             rows = [x+y for x, y in zip([0]+rows, rows+[0])]\\n         return rows\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         row = [1]\\n         for _ in range(rowIndex):\\n             row = [x + y for x, y in zip([0]+row, row+[0])]\\n         return row\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if rowIndex < 0: return []\\n         result = [0 for _ in range(rowIndex+1)]\\n         result[0] = 1\\n         for i in range(1, rowIndex+1):\\n             result[i] = 1\\n             for j in range(i-1, 0, -1):\\n                 result[j] = result[j] + result[j-1]\\n         return result\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         i=1\\n         res = [1]\\n         while rowIndex >=1:\\n             res.append(int(res[-1]*rowIndex/i))\\n             rowIndex,i = rowIndex-1,i+1\\n         return(res)\"]",
        "difficulty": "introductory",
        "input": [
            0
        ],
        "output": [
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "getRow",
        "starter_code": "\nclass Solution:\n    def getRow(self, rowIndex: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/pascals-triangle-ii/"
    },
    {
        "id": 1299,
        "task_id": 4534,
        "test_case_id": 3,
        "question": "Given a non-negative index k where k ≤ 33, return the kth index row of the Pascal's triangle.\n\nNote that the row index starts from 0.\n\n\nIn Pascal's triangle, each number is the sum of the two numbers directly above it.\n\nExample:\n\n\nInput: 3\nOutput: [1,3,3,1]\n\n\nFollow up:\n\nCould you optimize your algorithm to use only O(k) extra space?",
        "solutions": "[\"class Solution:\\n     def getRow(self, k):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = [1]\\n         cur = k\\n         for i in range(k//2):\\n             res += res[-1] * cur // (i+1),\\n             cur -= 1\\n         if k % 2 == 0:\\n             res = res + res[:-1][::-1]\\n         else:\\n             res = res + res[::-1]\\n         return res\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         a=[1]\\n         i=0\\n         b=[1]\\n         while i<rowIndex:\\n             a.append(0)\\n             b=a[:]\\n             for j in  range(i+2):\\n                 b[j]=a[j]+a[i-j+1]\\n             a=b\\n             i+=1\\n         return b\\n         \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         res = [1]\\n         for i in range(1, rowIndex + 1):\\n             res += [1]\\n             for j in range(len(res) - 2, -1, -1):\\n                 if j > 0:\\n                     res[j] = res[j] + res[j -1]\\n                 else:\\n                     res[j] = 1\\n                     \\n         return res\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         out = [1]\\n         for i in range(1,rowIndex+1):\\n             temp = []\\n             for j in range(0, i-1):\\n                 temp.append(out[j]+out[j+1])\\n             out = [1]+temp+[1]\\n         return out\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         l = [1]\\n         for i in range(rowIndex):\\n             l = [j+k for j, k in zip([0]+l, l+[0])]\\n         return l\\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result=[]\\n         for n in range(rowIndex+1):\\n             num=1\\n             for i in range(n):\\n                 num=int(num*(rowIndex-i)/(i+1))\\n             result=result+[num]\\n         return result\\n     \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if rowIndex < 0:\\n             return list()\\n         if rowIndex == 0:\\n             return list([1])\\n         l = list([1])\\n         for i in range(1,rowIndex+1):\\n             pre_value = l[0] #1\\n             for j in range(1,i):#j = 3\\n                 temp = l[j] #1\\n                 l[j] = pre_value+l[j] #4\\n                 pre_value = temp #1\\n             l.append(1)\\n         return l\\n                 \\n             \\n\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         rows = [1]\\n         for i in range(rowIndex):\\n             rows = [x+y for x, y in zip([0]+rows, rows+[0])]\\n         return rows\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         row = [1]\\n         for _ in range(rowIndex):\\n             row = [x + y for x, y in zip([0]+row, row+[0])]\\n         return row\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if rowIndex < 0: return []\\n         result = [0 for _ in range(rowIndex+1)]\\n         result[0] = 1\\n         for i in range(1, rowIndex+1):\\n             result[i] = 1\\n             for j in range(i-1, 0, -1):\\n                 result[j] = result[j] + result[j-1]\\n         return result\", \"class Solution:\\n     def getRow(self, rowIndex):\\n         \\\"\\\"\\\"\\n         :type rowIndex: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         i=1\\n         res = [1]\\n         while rowIndex >=1:\\n             res.append(int(res[-1]*rowIndex/i))\\n             rowIndex,i = rowIndex-1,i+1\\n         return(res)\"]",
        "difficulty": "introductory",
        "input": [
            1
        ],
        "output": [
            1,
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "getRow",
        "starter_code": "\nclass Solution:\n    def getRow(self, rowIndex: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/pascals-triangle-ii/"
    },
    {
        "id": 1300,
        "task_id": 4752,
        "test_case_id": 1,
        "question": "Given an array of integers, return indices of the two numbers such that they add up to a specific target.\n\nYou may assume that each input would have exactly one solution, and you may not use the same element twice.\n\nExample:\n\n\nGiven nums = [2, 7, 11, 15], target = 9,\n\nBecause nums[0] + nums[1] = 2 + 7 = 9,\nreturn [0, 1].",
        "solutions": "[\"class Solution:\\n     def twoSum(self, nums, target):\\n         tmp = {}\\n         for i in range(len(nums)):\\n             if target - nums[i] in tmp:\\n                 return(tmp[target - nums[i]], i)\\n             else:\\n                 tmp[nums[i]] = i;\\n         \\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = {}\\n         for i, num in enumerate(nums):\\n             if (target - num) in d:\\n                 return [d[target-num], i]\\n             d[num] = i\\n         \\n         new = sorted(nums)\\n         i,j = 0, -1\\n         for num in new:\\n             a, b = new[i], new[j]\\n             if a + b > target:\\n                 j = j - 1\\n             elif a + b < target:\\n                 i = i + 1\\n             elif a + b == target:\\n                 if a != b:\\n                     ans = [nums.index(a), nums.index(b)]\\n                 else:\\n                     m = nums.index(a)\\n                     nums.remove(a)\\n                     n = nums.index(b)\\n                     ans =[m, n+1]\\n                 return (ans)\\n         \\n \\n\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         hashdict = {}\\n         for i,num in enumerate(nums):\\n             if target-num in hashdict:\\n                 return [hashdict[target-num], i]\\n             else:\\n                 hashdict[num] = i\\n\", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         '''\\n         type nums:array\\n         type target:integer\\n         rtype :List\\n         '''\\n         d = {}\\n         for i, num in enumerate(nums):\\n             if target - num in d:\\n                 return[d[target - num], i]\\n             d[num] = i    \", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         t=dict()\\n         l=len(nums)\\n         if l<=1: return []\\n         for i in range(l):\\n             t[nums[i]]=i\\n         for i in range(l):\\n             comp=target-nums[i]\\n             if comp in t and t[comp]!=i:\\n                 return [i,t[comp]]\\n         return []\", \"class Solution(object):  \\n     def twoSum(self, nums, target):  \\n         \\\"\\\"\\\" \\n         :type nums: List[int] \\n         :type target: int \\n         :rtype: List[int] \\n         \\\"\\\"\\\"  \\n         arr = {}\\n         length = len(nums)\\n         for i in range(length):  \\n             if (target - nums[i]) in arr:\\n                 return [arr[target - nums[i]], i]\\n             arr[nums[i]] = i\", \"class Solution:\\n         \\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         hashmap = {}\\n         x = len(nums)\\n         for i in range(x):\\n             comp = target - nums[i]\\n             if comp in hashmap:\\n                 return [hashmap.get(comp), i]\\n             hashmap[nums[i]] = i\\n             \\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         sorted_nums = sorted(nums)\\n         start = 0\\n         end = len(nums) - 1\\n         while start < end:\\n             curr_sum = sorted_nums[start] + sorted_nums[end]\\n             if (curr_sum == target):\\n                 break\\n             if (curr_sum < target):\\n                 start += 1\\n             else:\\n                 end -= 1\\n         first_index = nums.index(sorted_nums[start])\\n         second_index = nums.index(sorted_nums[end])\\n         if sorted_nums[start] == sorted_nums[end]:\\n             nums.pop(first_index)\\n             second_index = nums.index(sorted_nums[end]) + 1\\n         return [first_index, second_index]\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         table = {}\\n         for i, e in enumerate(nums):\\n             print(\\\"fe\\\")\\n             if e in table:\\n                 table[e] = [table[e][0] + 1, table[e][1] + [i]]\\n             else:\\n                 table[e] = [1, [i]]\\n             \\n             sub_target = target - e\\n             if sub_target in table and (sub_target != e or table[sub_target][0] >= 2):\\n                 print(table)\\n                 first_idx = i\\n                 second_idx = None\\n                 for index in table[sub_target][1]:\\n                     if index != first_idx:\\n                         second_idx = index\\n                         break\\n                 \\n                 assert(second_idx, \\\"got shit\\\")\\n                 result = [first_idx, second_idx]\\n                 result.sort()\\n                 return result\\n         return []\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         hashdict = {}\\n         for i,num in enumerate(nums):\\n             if target-num in hashdict:\\n                 return [hashdict[target-num], i]\\n             else:\\n                 hashdict[num] = i\\n\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         nums_index = [(v, index) for index, v in enumerate(nums)]\\n         nums_index.sort()\\n         begin, end = 0, len(nums) - 1\\n         while begin < end:\\n             curr = nums_index[begin][0] + nums_index[end][0]\\n             if curr == target:\\n                 return [nums_index[begin][1], nums_index[end][1]]\\n             elif curr < target:\\n                 begin += 1\\n             else:\\n                 end -= 1\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         numsMap = {}\\n         for i, num in enumerate(nums):\\n             if target - num in numsMap:\\n                 return [i, numsMap[target-num]]\\n             numsMap[num] = i\\n         return [None, None]\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n #         length_nums = len(nums)\\n         \\n #         for i in range(length_nums):\\n #             for j in range(i+1, length_nums):\\n #                 if nums[i] + nums[j] == target:\\n #                     return [i, j]\\n \\n         # dic = {}\\n         # for i, num in enumerate(nums):\\n         #     if num in dic:\\n         #         return [dic[num], i]\\n         #     else:\\n         #         dic[target - num] = i\\n         \\n         num_dict = {}\\n         for i, num in enumerate(nums):\\n             rem = target - num\\n             if rem in num_dict:\\n                 return [num_dict[rem], i]\\n             num_dict[num] = i\\n         return None\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         dict = {}\\n         for i in range(0,len(nums)):\\n             if nums[i] in dict:\\n                 return [dict[nums[i]],i]\\n             else:\\n                 dict[target-nums[i]] = i\\n         return []\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         indexes = dict([(nums[i], i) for i in range(len(nums))])\\n         for i in range(len(nums)):\\n             if target-nums[i] in indexes and indexes[target - nums[i]] != i:\\n                 return [i, indexes[target - nums[i]]]\\n         return []\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         # record the sorted index (original position)\\n         nums_sorted_index = sorted(range(len(nums)), key = lambda k: nums[k])\\n         # sort the list\\n         nums.sort()\\n         for i in range(len(nums)):\\n             for j in range(i+1,len(nums)):\\n                 if nums[i]+nums[j] > target:\\n                     break\\n                 elif nums[i]+nums[j] == target:\\n                     return [nums_sorted_index[i], nums_sorted_index[j]]\\n         print('Can\\\\'t find a match!')\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\n         \\\"\\\"\\\"\\n         dic={}\\n         for i,now in enumerate(nums):\\n             dev = target - now\\n             if dev in dic:\\n                 return [dic[dev],i]\\n             dic[now]=i\\n         return None\\n         \\\"\\\"\\\"\\n     \\n         \\n     #   old\\n         a=sorted(nums)\\n         \\n         for j in range(len(nums)):\\n             for k in range(j+1,len(nums)):\\n                 s = a[j]+a[k]\\n                 if s == target:\\n                     if a[j]==a[k]:\\n                         return [i for i,x in enumerate(nums) if x == a[k]]\\n                     else:\\n                         b=nums.index(a[j])\\n                         c=nums.index(a[k])\\n                         return [b,c]\\n                         \\n                 elif s>target:\\n                     break\\n                \\n         \\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\"]",
        "difficulty": "introductory",
        "input": [
            [
                2,
                7,
                11,
                15
            ],
            [
                9
            ]
        ],
        "output": [
            0,
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "twoSum",
        "starter_code": "\nclass Solution:\n    def twoSum(self, nums: List[int], target: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/two-sum/"
    },
    {
        "id": 1301,
        "task_id": 4752,
        "test_case_id": 2,
        "question": "Given an array of integers, return indices of the two numbers such that they add up to a specific target.\n\nYou may assume that each input would have exactly one solution, and you may not use the same element twice.\n\nExample:\n\n\nGiven nums = [2, 7, 11, 15], target = 9,\n\nBecause nums[0] + nums[1] = 2 + 7 = 9,\nreturn [0, 1].",
        "solutions": "[\"class Solution:\\n     def twoSum(self, nums, target):\\n         tmp = {}\\n         for i in range(len(nums)):\\n             if target - nums[i] in tmp:\\n                 return(tmp[target - nums[i]], i)\\n             else:\\n                 tmp[nums[i]] = i;\\n         \\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = {}\\n         for i, num in enumerate(nums):\\n             if (target - num) in d:\\n                 return [d[target-num], i]\\n             d[num] = i\\n         \\n         new = sorted(nums)\\n         i,j = 0, -1\\n         for num in new:\\n             a, b = new[i], new[j]\\n             if a + b > target:\\n                 j = j - 1\\n             elif a + b < target:\\n                 i = i + 1\\n             elif a + b == target:\\n                 if a != b:\\n                     ans = [nums.index(a), nums.index(b)]\\n                 else:\\n                     m = nums.index(a)\\n                     nums.remove(a)\\n                     n = nums.index(b)\\n                     ans =[m, n+1]\\n                 return (ans)\\n         \\n \\n\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         hashdict = {}\\n         for i,num in enumerate(nums):\\n             if target-num in hashdict:\\n                 return [hashdict[target-num], i]\\n             else:\\n                 hashdict[num] = i\\n\", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         '''\\n         type nums:array\\n         type target:integer\\n         rtype :List\\n         '''\\n         d = {}\\n         for i, num in enumerate(nums):\\n             if target - num in d:\\n                 return[d[target - num], i]\\n             d[num] = i    \", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         t=dict()\\n         l=len(nums)\\n         if l<=1: return []\\n         for i in range(l):\\n             t[nums[i]]=i\\n         for i in range(l):\\n             comp=target-nums[i]\\n             if comp in t and t[comp]!=i:\\n                 return [i,t[comp]]\\n         return []\", \"class Solution(object):  \\n     def twoSum(self, nums, target):  \\n         \\\"\\\"\\\" \\n         :type nums: List[int] \\n         :type target: int \\n         :rtype: List[int] \\n         \\\"\\\"\\\"  \\n         arr = {}\\n         length = len(nums)\\n         for i in range(length):  \\n             if (target - nums[i]) in arr:\\n                 return [arr[target - nums[i]], i]\\n             arr[nums[i]] = i\", \"class Solution:\\n         \\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         hashmap = {}\\n         x = len(nums)\\n         for i in range(x):\\n             comp = target - nums[i]\\n             if comp in hashmap:\\n                 return [hashmap.get(comp), i]\\n             hashmap[nums[i]] = i\\n             \\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         sorted_nums = sorted(nums)\\n         start = 0\\n         end = len(nums) - 1\\n         while start < end:\\n             curr_sum = sorted_nums[start] + sorted_nums[end]\\n             if (curr_sum == target):\\n                 break\\n             if (curr_sum < target):\\n                 start += 1\\n             else:\\n                 end -= 1\\n         first_index = nums.index(sorted_nums[start])\\n         second_index = nums.index(sorted_nums[end])\\n         if sorted_nums[start] == sorted_nums[end]:\\n             nums.pop(first_index)\\n             second_index = nums.index(sorted_nums[end]) + 1\\n         return [first_index, second_index]\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         table = {}\\n         for i, e in enumerate(nums):\\n             print(\\\"fe\\\")\\n             if e in table:\\n                 table[e] = [table[e][0] + 1, table[e][1] + [i]]\\n             else:\\n                 table[e] = [1, [i]]\\n             \\n             sub_target = target - e\\n             if sub_target in table and (sub_target != e or table[sub_target][0] >= 2):\\n                 print(table)\\n                 first_idx = i\\n                 second_idx = None\\n                 for index in table[sub_target][1]:\\n                     if index != first_idx:\\n                         second_idx = index\\n                         break\\n                 \\n                 assert(second_idx, \\\"got shit\\\")\\n                 result = [first_idx, second_idx]\\n                 result.sort()\\n                 return result\\n         return []\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         hashdict = {}\\n         for i,num in enumerate(nums):\\n             if target-num in hashdict:\\n                 return [hashdict[target-num], i]\\n             else:\\n                 hashdict[num] = i\\n\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         nums_index = [(v, index) for index, v in enumerate(nums)]\\n         nums_index.sort()\\n         begin, end = 0, len(nums) - 1\\n         while begin < end:\\n             curr = nums_index[begin][0] + nums_index[end][0]\\n             if curr == target:\\n                 return [nums_index[begin][1], nums_index[end][1]]\\n             elif curr < target:\\n                 begin += 1\\n             else:\\n                 end -= 1\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         numsMap = {}\\n         for i, num in enumerate(nums):\\n             if target - num in numsMap:\\n                 return [i, numsMap[target-num]]\\n             numsMap[num] = i\\n         return [None, None]\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n #         length_nums = len(nums)\\n         \\n #         for i in range(length_nums):\\n #             for j in range(i+1, length_nums):\\n #                 if nums[i] + nums[j] == target:\\n #                     return [i, j]\\n \\n         # dic = {}\\n         # for i, num in enumerate(nums):\\n         #     if num in dic:\\n         #         return [dic[num], i]\\n         #     else:\\n         #         dic[target - num] = i\\n         \\n         num_dict = {}\\n         for i, num in enumerate(nums):\\n             rem = target - num\\n             if rem in num_dict:\\n                 return [num_dict[rem], i]\\n             num_dict[num] = i\\n         return None\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         dict = {}\\n         for i in range(0,len(nums)):\\n             if nums[i] in dict:\\n                 return [dict[nums[i]],i]\\n             else:\\n                 dict[target-nums[i]] = i\\n         return []\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         indexes = dict([(nums[i], i) for i in range(len(nums))])\\n         for i in range(len(nums)):\\n             if target-nums[i] in indexes and indexes[target - nums[i]] != i:\\n                 return [i, indexes[target - nums[i]]]\\n         return []\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         # record the sorted index (original position)\\n         nums_sorted_index = sorted(range(len(nums)), key = lambda k: nums[k])\\n         # sort the list\\n         nums.sort()\\n         for i in range(len(nums)):\\n             for j in range(i+1,len(nums)):\\n                 if nums[i]+nums[j] > target:\\n                     break\\n                 elif nums[i]+nums[j] == target:\\n                     return [nums_sorted_index[i], nums_sorted_index[j]]\\n         print('Can\\\\'t find a match!')\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\n         \\\"\\\"\\\"\\n         dic={}\\n         for i,now in enumerate(nums):\\n             dev = target - now\\n             if dev in dic:\\n                 return [dic[dev],i]\\n             dic[now]=i\\n         return None\\n         \\\"\\\"\\\"\\n     \\n         \\n     #   old\\n         a=sorted(nums)\\n         \\n         for j in range(len(nums)):\\n             for k in range(j+1,len(nums)):\\n                 s = a[j]+a[k]\\n                 if s == target:\\n                     if a[j]==a[k]:\\n                         return [i for i,x in enumerate(nums) if x == a[k]]\\n                     else:\\n                         b=nums.index(a[j])\\n                         c=nums.index(a[k])\\n                         return [b,c]\\n                         \\n                 elif s>target:\\n                     break\\n                \\n         \\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\"]",
        "difficulty": "introductory",
        "input": [
            [
                3,
                2,
                4
            ],
            [
                6
            ]
        ],
        "output": [
            1,
            2
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "twoSum",
        "starter_code": "\nclass Solution:\n    def twoSum(self, nums: List[int], target: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/two-sum/"
    },
    {
        "id": 1302,
        "task_id": 4752,
        "test_case_id": 3,
        "question": "Given an array of integers, return indices of the two numbers such that they add up to a specific target.\n\nYou may assume that each input would have exactly one solution, and you may not use the same element twice.\n\nExample:\n\n\nGiven nums = [2, 7, 11, 15], target = 9,\n\nBecause nums[0] + nums[1] = 2 + 7 = 9,\nreturn [0, 1].",
        "solutions": "[\"class Solution:\\n     def twoSum(self, nums, target):\\n         tmp = {}\\n         for i in range(len(nums)):\\n             if target - nums[i] in tmp:\\n                 return(tmp[target - nums[i]], i)\\n             else:\\n                 tmp[nums[i]] = i;\\n         \\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = {}\\n         for i, num in enumerate(nums):\\n             if (target - num) in d:\\n                 return [d[target-num], i]\\n             d[num] = i\\n         \\n         new = sorted(nums)\\n         i,j = 0, -1\\n         for num in new:\\n             a, b = new[i], new[j]\\n             if a + b > target:\\n                 j = j - 1\\n             elif a + b < target:\\n                 i = i + 1\\n             elif a + b == target:\\n                 if a != b:\\n                     ans = [nums.index(a), nums.index(b)]\\n                 else:\\n                     m = nums.index(a)\\n                     nums.remove(a)\\n                     n = nums.index(b)\\n                     ans =[m, n+1]\\n                 return (ans)\\n         \\n \\n\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         hashdict = {}\\n         for i,num in enumerate(nums):\\n             if target-num in hashdict:\\n                 return [hashdict[target-num], i]\\n             else:\\n                 hashdict[num] = i\\n\", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         '''\\n         type nums:array\\n         type target:integer\\n         rtype :List\\n         '''\\n         d = {}\\n         for i, num in enumerate(nums):\\n             if target - num in d:\\n                 return[d[target - num], i]\\n             d[num] = i    \", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         t=dict()\\n         l=len(nums)\\n         if l<=1: return []\\n         for i in range(l):\\n             t[nums[i]]=i\\n         for i in range(l):\\n             comp=target-nums[i]\\n             if comp in t and t[comp]!=i:\\n                 return [i,t[comp]]\\n         return []\", \"class Solution(object):  \\n     def twoSum(self, nums, target):  \\n         \\\"\\\"\\\" \\n         :type nums: List[int] \\n         :type target: int \\n         :rtype: List[int] \\n         \\\"\\\"\\\"  \\n         arr = {}\\n         length = len(nums)\\n         for i in range(length):  \\n             if (target - nums[i]) in arr:\\n                 return [arr[target - nums[i]], i]\\n             arr[nums[i]] = i\", \"class Solution:\\n         \\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         hashmap = {}\\n         x = len(nums)\\n         for i in range(x):\\n             comp = target - nums[i]\\n             if comp in hashmap:\\n                 return [hashmap.get(comp), i]\\n             hashmap[nums[i]] = i\\n             \\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         sorted_nums = sorted(nums)\\n         start = 0\\n         end = len(nums) - 1\\n         while start < end:\\n             curr_sum = sorted_nums[start] + sorted_nums[end]\\n             if (curr_sum == target):\\n                 break\\n             if (curr_sum < target):\\n                 start += 1\\n             else:\\n                 end -= 1\\n         first_index = nums.index(sorted_nums[start])\\n         second_index = nums.index(sorted_nums[end])\\n         if sorted_nums[start] == sorted_nums[end]:\\n             nums.pop(first_index)\\n             second_index = nums.index(sorted_nums[end]) + 1\\n         return [first_index, second_index]\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         table = {}\\n         for i, e in enumerate(nums):\\n             print(\\\"fe\\\")\\n             if e in table:\\n                 table[e] = [table[e][0] + 1, table[e][1] + [i]]\\n             else:\\n                 table[e] = [1, [i]]\\n             \\n             sub_target = target - e\\n             if sub_target in table and (sub_target != e or table[sub_target][0] >= 2):\\n                 print(table)\\n                 first_idx = i\\n                 second_idx = None\\n                 for index in table[sub_target][1]:\\n                     if index != first_idx:\\n                         second_idx = index\\n                         break\\n                 \\n                 assert(second_idx, \\\"got shit\\\")\\n                 result = [first_idx, second_idx]\\n                 result.sort()\\n                 return result\\n         return []\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         hashdict = {}\\n         for i,num in enumerate(nums):\\n             if target-num in hashdict:\\n                 return [hashdict[target-num], i]\\n             else:\\n                 hashdict[num] = i\\n\", \"class Solution(object):\\n     def twoSum(self, nums, target):\\n         nums_index = [(v, index) for index, v in enumerate(nums)]\\n         nums_index.sort()\\n         begin, end = 0, len(nums) - 1\\n         while begin < end:\\n             curr = nums_index[begin][0] + nums_index[end][0]\\n             if curr == target:\\n                 return [nums_index[begin][1], nums_index[end][1]]\\n             elif curr < target:\\n                 begin += 1\\n             else:\\n                 end -= 1\\n\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         numsMap = {}\\n         for i, num in enumerate(nums):\\n             if target - num in numsMap:\\n                 return [i, numsMap[target-num]]\\n             numsMap[num] = i\\n         return [None, None]\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n #         length_nums = len(nums)\\n         \\n #         for i in range(length_nums):\\n #             for j in range(i+1, length_nums):\\n #                 if nums[i] + nums[j] == target:\\n #                     return [i, j]\\n \\n         # dic = {}\\n         # for i, num in enumerate(nums):\\n         #     if num in dic:\\n         #         return [dic[num], i]\\n         #     else:\\n         #         dic[target - num] = i\\n         \\n         num_dict = {}\\n         for i, num in enumerate(nums):\\n             rem = target - num\\n             if rem in num_dict:\\n                 return [num_dict[rem], i]\\n             num_dict[num] = i\\n         return None\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         dict = {}\\n         for i in range(0,len(nums)):\\n             if nums[i] in dict:\\n                 return [dict[nums[i]],i]\\n             else:\\n                 dict[target-nums[i]] = i\\n         return []\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         indexes = dict([(nums[i], i) for i in range(len(nums))])\\n         for i in range(len(nums)):\\n             if target-nums[i] in indexes and indexes[target - nums[i]] != i:\\n                 return [i, indexes[target - nums[i]]]\\n         return []\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         # record the sorted index (original position)\\n         nums_sorted_index = sorted(range(len(nums)), key = lambda k: nums[k])\\n         # sort the list\\n         nums.sort()\\n         for i in range(len(nums)):\\n             for j in range(i+1,len(nums)):\\n                 if nums[i]+nums[j] > target:\\n                     break\\n                 elif nums[i]+nums[j] == target:\\n                     return [nums_sorted_index[i], nums_sorted_index[j]]\\n         print('Can\\\\'t find a match!')\", \"class Solution:\\n     def twoSum(self, nums, target):\\n         \\n         \\\"\\\"\\\"\\n         dic={}\\n         for i,now in enumerate(nums):\\n             dev = target - now\\n             if dev in dic:\\n                 return [dic[dev],i]\\n             dic[now]=i\\n         return None\\n         \\\"\\\"\\\"\\n     \\n         \\n     #   old\\n         a=sorted(nums)\\n         \\n         for j in range(len(nums)):\\n             for k in range(j+1,len(nums)):\\n                 s = a[j]+a[k]\\n                 if s == target:\\n                     if a[j]==a[k]:\\n                         return [i for i,x in enumerate(nums) if x == a[k]]\\n                     else:\\n                         b=nums.index(a[j])\\n                         c=nums.index(a[k])\\n                         return [b,c]\\n                         \\n                 elif s>target:\\n                     break\\n                \\n         \\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n\", \"# Using 'dict', like hash, to find the pair number\\n class Solution:\\n     def twoSum(self, numbers, target):\\n         ans = []\\n         dir = {}\\n         ll = len(numbers)\\n         for i in range(ll):\\n             dir[numbers[i]] = i\\n         for i in range(ll):\\n             o2 = target-numbers[i]\\n             if o2 in dir:\\n                 if (dir[o2] != i):\\n                     ans.append(i)\\n                     ans.append(dir[o2])\\n                     return ans\"]",
        "difficulty": "introductory",
        "input": [
            [
                3,
                3
            ],
            [
                6
            ]
        ],
        "output": [
            0,
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "twoSum",
        "starter_code": "\nclass Solution:\n    def twoSum(self, nums: List[int], target: int) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/two-sum/"
    },
    {
        "id": 1303,
        "task_id": 2466,
        "test_case_id": 1,
        "question": "Given a collection of distinct integers, return all possible permutations.\n\nExample:\n\n\nInput: [1,2,3]\nOutput:\n[\n  [1,2,3],\n  [1,3,2],\n  [2,1,3],\n  [2,3,1],\n  [3,1,2],\n  [3,2,1]\n]",
        "solutions": "[\"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         all_permutes = []\\n         self.permute_nums(all_permutes, nums, [])\\n         return all_permutes\\n     \\n     def permute_nums(self, all_permutes, nums, cur_permute):\\n         if len(nums) == 0:\\n             all_permutes.append(cur_permute)\\n             return\\n \\n         for i in range(len(nums)):\\n             num = nums[i]\\n \\n             self.permute_nums(all_permutes, nums[0:i] + nums[i+1:len(nums)], cur_permute + [num])\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return []\\n \\n         nums.sort()\\n         res = [nums[:]]\\n         n = len(nums)\\n         i = n-1\\n         while i > 0:\\n             if nums[i-1] < nums[i]:\\n                 j = n-1\\n                 while nums[j] < nums[i-1]:\\n                     j -= 1\\n                 nums[i-1], nums[j] = nums[j], nums[i-1]\\n                 nums[i:] = sorted(nums[i:])\\n                 res.append(nums[:])\\n                 i = n-1\\n             else:\\n                 i -= 1\\n \\n         return res\\n     \\n     \\n\", \"class Solution:\\n \\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         res = []\\n         \\n         self.dfs(nums, [], res)\\n         \\n         return res\\n             \\n     def dfs(self, nums, path, res):\\n         if not nums:\\n             res.append(path)\\n         \\n         for i in range(len(nums)):\\n             self.dfs(nums[:i] + nums[i+1:], path+[nums[i]], res)\\n\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         self.res = []\\n         if not nums or len(nums) == 0:\\n             return self.res\\n         self.dfs(0, [], nums)\\n         return self.res\\n         \\n     def dfs(self, i, path, remaining):\\n         if len(remaining) == 0:\\n             self.res.append(path)\\n         for i in range(len(remaining)):\\n             self.dfs(i+1, path + [remaining[i]], remaining[0:i]+remaining[i+1:])\\n\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         print(nums)\\n         \\n         def swap(a, i, j):\\n             temp = a[i]\\n             a[i] = a[j]\\n             a[j] = temp\\n         \\n         def helper(index, path):\\n             if index == len(nums) - 1:\\n                 res.append(path.copy())\\n             for i in range(index, len(nums)):\\n                 swap(path, index, i)\\n                 helper(index + 1, path.copy())\\n             \\n         helper(0, nums)\\n         print(nums)\\n         return res\"]",
        "difficulty": "interview",
        "input": [
            [
                1,
                2,
                3
            ]
        ],
        "output": [
            [
                1,
                2,
                3
            ],
            [
                1,
                3,
                2
            ],
            [
                2,
                1,
                3
            ],
            [
                2,
                3,
                1
            ],
            [
                3,
                1,
                2
            ],
            [
                3,
                2,
                1
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "permute",
        "starter_code": "\nclass Solution:\n    def permute(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/permutations/"
    },
    {
        "id": 1304,
        "task_id": 2466,
        "test_case_id": 2,
        "question": "Given a collection of distinct integers, return all possible permutations.\n\nExample:\n\n\nInput: [1,2,3]\nOutput:\n[\n  [1,2,3],\n  [1,3,2],\n  [2,1,3],\n  [2,3,1],\n  [3,1,2],\n  [3,2,1]\n]",
        "solutions": "[\"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         all_permutes = []\\n         self.permute_nums(all_permutes, nums, [])\\n         return all_permutes\\n     \\n     def permute_nums(self, all_permutes, nums, cur_permute):\\n         if len(nums) == 0:\\n             all_permutes.append(cur_permute)\\n             return\\n \\n         for i in range(len(nums)):\\n             num = nums[i]\\n \\n             self.permute_nums(all_permutes, nums[0:i] + nums[i+1:len(nums)], cur_permute + [num])\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return []\\n \\n         nums.sort()\\n         res = [nums[:]]\\n         n = len(nums)\\n         i = n-1\\n         while i > 0:\\n             if nums[i-1] < nums[i]:\\n                 j = n-1\\n                 while nums[j] < nums[i-1]:\\n                     j -= 1\\n                 nums[i-1], nums[j] = nums[j], nums[i-1]\\n                 nums[i:] = sorted(nums[i:])\\n                 res.append(nums[:])\\n                 i = n-1\\n             else:\\n                 i -= 1\\n \\n         return res\\n     \\n     \\n\", \"class Solution:\\n \\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         res = []\\n         \\n         self.dfs(nums, [], res)\\n         \\n         return res\\n             \\n     def dfs(self, nums, path, res):\\n         if not nums:\\n             res.append(path)\\n         \\n         for i in range(len(nums)):\\n             self.dfs(nums[:i] + nums[i+1:], path+[nums[i]], res)\\n\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         self.res = []\\n         if not nums or len(nums) == 0:\\n             return self.res\\n         self.dfs(0, [], nums)\\n         return self.res\\n         \\n     def dfs(self, i, path, remaining):\\n         if len(remaining) == 0:\\n             self.res.append(path)\\n         for i in range(len(remaining)):\\n             self.dfs(i+1, path + [remaining[i]], remaining[0:i]+remaining[i+1:])\\n\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         print(nums)\\n         \\n         def swap(a, i, j):\\n             temp = a[i]\\n             a[i] = a[j]\\n             a[j] = temp\\n         \\n         def helper(index, path):\\n             if index == len(nums) - 1:\\n                 res.append(path.copy())\\n             for i in range(index, len(nums)):\\n                 swap(path, index, i)\\n                 helper(index + 1, path.copy())\\n             \\n         helper(0, nums)\\n         print(nums)\\n         return res\"]",
        "difficulty": "interview",
        "input": [
            [
                0,
                1
            ]
        ],
        "output": [
            [
                0,
                1
            ],
            [
                1,
                0
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "permute",
        "starter_code": "\nclass Solution:\n    def permute(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/permutations/"
    },
    {
        "id": 1305,
        "task_id": 2466,
        "test_case_id": 3,
        "question": "Given a collection of distinct integers, return all possible permutations.\n\nExample:\n\n\nInput: [1,2,3]\nOutput:\n[\n  [1,2,3],\n  [1,3,2],\n  [2,1,3],\n  [2,3,1],\n  [3,1,2],\n  [3,2,1]\n]",
        "solutions": "[\"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         all_permutes = []\\n         self.permute_nums(all_permutes, nums, [])\\n         return all_permutes\\n     \\n     def permute_nums(self, all_permutes, nums, cur_permute):\\n         if len(nums) == 0:\\n             all_permutes.append(cur_permute)\\n             return\\n \\n         for i in range(len(nums)):\\n             num = nums[i]\\n \\n             self.permute_nums(all_permutes, nums[0:i] + nums[i+1:len(nums)], cur_permute + [num])\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return []\\n \\n         nums.sort()\\n         res = [nums[:]]\\n         n = len(nums)\\n         i = n-1\\n         while i > 0:\\n             if nums[i-1] < nums[i]:\\n                 j = n-1\\n                 while nums[j] < nums[i-1]:\\n                     j -= 1\\n                 nums[i-1], nums[j] = nums[j], nums[i-1]\\n                 nums[i:] = sorted(nums[i:])\\n                 res.append(nums[:])\\n                 i = n-1\\n             else:\\n                 i -= 1\\n \\n         return res\\n     \\n     \\n\", \"class Solution:\\n \\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         res = []\\n         \\n         self.dfs(nums, [], res)\\n         \\n         return res\\n             \\n     def dfs(self, nums, path, res):\\n         if not nums:\\n             res.append(path)\\n         \\n         for i in range(len(nums)):\\n             self.dfs(nums[:i] + nums[i+1:], path+[nums[i]], res)\\n\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         self.res = []\\n         if not nums or len(nums) == 0:\\n             return self.res\\n         self.dfs(0, [], nums)\\n         return self.res\\n         \\n     def dfs(self, i, path, remaining):\\n         if len(remaining) == 0:\\n             self.res.append(path)\\n         for i in range(len(remaining)):\\n             self.dfs(i+1, path + [remaining[i]], remaining[0:i]+remaining[i+1:])\\n\", \"class Solution:\\n     def permute(self, nums):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         print(nums)\\n         \\n         def swap(a, i, j):\\n             temp = a[i]\\n             a[i] = a[j]\\n             a[j] = temp\\n         \\n         def helper(index, path):\\n             if index == len(nums) - 1:\\n                 res.append(path.copy())\\n             for i in range(index, len(nums)):\\n                 swap(path, index, i)\\n                 helper(index + 1, path.copy())\\n             \\n         helper(0, nums)\\n         print(nums)\\n         return res\"]",
        "difficulty": "interview",
        "input": [
            [
                1
            ]
        ],
        "output": [
            [
                1
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "permute",
        "starter_code": "\nclass Solution:\n    def permute(self, nums: List[int]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/permutations/"
    },
    {
        "id": 1306,
        "task_id": 2467,
        "test_case_id": 1,
        "question": "Find all possible combinations of k numbers that add up to a number n, given that only numbers from 1 to 9 can be used and each combination should be a unique set of numbers.\n\nNote:\n\n\n       All numbers will be positive integers.\n       The solution set must not contain duplicate combinations.\n\n\nExample 1:\n\n\nInput: k = 3, n = 7\nOutput: [[1,2,4]]\n\n\nExample 2:\n\n\nInput: k = 3, n = 9\nOutput: [[1,2,6], [1,3,5], [2,3,4]]",
        "solutions": "[\"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         to_return = []\\n         self.backtrack(to_return, [], k, n, 1)\\n         return to_return\\n     \\n     def backtrack(self, to_return, temp, k, n, start):\\n         total = sum(temp)\\n         \\n         if total > n:\\n             return\\n         if len(temp) == k and total == n:\\n             to_return.append(temp[:])\\n             return\\n         \\n         for i in range(start, 10):\\n             temp.append(i)\\n             self.backtrack(to_return, temp, k, n, i + 1)\\n             temp.pop()\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         self.dfs(range(1,10), k, n, 0, [], res)\\n         return res\\n \\n     def dfs(self, nums, k, n, index, path, res):\\n         if k == 0 and n == 0:\\n             res.append(path)\\n             return \\n         for i in range(index, len(nums)):\\n             self.dfs(nums, k-1, n- nums[i], i + 1, path+[nums[i]], res)\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n \\n         res = []\\n         temp = []\\n         idx = 1\\n         self.backtracking(res, temp, n, k, idx)\\n         \\n         return res\\n     \\n     def backtracking(self, res, temp, n, k, idx):\\n         if len(temp) == k and n == 0:\\n             res.append(temp[:])\\n             return\\n         \\n         for i in range(idx, 10):\\n             temp.append(i)\\n             self.backtracking(res, temp, n - i, k, i + 1)\\n             temp.pop()\\n             \\n\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         res, com = [], []\\n         self.combination(n, res, com, 1, k)\\n         return res\\n     \\n     def combination(self, target, res, com, begin, k):\\n         for i in range(begin, target + 1 if target < 9 else 10):\\n             com.append(i)\\n             if i == target and k == 1: res.append(com[:])\\n             self.combination(target - i, res, com, i + 1, k - 1)\\n             com.pop()\\n\", \"class Solution:\\n     result = []\\n     def combinationSum3Util(self, candidates, current_subset, k, target, start):\\n         if target == 0 and k == 0:\\n             Solution.result.append(current_subset)\\n         \\n         elif target != 0 and k != 0:\\n             for i in range(start, len(candidates)):\\n                 current_candidate_num = candidates[i]\\n                 if i > start and candidates[i] == candidates[i-1]:\\n                     continue\\n                     \\n                 if target - current_candidate_num >= 0:\\n                     next_subset = current_subset.copy()\\n                     next_subset.append(current_candidate_num)\\n                     self.combinationSum3Util(candidates, next_subset, k-1, target - current_candidate_num, i+1)\\n                 else:\\n                     return\\n         \\n         else:   # target == 0 and k != 0     OR     target != 0 and k == 0\\n             return\\n                 \\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         Solution.result = []\\n         \\n         if k == 0:\\n             return [[]]\\n         \\n         candidates = [x for x in range(1, 10)]\\n         current_subset = []\\n         self.combinationSum3Util(candidates, current_subset, k, n, 0)\\n         return Solution.result\\n\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         if(n > 45): return []\\n         res = []\\n         def rev(start, s, count, one):\\n             if(count == k and s == n):\\n                 res.append(one[:])\\n                 return\\n             if(count == k or s == n):\\n                 return\\n             for i in range(start, 10):\\n                 if(s + i > n): return\\n                 rev(i+1, s+i, count+1, one+[i])\\n         rev(1, 0, 0, [])\\n         return res\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         self.dfs(res, [], 0, 0, 1, k, n)\\n         return res\\n     \\n     def dfs(self, res, cur, d, s, b, k, n):\\n         if d == k and s == n:\\n             res.append(cur)\\n             return\\n         \\n         for i in range(b, 10):\\n             self.dfs(res, cur + [i], d + 1, s + i, i + 1, k, n)\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         import functools;\\n         return [ c for c in functools.reduce(lambda nexts, _ : [next_+[first] for next_ in nexts for first in range(next_[-1]+1 if next_ else 1, 10)], range(k), [[]]) if sum(c) == n]\"]",
        "difficulty": "interview",
        "input": [
            3,
            7
        ],
        "output": [
            [
                1,
                2,
                4
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "combinationSum3",
        "starter_code": "\nclass Solution:\n    def combinationSum3(self, k: int, n: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/combination-sum-iii/"
    },
    {
        "id": 1307,
        "task_id": 2467,
        "test_case_id": 2,
        "question": "Find all possible combinations of k numbers that add up to a number n, given that only numbers from 1 to 9 can be used and each combination should be a unique set of numbers.\n\nNote:\n\n\n       All numbers will be positive integers.\n       The solution set must not contain duplicate combinations.\n\n\nExample 1:\n\n\nInput: k = 3, n = 7\nOutput: [[1,2,4]]\n\n\nExample 2:\n\n\nInput: k = 3, n = 9\nOutput: [[1,2,6], [1,3,5], [2,3,4]]",
        "solutions": "[\"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         to_return = []\\n         self.backtrack(to_return, [], k, n, 1)\\n         return to_return\\n     \\n     def backtrack(self, to_return, temp, k, n, start):\\n         total = sum(temp)\\n         \\n         if total > n:\\n             return\\n         if len(temp) == k and total == n:\\n             to_return.append(temp[:])\\n             return\\n         \\n         for i in range(start, 10):\\n             temp.append(i)\\n             self.backtrack(to_return, temp, k, n, i + 1)\\n             temp.pop()\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         self.dfs(range(1,10), k, n, 0, [], res)\\n         return res\\n \\n     def dfs(self, nums, k, n, index, path, res):\\n         if k == 0 and n == 0:\\n             res.append(path)\\n             return \\n         for i in range(index, len(nums)):\\n             self.dfs(nums, k-1, n- nums[i], i + 1, path+[nums[i]], res)\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n \\n         res = []\\n         temp = []\\n         idx = 1\\n         self.backtracking(res, temp, n, k, idx)\\n         \\n         return res\\n     \\n     def backtracking(self, res, temp, n, k, idx):\\n         if len(temp) == k and n == 0:\\n             res.append(temp[:])\\n             return\\n         \\n         for i in range(idx, 10):\\n             temp.append(i)\\n             self.backtracking(res, temp, n - i, k, i + 1)\\n             temp.pop()\\n             \\n\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         res, com = [], []\\n         self.combination(n, res, com, 1, k)\\n         return res\\n     \\n     def combination(self, target, res, com, begin, k):\\n         for i in range(begin, target + 1 if target < 9 else 10):\\n             com.append(i)\\n             if i == target and k == 1: res.append(com[:])\\n             self.combination(target - i, res, com, i + 1, k - 1)\\n             com.pop()\\n\", \"class Solution:\\n     result = []\\n     def combinationSum3Util(self, candidates, current_subset, k, target, start):\\n         if target == 0 and k == 0:\\n             Solution.result.append(current_subset)\\n         \\n         elif target != 0 and k != 0:\\n             for i in range(start, len(candidates)):\\n                 current_candidate_num = candidates[i]\\n                 if i > start and candidates[i] == candidates[i-1]:\\n                     continue\\n                     \\n                 if target - current_candidate_num >= 0:\\n                     next_subset = current_subset.copy()\\n                     next_subset.append(current_candidate_num)\\n                     self.combinationSum3Util(candidates, next_subset, k-1, target - current_candidate_num, i+1)\\n                 else:\\n                     return\\n         \\n         else:   # target == 0 and k != 0     OR     target != 0 and k == 0\\n             return\\n                 \\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         Solution.result = []\\n         \\n         if k == 0:\\n             return [[]]\\n         \\n         candidates = [x for x in range(1, 10)]\\n         current_subset = []\\n         self.combinationSum3Util(candidates, current_subset, k, n, 0)\\n         return Solution.result\\n\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         if(n > 45): return []\\n         res = []\\n         def rev(start, s, count, one):\\n             if(count == k and s == n):\\n                 res.append(one[:])\\n                 return\\n             if(count == k or s == n):\\n                 return\\n             for i in range(start, 10):\\n                 if(s + i > n): return\\n                 rev(i+1, s+i, count+1, one+[i])\\n         rev(1, 0, 0, [])\\n         return res\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         self.dfs(res, [], 0, 0, 1, k, n)\\n         return res\\n     \\n     def dfs(self, res, cur, d, s, b, k, n):\\n         if d == k and s == n:\\n             res.append(cur)\\n             return\\n         \\n         for i in range(b, 10):\\n             self.dfs(res, cur + [i], d + 1, s + i, i + 1, k, n)\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         import functools;\\n         return [ c for c in functools.reduce(lambda nexts, _ : [next_+[first] for next_ in nexts for first in range(next_[-1]+1 if next_ else 1, 10)], range(k), [[]]) if sum(c) == n]\"]",
        "difficulty": "interview",
        "input": [
            3,
            9
        ],
        "output": [
            [
                1,
                2,
                6
            ],
            [
                1,
                3,
                5
            ],
            [
                2,
                3,
                4
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "combinationSum3",
        "starter_code": "\nclass Solution:\n    def combinationSum3(self, k: int, n: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/combination-sum-iii/"
    },
    {
        "id": 1308,
        "task_id": 2467,
        "test_case_id": 3,
        "question": "Find all possible combinations of k numbers that add up to a number n, given that only numbers from 1 to 9 can be used and each combination should be a unique set of numbers.\n\nNote:\n\n\n       All numbers will be positive integers.\n       The solution set must not contain duplicate combinations.\n\n\nExample 1:\n\n\nInput: k = 3, n = 7\nOutput: [[1,2,4]]\n\n\nExample 2:\n\n\nInput: k = 3, n = 9\nOutput: [[1,2,6], [1,3,5], [2,3,4]]",
        "solutions": "[\"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         to_return = []\\n         self.backtrack(to_return, [], k, n, 1)\\n         return to_return\\n     \\n     def backtrack(self, to_return, temp, k, n, start):\\n         total = sum(temp)\\n         \\n         if total > n:\\n             return\\n         if len(temp) == k and total == n:\\n             to_return.append(temp[:])\\n             return\\n         \\n         for i in range(start, 10):\\n             temp.append(i)\\n             self.backtrack(to_return, temp, k, n, i + 1)\\n             temp.pop()\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         self.dfs(range(1,10), k, n, 0, [], res)\\n         return res\\n \\n     def dfs(self, nums, k, n, index, path, res):\\n         if k == 0 and n == 0:\\n             res.append(path)\\n             return \\n         for i in range(index, len(nums)):\\n             self.dfs(nums, k-1, n- nums[i], i + 1, path+[nums[i]], res)\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n \\n         res = []\\n         temp = []\\n         idx = 1\\n         self.backtracking(res, temp, n, k, idx)\\n         \\n         return res\\n     \\n     def backtracking(self, res, temp, n, k, idx):\\n         if len(temp) == k and n == 0:\\n             res.append(temp[:])\\n             return\\n         \\n         for i in range(idx, 10):\\n             temp.append(i)\\n             self.backtracking(res, temp, n - i, k, i + 1)\\n             temp.pop()\\n             \\n\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         res, com = [], []\\n         self.combination(n, res, com, 1, k)\\n         return res\\n     \\n     def combination(self, target, res, com, begin, k):\\n         for i in range(begin, target + 1 if target < 9 else 10):\\n             com.append(i)\\n             if i == target and k == 1: res.append(com[:])\\n             self.combination(target - i, res, com, i + 1, k - 1)\\n             com.pop()\\n\", \"class Solution:\\n     result = []\\n     def combinationSum3Util(self, candidates, current_subset, k, target, start):\\n         if target == 0 and k == 0:\\n             Solution.result.append(current_subset)\\n         \\n         elif target != 0 and k != 0:\\n             for i in range(start, len(candidates)):\\n                 current_candidate_num = candidates[i]\\n                 if i > start and candidates[i] == candidates[i-1]:\\n                     continue\\n                     \\n                 if target - current_candidate_num >= 0:\\n                     next_subset = current_subset.copy()\\n                     next_subset.append(current_candidate_num)\\n                     self.combinationSum3Util(candidates, next_subset, k-1, target - current_candidate_num, i+1)\\n                 else:\\n                     return\\n         \\n         else:   # target == 0 and k != 0     OR     target != 0 and k == 0\\n             return\\n                 \\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         Solution.result = []\\n         \\n         if k == 0:\\n             return [[]]\\n         \\n         candidates = [x for x in range(1, 10)]\\n         current_subset = []\\n         self.combinationSum3Util(candidates, current_subset, k, n, 0)\\n         return Solution.result\\n\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n         if(n > 45): return []\\n         res = []\\n         def rev(start, s, count, one):\\n             if(count == k and s == n):\\n                 res.append(one[:])\\n                 return\\n             if(count == k or s == n):\\n                 return\\n             for i in range(start, 10):\\n                 if(s + i > n): return\\n                 rev(i+1, s+i, count+1, one+[i])\\n         rev(1, 0, 0, [])\\n         return res\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         res = []\\n         self.dfs(res, [], 0, 0, 1, k, n)\\n         return res\\n     \\n     def dfs(self, res, cur, d, s, b, k, n):\\n         if d == k and s == n:\\n             res.append(cur)\\n             return\\n         \\n         for i in range(b, 10):\\n             self.dfs(res, cur + [i], d + 1, s + i, i + 1, k, n)\", \"class Solution:\\n     def combinationSum3(self, k, n):\\n         \\\"\\\"\\\"\\n         :type k: int\\n         :type n: int\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         import functools;\\n         return [ c for c in functools.reduce(lambda nexts, _ : [next_+[first] for next_ in nexts for first in range(next_[-1]+1 if next_ else 1, 10)], range(k), [[]]) if sum(c) == n]\"]",
        "difficulty": "interview",
        "input": [
            9,
            45
        ],
        "output": [
            [
                1,
                2,
                3,
                4,
                5,
                6,
                7,
                8,
                9
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "combinationSum3",
        "starter_code": "\nclass Solution:\n    def combinationSum3(self, k: int, n: int) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/combination-sum-iii/"
    },
    {
        "id": 1309,
        "task_id": 2538,
        "test_case_id": 1,
        "question": "You play your favourite game yet another time. You chose the character you didn't play before. It has $str$ points of strength and $int$ points of intelligence. Also, at start, the character has $exp$ free experience points you can invest either in strength or in intelligence (by investing one point you can either raise strength by $1$ or raise intelligence by $1$).\n\nSince you'd like to make some fun you want to create a jock character, so it has more strength than intelligence points (resulting strength is strictly greater than the resulting intelligence).\n\nCalculate the number of different character builds you can create (for the purpose of replayability) if you must invest all free points. Two character builds are different if their strength and/or intellect are different.\n\n\n-----Input-----\n\nThe first line contains the single integer $T$ ($1 \\le T \\le 100$) — the number of queries. Next $T$ lines contain descriptions of queries — one per line.\n\nThis line contains three integers $str$, $int$ and $exp$ ($1 \\le str, int \\le 10^8$, $0 \\le exp \\le 10^8$) — the initial strength and intelligence of the character and the number of free points, respectively.\n\n\n-----Output-----\n\nPrint $T$ integers — one per query. For each query print the number of different character builds you can create.\n\n\n-----Example-----\nInput\n4\n5 3 4\n2 1 0\n3 5 5\n4 10 6\n\nOutput\n3\n1\n2\n0\n\n\n\n-----Note-----\n\nIn the first query there are only three appropriate character builds: $(str = 7, int = 5)$, $(8, 4)$ and $(9, 3)$. All other builds are either too smart or don't use all free points.\n\nIn the second query there is only one possible build: $(2, 1)$.\n\nIn the third query there are two appropriate builds: $(7, 6)$, $(8, 5)$.\n\nIn the fourth query all builds have too much brains.",
        "solutions": "[\"for TT in range(1, int(input()) + 1):\\n    a, b, c = map(int, input().split())\\n    l = max(-1, (b + c - a) // 2)\\n    r = c\\n    print(r - l if r >= l else 0)\", \"for _ in range(int(input())):\\n    a,b,c=map(int,input().split())\\n    total=1+(a+b+c)//2\\n    print(max(0,a+c-max(total,a)+1))\", \"t = int(input())\\n\\nfor _ in range(t):\\n    s, i, e = list(map(int, input().split()))\\n\\n    half = (s + i + e) // 2\\n    maxeven = half + 1\\n    maxeven = max(maxeven, s)\\n\\n    print(max(s + e - maxeven + 1, 0))\\n\", \"T = int(input())\\nfor i in range(T):\\n    s, k, f = map(int, input().split())\\n    print(min(f + 1, max(0, (s + f - k + 1) // 2)))\", \"T = int(input())\\nfor t in range(T):\\n    s, i, e = list(map(int, input().split()))\\n    min_s = max((s + i + e)//2 + 1, s)\\n    max_s = s + e\\n    c = max(0, max_s-min_s+1)\\n    print(c)\\n\", \"for cas in range(int(input())):\\n    a, b, c = map(int, input().split())\\n    l = (b + c - a) // 2 + 1\\n    r = c\\n    print(max(0, r - max(l, 0) + 1))\", \"for i in range(int(input())):\\n    a,b,c = list(map(int,input().split()))\\n    if a+c <= b:\\n        print(0)\\n    else:\\n        print(min((a+c-b+1)//2,c+1))\\n\", \"T = int(input())\\nfor i in range(T):\\n    s, i, e = list(map(int, input().split()))\\n    if s > i + e:\\n        a = e + 1\\n    else:\\n        a = min((s+e-i-1)//2 + 1, e)\\n    if a <= 0:\\n        print(0)\\n    else:\\n        print(a)\\n\", \"for _ in range(int(input())):\\n    s, i, e = list(map(int, input().split()))\\n    if s + e <= i:\\n        print('0')\\n    elif s > i + e:\\n        print(e + 1)\\n    else:\\n        print((e + s + 1 - i) // 2)\\n\", \"T = int(input())\\nfor _ in range(T):\\n    a, b, c = list(map(int, input().split()))\\n    print(max(c+1 - max((-a+b+c+2)//2, 0), 0))\\n\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nfor _ in range(int(input())):\\n    s, i, e = [int(i) for i in input().split()]\\n    \\n    x = (i+e-s)//2\\n    \\n    if x < 0: print(e+1)\\n    elif x > e: print(0)\\n    else: print(e-x)\", \"n=int(input())\\nfor i in range(n):\\n    a,b,c=[int(x) for x in input().split()]\\n    l=0\\n    r=c\\n    while r-l>1:\\n        mid=(l+r)//2\\n        if a + mid >b+c-mid:\\n            r=mid\\n        else:\\n            l=mid\\n    if a>b+c:\\n        print(c+1)\\n    elif a+ r >b+c-r:\\n        print(c-r+1)\\n    \\n    else:\\n        print(0)\\n    \\n            \\n\", \"for _ in range(int(input())):\\n    ans = 0\\n    cnt = 0\\n    s, i, e = list(map(int, input().split()))\\n    b = (s - i + e - 1) // 2\\n    if b < 0:\\n        print(0)\\n    elif b == 0:\\n        print(1)\\n    elif b >= e:\\n        print(e + 1)\\n    else:\\n        print(b + 1)\\n\\n\", \"n = int(input())\\nfor i in range(n):\\n    s, i, e = (int(i) for i in input().split())\\n    if s > i + e:\\n        print(e + 1)\\n    elif s + e <= i:\\n        print(0)\\n    else:\\n        r = s - i\\n        r = e + r\\n        t = 0\\n        if r % 2:\\n            t = 1\\n        r = r // 2\\n        if t:\\n            r += 1\\n        print(r)\\n\", \"T = int(input())\\nfor i in range(0, T):\\n    s, i, exp = (int(i) for i in input().split())\\n    if (s + exp <= i):\\n        print(0)\\n    elif (i + exp < s):\\n        print(exp + 1)\\n    else:\\n        print(exp + 1 - (i + exp - s + 2) // 2)\", \"import math\\nn = int(input())\\nfor i in range(n):\\n    a, b, c = [int(item) for item in input().split()]\\n    ans = min(c+1, max(0, math.ceil(((a - b) + c) / 2)))\\n    print(ans)\", \"for _ in range(int(input())):\\n    \\n    s, i, e = map(int, input().split())\\n    \\n    lo, hi, res = 0, e, 0\\n\\n    if s + e <= i:\\n        print(0)\\n        continue\\n\\n    while lo <= hi:\\n\\n        mi = (lo + hi) // 2\\n\\n        if s + mi > i + (e - mi):\\n            res = mi\\n            hi = mi - 1\\n        else:\\n            lo = mi + 1\\n    \\n    print(e - res + 1)\", \"t = int(input())\\nfor _ in range(t):\\n    s, i, e = list(map(int, input().split()))\\n    if s + e <= i:\\n        print(0)\\n        continue\\n    if i + e < s:\\n        print(e + 1)\\n        continue\\n    mx = s + e\\n    dif = abs(s - i)\\n    if s > i:\\n        i += dif\\n    else:\\n        s += dif\\n    e -= dif\\n    mn = s + e // 2 + 1\\n    print(mx - mn + 1)\\n\", \"import math\\nntc = int(input())\\nfor tcs in range(ntc):\\n    s, i, e = list(map(int, input().split()))\\n    if s + e > i:\\n        res = math.ceil(((s + e) - i) / 2)\\n        print(min(res, e+1))\\n    else:\\n        print(0)\", \"''' CODED WITH LOVE BY SATYAM KUMAR '''\\n\\nfrom sys import stdin, stdout\\nimport heapq\\nimport cProfile, math\\nfrom collections import Counter, defaultdict, deque\\nfrom bisect import bisect_left, bisect, bisect_right\\nimport itertools\\nfrom copy import deepcopy\\nfrom fractions import Fraction\\nimport sys, threading\\nimport operator as op\\nfrom functools import reduce\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)  # max depth of recursion\\nthreading.stack_size(2 ** 27)  # new thread will get stack of such size\\nfac_warm_up = False\\nprintHeap = str()\\nmemory_constrained = False\\nP = 10 ** 9 + 7\\n\\n\\nclass MergeFind:\\n    def __init__(self, n):\\n        self.parent = list(range(n))\\n        self.size = [1] * n\\n        self.num_sets = n\\n        self.lista = [[_] for _ in range(n)]\\n\\n    def find(self, a):\\n        to_update = []\\n        while a != self.parent[a]:\\n            to_update.append(a)\\n            a = self.parent[a]\\n        for b in to_update:\\n            self.parent[b] = a\\n        return self.parent[a]\\n\\n    def merge(self, a, b):\\n        a = self.find(a)\\n        b = self.find(b)\\n        if a == b:\\n            return\\n        if self.size[a] < self.size[b]:\\n            a, b = b, a\\n        self.num_sets -= 1\\n        self.parent[b] = a\\n        self.size[a] += self.size[b]\\n        self.lista[a] += self.lista[b]\\n\\n    def set_size(self, a):\\n        return self.size[self.find(a)]\\n\\n    def __len__(self):\\n        return self.num_sets\\n\\n\\ndef display(string_to_print):\\n    stdout.write(str(string_to_print) + \\\"\\\\n\\\")\\n\\n\\ndef prime_factors(n):  # n**0.5 complex\\n    factors = dict()\\n    for i in range(2, math.ceil(math.sqrt(n)) + 1):\\n        while n % i == 0:\\n            if i in factors:\\n                factors[i] += 1\\n            else:\\n                factors[i] = 1\\n            n = n // i\\n    if n > 2:\\n        factors[n] = 1\\n    return (factors)\\n\\n\\ndef all_factors(n):\\n    return set(reduce(list.__add__,\\n                      ([i, n // i] for i in range(1, int(n ** 0.5) + 1) if n % i == 0)))\\n\\n\\ndef fibonacci_modP(n, MOD):\\n    if n < 2: return 1\\n    return (cached_fn(fibonacci_modP, (n + 1) // 2, MOD) * cached_fn(fibonacci_modP, n // 2, MOD) + cached_fn(\\n        fibonacci_modP, (n - 1) // 2, MOD) * cached_fn(fibonacci_modP, (n - 2) // 2, MOD)) % MOD\\n\\n\\ndef factorial_modP_Wilson(n, p):\\n    if (p <= n):\\n        return 0\\n    res = (p - 1)\\n    for i in range(n + 1, p):\\n        res = (res * cached_fn(InverseEuler, i, p)) % p\\n    return res\\n\\n\\ndef binary(n, digits=20):\\n    b = bin(n)[2:]\\n    b = '0' * (digits - len(b)) + b\\n    return b\\n\\n\\ndef is_prime(n):\\n    \\\"\\\"\\\"Returns True if n is prime.\\\"\\\"\\\"\\n    if n < 4:\\n        return True\\n    if n % 2 == 0:\\n        return False\\n    if n % 3 == 0:\\n        return False\\n    i = 5\\n    w = 2\\n    while i * i <= n:\\n        if n % i == 0:\\n            return False\\n        i += w\\n        w = 6 - w\\n    return True\\n\\n\\ndef generate_primes(n):\\n    prime = [True for i in range(n + 1)]\\n    p = 2\\n    while p * p <= n:\\n        if prime[p]:\\n            for i in range(p * 2, n + 1, p):\\n                prime[i] = False\\n        p += 1\\n    return prime\\n\\n\\nfactorial_modP = []\\n\\n\\ndef warm_up_fac(MOD):\\n    nonlocal factorial_modP, fac_warm_up\\n    if fac_warm_up: return\\n    factorial_modP = [1 for _ in range(fac_warm_up_size + 1)]\\n    for i in range(2, fac_warm_up_size):\\n        factorial_modP[i] = (factorial_modP[i - 1] * i) % MOD\\n    fac_warm_up = True\\n\\n\\ndef InverseEuler(n, MOD):\\n    return pow(n, MOD - 2, MOD)\\n\\n\\ndef nCr(n, r, MOD):\\n    nonlocal fac_warm_up, factorial_modP\\n    if not fac_warm_up:\\n        warm_up_fac(MOD)\\n        fac_warm_up = True\\n    return (factorial_modP[n] * (\\n                (pow(factorial_modP[r], MOD - 2, MOD) * pow(factorial_modP[n - r], MOD - 2, MOD)) % MOD)) % MOD\\n\\n\\ndef test_print(*args):\\n    if testingMode:\\n        print(args)\\n\\n\\ndef display_list(list1, sep=\\\" \\\"):\\n    stdout.write(sep.join(map(str, list1)) + \\\"\\\\n\\\")\\n\\n\\ndef display_2D_list(li):\\n    for i in li:\\n        print(i)\\n\\n\\ndef prefix_sum(li):\\n    sm = 0\\n    res = []\\n    for i in li:\\n        sm += i\\n        res.append(sm)\\n    return res\\n\\n\\ndef get_int():\\n    return int(stdin.readline().strip())\\n\\n\\ndef get_tuple():\\n    return list(map(int, stdin.readline().split()))\\n\\n\\ndef get_list():\\n    return list(map(int, stdin.readline().split()))\\n\\n\\nmemory = dict()\\n\\n\\ndef clear_cache():\\n    nonlocal memory\\n    memory = dict()\\n\\n\\ndef cached_fn(fn, *args):\\n    nonlocal memory\\n    if args in memory:\\n        return memory[args]\\n    else:\\n        result = fn(*args)\\n        memory[args] = result\\n        return result\\n\\n\\ndef ncr(n, r):\\n    return math.factorial(n) / (math.factorial(n - r) * math.factorial(r))\\n\\n\\ndef binary_search(i, li):\\n    fn = lambda x: li[x] - x // i\\n    x = -1\\n    b = len(li)\\n    while b >= 1:\\n        while b + x < len(li) and fn(b + x) > 0:  # Change this condition 2 to whatever you like\\n            x += b\\n        b = b // 2\\n    return x\\n\\n\\n# -------------------------------------------------------------- MAIN PROGRAM\\n\\n\\nTestCases = True\\nfac_warm_up_size = 10 ** 5 + 100\\noptimise_for_recursion = False  # Can not be used clubbed with TestCases WHen using recursive functions, use Python 3\\n\\n\\ndef main():\\n    s, i, e = get_tuple()\\n    print(min(max(0, (s+e+1-i)//2), e+1))\\n# --------------------------------------------------------------------- END=\\n\\n\\nif TestCases:\\n    for i in range(get_int()):\\n        main()\\nelse:\\n    main() if not optimise_for_recursion else threading.Thread(target=main).start()\\n\", \"n = int(input())\\nfor i in range(n):\\n    s, i, e = map(int, input().split())\\n    print(min(max(0, (s - i + e + 1) // 2), e + 1))\", \"import sys \\nimport math \\nfrom collections import defaultdict\\ninput = lambda : sys.stdin.readline().rstrip()\\n\\nfor t in range(int(input())):\\n  s, i, e = map(int, input().split())\\n  \\n  x = (i + e - s) / 2\\n\\n  if x < 0:\\n    print(e + 1)\\n  \\n  elif x == 0:\\n    print(e)\\n\\n  elif x >= e:\\n    print(0)\\n  \\n  else:\\n    print(e - int(x))\", \"t = int(input())\\nwhile (t > 0):\\n    t -= 1\\n    s, i, e = [int(x) for x in input().split()]\\n    ms = s + e\\n    if (ms <= i):\\n        print(0)\\n    else:\\n        c = (ms - i + 1) // 2\\n        print(min(c, e + 1))\\n\", \"for q in range(int(input())):\\n    x, y, z = list(map(int, input().split()))\\n    if x <= y:\\n        z -= y-x+1\\n        if z < 0:\\n            print(0)\\n        else:\\n            print(z//2+1)\\n    elif z < x-y:\\n        print(z+1)\\n    else:\\n        z -= x-y+1\\n        print(x-y+z//2+1)\\n\"]",
        "difficulty": "interview",
        "input": "4\n5 3 4\n2 1 0\n3 5 5\n4 10 6\n",
        "output": "3\n1\n2\n0\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1217/A"
    },
    {
        "id": 1310,
        "task_id": 2538,
        "test_case_id": 2,
        "question": "You play your favourite game yet another time. You chose the character you didn't play before. It has $str$ points of strength and $int$ points of intelligence. Also, at start, the character has $exp$ free experience points you can invest either in strength or in intelligence (by investing one point you can either raise strength by $1$ or raise intelligence by $1$).\n\nSince you'd like to make some fun you want to create a jock character, so it has more strength than intelligence points (resulting strength is strictly greater than the resulting intelligence).\n\nCalculate the number of different character builds you can create (for the purpose of replayability) if you must invest all free points. Two character builds are different if their strength and/or intellect are different.\n\n\n-----Input-----\n\nThe first line contains the single integer $T$ ($1 \\le T \\le 100$) — the number of queries. Next $T$ lines contain descriptions of queries — one per line.\n\nThis line contains three integers $str$, $int$ and $exp$ ($1 \\le str, int \\le 10^8$, $0 \\le exp \\le 10^8$) — the initial strength and intelligence of the character and the number of free points, respectively.\n\n\n-----Output-----\n\nPrint $T$ integers — one per query. For each query print the number of different character builds you can create.\n\n\n-----Example-----\nInput\n4\n5 3 4\n2 1 0\n3 5 5\n4 10 6\n\nOutput\n3\n1\n2\n0\n\n\n\n-----Note-----\n\nIn the first query there are only three appropriate character builds: $(str = 7, int = 5)$, $(8, 4)$ and $(9, 3)$. All other builds are either too smart or don't use all free points.\n\nIn the second query there is only one possible build: $(2, 1)$.\n\nIn the third query there are two appropriate builds: $(7, 6)$, $(8, 5)$.\n\nIn the fourth query all builds have too much brains.",
        "solutions": "[\"for TT in range(1, int(input()) + 1):\\n    a, b, c = map(int, input().split())\\n    l = max(-1, (b + c - a) // 2)\\n    r = c\\n    print(r - l if r >= l else 0)\", \"for _ in range(int(input())):\\n    a,b,c=map(int,input().split())\\n    total=1+(a+b+c)//2\\n    print(max(0,a+c-max(total,a)+1))\", \"t = int(input())\\n\\nfor _ in range(t):\\n    s, i, e = list(map(int, input().split()))\\n\\n    half = (s + i + e) // 2\\n    maxeven = half + 1\\n    maxeven = max(maxeven, s)\\n\\n    print(max(s + e - maxeven + 1, 0))\\n\", \"T = int(input())\\nfor i in range(T):\\n    s, k, f = map(int, input().split())\\n    print(min(f + 1, max(0, (s + f - k + 1) // 2)))\", \"T = int(input())\\nfor t in range(T):\\n    s, i, e = list(map(int, input().split()))\\n    min_s = max((s + i + e)//2 + 1, s)\\n    max_s = s + e\\n    c = max(0, max_s-min_s+1)\\n    print(c)\\n\", \"for cas in range(int(input())):\\n    a, b, c = map(int, input().split())\\n    l = (b + c - a) // 2 + 1\\n    r = c\\n    print(max(0, r - max(l, 0) + 1))\", \"for i in range(int(input())):\\n    a,b,c = list(map(int,input().split()))\\n    if a+c <= b:\\n        print(0)\\n    else:\\n        print(min((a+c-b+1)//2,c+1))\\n\", \"T = int(input())\\nfor i in range(T):\\n    s, i, e = list(map(int, input().split()))\\n    if s > i + e:\\n        a = e + 1\\n    else:\\n        a = min((s+e-i-1)//2 + 1, e)\\n    if a <= 0:\\n        print(0)\\n    else:\\n        print(a)\\n\", \"for _ in range(int(input())):\\n    s, i, e = list(map(int, input().split()))\\n    if s + e <= i:\\n        print('0')\\n    elif s > i + e:\\n        print(e + 1)\\n    else:\\n        print((e + s + 1 - i) // 2)\\n\", \"T = int(input())\\nfor _ in range(T):\\n    a, b, c = list(map(int, input().split()))\\n    print(max(c+1 - max((-a+b+c+2)//2, 0), 0))\\n\\n\", \"from sys import stdin\\ninput = stdin.readline\\n\\nfor _ in range(int(input())):\\n    s, i, e = [int(i) for i in input().split()]\\n    \\n    x = (i+e-s)//2\\n    \\n    if x < 0: print(e+1)\\n    elif x > e: print(0)\\n    else: print(e-x)\", \"n=int(input())\\nfor i in range(n):\\n    a,b,c=[int(x) for x in input().split()]\\n    l=0\\n    r=c\\n    while r-l>1:\\n        mid=(l+r)//2\\n        if a + mid >b+c-mid:\\n            r=mid\\n        else:\\n            l=mid\\n    if a>b+c:\\n        print(c+1)\\n    elif a+ r >b+c-r:\\n        print(c-r+1)\\n    \\n    else:\\n        print(0)\\n    \\n            \\n\", \"for _ in range(int(input())):\\n    ans = 0\\n    cnt = 0\\n    s, i, e = list(map(int, input().split()))\\n    b = (s - i + e - 1) // 2\\n    if b < 0:\\n        print(0)\\n    elif b == 0:\\n        print(1)\\n    elif b >= e:\\n        print(e + 1)\\n    else:\\n        print(b + 1)\\n\\n\", \"n = int(input())\\nfor i in range(n):\\n    s, i, e = (int(i) for i in input().split())\\n    if s > i + e:\\n        print(e + 1)\\n    elif s + e <= i:\\n        print(0)\\n    else:\\n        r = s - i\\n        r = e + r\\n        t = 0\\n        if r % 2:\\n            t = 1\\n        r = r // 2\\n        if t:\\n            r += 1\\n        print(r)\\n\", \"T = int(input())\\nfor i in range(0, T):\\n    s, i, exp = (int(i) for i in input().split())\\n    if (s + exp <= i):\\n        print(0)\\n    elif (i + exp < s):\\n        print(exp + 1)\\n    else:\\n        print(exp + 1 - (i + exp - s + 2) // 2)\", \"import math\\nn = int(input())\\nfor i in range(n):\\n    a, b, c = [int(item) for item in input().split()]\\n    ans = min(c+1, max(0, math.ceil(((a - b) + c) / 2)))\\n    print(ans)\", \"for _ in range(int(input())):\\n    \\n    s, i, e = map(int, input().split())\\n    \\n    lo, hi, res = 0, e, 0\\n\\n    if s + e <= i:\\n        print(0)\\n        continue\\n\\n    while lo <= hi:\\n\\n        mi = (lo + hi) // 2\\n\\n        if s + mi > i + (e - mi):\\n            res = mi\\n            hi = mi - 1\\n        else:\\n            lo = mi + 1\\n    \\n    print(e - res + 1)\", \"t = int(input())\\nfor _ in range(t):\\n    s, i, e = list(map(int, input().split()))\\n    if s + e <= i:\\n        print(0)\\n        continue\\n    if i + e < s:\\n        print(e + 1)\\n        continue\\n    mx = s + e\\n    dif = abs(s - i)\\n    if s > i:\\n        i += dif\\n    else:\\n        s += dif\\n    e -= dif\\n    mn = s + e // 2 + 1\\n    print(mx - mn + 1)\\n\", \"import math\\nntc = int(input())\\nfor tcs in range(ntc):\\n    s, i, e = list(map(int, input().split()))\\n    if s + e > i:\\n        res = math.ceil(((s + e) - i) / 2)\\n        print(min(res, e+1))\\n    else:\\n        print(0)\", \"''' CODED WITH LOVE BY SATYAM KUMAR '''\\n\\nfrom sys import stdin, stdout\\nimport heapq\\nimport cProfile, math\\nfrom collections import Counter, defaultdict, deque\\nfrom bisect import bisect_left, bisect, bisect_right\\nimport itertools\\nfrom copy import deepcopy\\nfrom fractions import Fraction\\nimport sys, threading\\nimport operator as op\\nfrom functools import reduce\\nimport sys\\n\\nsys.setrecursionlimit(10 ** 6)  # max depth of recursion\\nthreading.stack_size(2 ** 27)  # new thread will get stack of such size\\nfac_warm_up = False\\nprintHeap = str()\\nmemory_constrained = False\\nP = 10 ** 9 + 7\\n\\n\\nclass MergeFind:\\n    def __init__(self, n):\\n        self.parent = list(range(n))\\n        self.size = [1] * n\\n        self.num_sets = n\\n        self.lista = [[_] for _ in range(n)]\\n\\n    def find(self, a):\\n        to_update = []\\n        while a != self.parent[a]:\\n            to_update.append(a)\\n            a = self.parent[a]\\n        for b in to_update:\\n            self.parent[b] = a\\n        return self.parent[a]\\n\\n    def merge(self, a, b):\\n        a = self.find(a)\\n        b = self.find(b)\\n        if a == b:\\n            return\\n        if self.size[a] < self.size[b]:\\n            a, b = b, a\\n        self.num_sets -= 1\\n        self.parent[b] = a\\n        self.size[a] += self.size[b]\\n        self.lista[a] += self.lista[b]\\n\\n    def set_size(self, a):\\n        return self.size[self.find(a)]\\n\\n    def __len__(self):\\n        return self.num_sets\\n\\n\\ndef display(string_to_print):\\n    stdout.write(str(string_to_print) + \\\"\\\\n\\\")\\n\\n\\ndef prime_factors(n):  # n**0.5 complex\\n    factors = dict()\\n    for i in range(2, math.ceil(math.sqrt(n)) + 1):\\n        while n % i == 0:\\n            if i in factors:\\n                factors[i] += 1\\n            else:\\n                factors[i] = 1\\n            n = n // i\\n    if n > 2:\\n        factors[n] = 1\\n    return (factors)\\n\\n\\ndef all_factors(n):\\n    return set(reduce(list.__add__,\\n                      ([i, n // i] for i in range(1, int(n ** 0.5) + 1) if n % i == 0)))\\n\\n\\ndef fibonacci_modP(n, MOD):\\n    if n < 2: return 1\\n    return (cached_fn(fibonacci_modP, (n + 1) // 2, MOD) * cached_fn(fibonacci_modP, n // 2, MOD) + cached_fn(\\n        fibonacci_modP, (n - 1) // 2, MOD) * cached_fn(fibonacci_modP, (n - 2) // 2, MOD)) % MOD\\n\\n\\ndef factorial_modP_Wilson(n, p):\\n    if (p <= n):\\n        return 0\\n    res = (p - 1)\\n    for i in range(n + 1, p):\\n        res = (res * cached_fn(InverseEuler, i, p)) % p\\n    return res\\n\\n\\ndef binary(n, digits=20):\\n    b = bin(n)[2:]\\n    b = '0' * (digits - len(b)) + b\\n    return b\\n\\n\\ndef is_prime(n):\\n    \\\"\\\"\\\"Returns True if n is prime.\\\"\\\"\\\"\\n    if n < 4:\\n        return True\\n    if n % 2 == 0:\\n        return False\\n    if n % 3 == 0:\\n        return False\\n    i = 5\\n    w = 2\\n    while i * i <= n:\\n        if n % i == 0:\\n            return False\\n        i += w\\n        w = 6 - w\\n    return True\\n\\n\\ndef generate_primes(n):\\n    prime = [True for i in range(n + 1)]\\n    p = 2\\n    while p * p <= n:\\n        if prime[p]:\\n            for i in range(p * 2, n + 1, p):\\n                prime[i] = False\\n        p += 1\\n    return prime\\n\\n\\nfactorial_modP = []\\n\\n\\ndef warm_up_fac(MOD):\\n    nonlocal factorial_modP, fac_warm_up\\n    if fac_warm_up: return\\n    factorial_modP = [1 for _ in range(fac_warm_up_size + 1)]\\n    for i in range(2, fac_warm_up_size):\\n        factorial_modP[i] = (factorial_modP[i - 1] * i) % MOD\\n    fac_warm_up = True\\n\\n\\ndef InverseEuler(n, MOD):\\n    return pow(n, MOD - 2, MOD)\\n\\n\\ndef nCr(n, r, MOD):\\n    nonlocal fac_warm_up, factorial_modP\\n    if not fac_warm_up:\\n        warm_up_fac(MOD)\\n        fac_warm_up = True\\n    return (factorial_modP[n] * (\\n                (pow(factorial_modP[r], MOD - 2, MOD) * pow(factorial_modP[n - r], MOD - 2, MOD)) % MOD)) % MOD\\n\\n\\ndef test_print(*args):\\n    if testingMode:\\n        print(args)\\n\\n\\ndef display_list(list1, sep=\\\" \\\"):\\n    stdout.write(sep.join(map(str, list1)) + \\\"\\\\n\\\")\\n\\n\\ndef display_2D_list(li):\\n    for i in li:\\n        print(i)\\n\\n\\ndef prefix_sum(li):\\n    sm = 0\\n    res = []\\n    for i in li:\\n        sm += i\\n        res.append(sm)\\n    return res\\n\\n\\ndef get_int():\\n    return int(stdin.readline().strip())\\n\\n\\ndef get_tuple():\\n    return list(map(int, stdin.readline().split()))\\n\\n\\ndef get_list():\\n    return list(map(int, stdin.readline().split()))\\n\\n\\nmemory = dict()\\n\\n\\ndef clear_cache():\\n    nonlocal memory\\n    memory = dict()\\n\\n\\ndef cached_fn(fn, *args):\\n    nonlocal memory\\n    if args in memory:\\n        return memory[args]\\n    else:\\n        result = fn(*args)\\n        memory[args] = result\\n        return result\\n\\n\\ndef ncr(n, r):\\n    return math.factorial(n) / (math.factorial(n - r) * math.factorial(r))\\n\\n\\ndef binary_search(i, li):\\n    fn = lambda x: li[x] - x // i\\n    x = -1\\n    b = len(li)\\n    while b >= 1:\\n        while b + x < len(li) and fn(b + x) > 0:  # Change this condition 2 to whatever you like\\n            x += b\\n        b = b // 2\\n    return x\\n\\n\\n# -------------------------------------------------------------- MAIN PROGRAM\\n\\n\\nTestCases = True\\nfac_warm_up_size = 10 ** 5 + 100\\noptimise_for_recursion = False  # Can not be used clubbed with TestCases WHen using recursive functions, use Python 3\\n\\n\\ndef main():\\n    s, i, e = get_tuple()\\n    print(min(max(0, (s+e+1-i)//2), e+1))\\n# --------------------------------------------------------------------- END=\\n\\n\\nif TestCases:\\n    for i in range(get_int()):\\n        main()\\nelse:\\n    main() if not optimise_for_recursion else threading.Thread(target=main).start()\\n\", \"n = int(input())\\nfor i in range(n):\\n    s, i, e = map(int, input().split())\\n    print(min(max(0, (s - i + e + 1) // 2), e + 1))\", \"import sys \\nimport math \\nfrom collections import defaultdict\\ninput = lambda : sys.stdin.readline().rstrip()\\n\\nfor t in range(int(input())):\\n  s, i, e = map(int, input().split())\\n  \\n  x = (i + e - s) / 2\\n\\n  if x < 0:\\n    print(e + 1)\\n  \\n  elif x == 0:\\n    print(e)\\n\\n  elif x >= e:\\n    print(0)\\n  \\n  else:\\n    print(e - int(x))\", \"t = int(input())\\nwhile (t > 0):\\n    t -= 1\\n    s, i, e = [int(x) for x in input().split()]\\n    ms = s + e\\n    if (ms <= i):\\n        print(0)\\n    else:\\n        c = (ms - i + 1) // 2\\n        print(min(c, e + 1))\\n\", \"for q in range(int(input())):\\n    x, y, z = list(map(int, input().split()))\\n    if x <= y:\\n        z -= y-x+1\\n        if z < 0:\\n            print(0)\\n        else:\\n            print(z//2+1)\\n    elif z < x-y:\\n        print(z+1)\\n    else:\\n        z -= x-y+1\\n        print(x-y+z//2+1)\\n\"]",
        "difficulty": "interview",
        "input": "5\n1 1 100000000\n100000000 100000000 100000000\n100000000 1 100000000\n1 100000000 100000000\n100000000 100000000 1\n",
        "output": "50000000\n50000000\n100000000\n1\n1\n",
        "halu_type": "Identification Hallucination",
        "fn_name": null,
        "starter_code": "",
        "url": "https://codeforces.com/problemset/problem/1217/A"
    },
    {
        "id": 1311,
        "task_id": 2633,
        "test_case_id": 1,
        "question": "table.dungeon, .dungeon th, .dungeon td {\n  border:3px solid black;\n}\n\n .dungeon th, .dungeon td {\n    text-align: center;\n    height: 70px;\n    width: 70px;\n}\n\nThe demons had captured the princess (P) and imprisoned her in the bottom-right corner of a dungeon. The dungeon consists of M x N rooms laid out in a 2D grid. Our valiant knight (K) was initially positioned in the top-left room and must fight his way through the dungeon to rescue the princess.\n\nThe knight has an initial health point represented by a positive integer. If at any point his health point drops to 0 or below, he dies immediately.\n\nSome of the rooms are guarded by demons, so the knight loses health (negative integers) upon entering these rooms; other rooms are either empty (0's) or contain magic orbs that increase the knight's health (positive integers).\n\nIn order to reach the princess as quickly as possible, the knight decides to move only rightward or downward in each step.\n\n \n\nWrite a function to determine the knight's minimum initial health so that he is able to rescue the princess.\n\nFor example, given the dungeon below, the initial health of the knight must be at least 7 if he follows the optimal path RIGHT-> RIGHT -> DOWN -> DOWN.\n\n\n       \n               \n                       -2 (K)\n                       -3\n                       3\n               \n               \n                       -5\n                       -10\n                       1\n               \n               \n                       10\n                       30\n                       -5 (P)\n               \n       \n\n\n \n\nNote:\n\n\n       The knight's health has no upper bound.\n       Any room can contain threats or power-ups, even the first room the knight enters and the bottom-right room where the princess is imprisoned.",
        "solutions": "[\"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         '''\\n         \\u9006\\u56de\\uff0c\\u5f53\\u662f\\u5c0f\\u4e8e\\u7b49\\u4e8e0\\u65f6\\uff0c\\u662f\\u76f8\\u53cd\\u6570+1.   dp\\u4e3a\\u5230\\u5f53\\u524di,j\\u53ea\\u8981\\u9700\\u8981\\u7684\\u70b9\\u6570\\uff0c \\u6700\\u5c0f\\u4e3a1\\uff0c\\u4fdd\\u8bc1 \\u6d3b\\u7740\\n         \\n         r,  c   r   ,c+1\\n         r+1,c   r+1,c+1\\n         \\n         r,c\\u5904\\u81f3\\u5c11\\u7684\\u70b9\\u6570 - r,c\\u5904\\u60e9\\u7f5a\\u70b9\\u6570  \\u662f  \\u5176\\u4e0b\\u9762\\u548c\\u53f3\\u9762\\u6700\\u5c11\\u9700\\u8981\\u7684\\u70b9\\u6570\\uff0c  \\u4e5f\\u5c31\\u662f\\u7b2c33\\u884c\\n         \\n         '''\\n         row = len(dungeon)\\n         col = len(dungeon[0])\\n         dp = [[0 for  c in range(col)] for r in range(row)]\\n         if dungeon[-1][-1] <= 0:\\n             dp[-1][-1] = -dungeon[-1][-1] + 1\\n         else:\\n             dp[-1][-1] = 1\\n         for r in range(row-2,-1,-1):\\n             dp[r][-1] = dp[r+1][-1]  - dungeon[r][-1]\\n             if dp[r][-1] <= 0:\\n                 dp[r][-1] = 1\\n         for c in range(col-2,-1,-1):\\n             dp[-1][c] = dp[-1][c+1]  - dungeon[-1][c]\\n             if dp[-1][c] <= 0:\\n                 dp[-1][c] = 1\\n         for r in range(row-2,-1,-1):\\n             for c in range(col-2,-1,-1):\\n                 dp[r][c] = min(dp[r+1][c],dp[r][c+1]) - dungeon[r][c]\\n                 if dp[r][c] <= 0:\\n                     dp[r][c] = 1\\n         return dp[0][0]\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n \\n         # sol1: DP, choose the safest path\\n         # http://leetcodesolution.blogspot.kr/2015/01/leetcode-dungeon-game.html\\n         # minInitHealth[i][j] = min(minInitHealth[i+1][j], minInitHealth[i][j+1]) - dungeon[i][j]\\n         # set minInitHealth[i][j] to 1 if < 1\\n         M, N = len(dungeon), len(dungeon[0])\\n         dp = [[0] * N for _ in range(M)]\\n         for i in reversed(list(range(M))):\\n             for j in reversed(list(range(N))):\\n                 v = dungeon[i][j]\\n                 # choose right/down, which requires less\\n                 if i == M - 1 and j == N - 1:\\n                     dp[i][j] = 1 - v\\n                 elif i == M - 1:\\n                     dp[i][j] = dp[i][j + 1] - v\\n                 elif j == N - 1:\\n                     dp[i][j] = dp[i + 1][j] - v\\n                 else:\\n                     dp[i][j] = min(dp[i + 1][j], dp[i][j + 1]) - v\\n                 # always ensure >= 1\\n                 dp[i][j] = max(1, dp[i][j])\\n         return dp[0][0]\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m=len( dungeon)  \\n         n=len( dungeon[0])\\n         \\n         dp=[[float('inf') for _ in range(n+1)] for _ in range(m+1)]\\n         dp[m][n-1]=1\\n         dp[m-1][n]=1\\n         \\n         for i in range(m-1,-1,-1):\\n             for j in range(n-1,-1,-1):\\n                 \\n                 need =min(dp[i+1][j],dp[i][j+1])-dungeon[i][j]\\n                 if need <=0:\\n                     need=1\\n                 dp[i][j]=need \\n         return dp[0][0]        \", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         n = len(dungeon[0])\\n         need = [2**31] * (n-1) + [1]\\n         for row in dungeon[::-1]:\\n             for j in range(n)[::-1]:\\n                 need[j] = max(min(need[j:j+2]) - row[j], 1)\\n         return need[0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         if not dungeon or not dungeon[0]:\\n             return 1\\n         dp = [[0 for i in range(len(dungeon[0]))] for i in range(len(dungeon))]\\n         dp[-1][-1] = 1 - dungeon[-1][-1]\\n         row = len(dungeon)\\n         column = len(dungeon[0])\\n         for i in range(1,row):\\n             dp[row-1-i][-1] = max(1 - dungeon[row-1-i][-1], dp[row-i][-1] - dungeon[row-1-i][-1])\\n         for i in range(1,column):\\n             dp[-1][column-1-i] = max(1 - dungeon[-1][column-1-i], dp[-1][column-i] - dungeon[-1][column-1-i])\\n         for i in range(1,row):\\n             for j in range(1,column):\\n                 dp[row-1-i][column-1-j] = max(1 - dungeon[row-1-i][column-1-j], min(dp[row-i][column-1-j],dp[row-1-i][column-j]) - dungeon[row-1-i][column-1-j])\\n         if dp[0][0] <= 0:\\n             return 1\\n         return dp[0][0]\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         num_rows = len(dungeon)\\n         num_cols = len(dungeon[0])\\n         INF = float('inf')\\n         for ri in range(num_rows - 1, -1, -1):\\n             for ci in range(num_cols - 1, -1, -1):\\n                 if ri == num_rows - 1 and ci == num_cols - 1:\\n                     need = -dungeon[ri][ci] + 1\\n                 else:\\n                     down = INF if ri == num_rows - 1 else dungeon[ri+1][ci]\\n                     right = INF if ci == num_cols - 1 else dungeon[ri][ci + 1]\\n                     need = min(down, right) - dungeon[ri][ci]\\n                 dungeon[ri][ci] = max(need, 1)\\n         return dungeon[0][0]\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         best = [[0] * len(dungeon[0]) for x in range(0, len(dungeon))]\\n         for row in range(-1, -len(dungeon) - 1, -1):\\n             for col in range(-1, -len(dungeon[0]) - 1, -1):\\n                 needed = -dungeon[row][col]\\n                 nextSteps = []\\n                 if row < -1:\\n                     nextSteps.append(best[row + 1][col])\\n                 if col < -1:\\n                     nextSteps.append(best[row][col + 1])\\n                 if len(nextSteps) > 0:\\n                     needed += min(nextSteps)\\n                 needed = max(0, needed)\\n                 best[row][col] = needed\\n                 \\n         return best[0][0] + 1\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m,n =len(dungeon), len(dungeon[0])\\n         record=[[None]*n for i in range(m)]\\n         record[-1][-1]=max(1-dungeon[-1][-1], 1)\\n         for i in range(n-1)[::-1]:\\n             record[m-1][i]=max(1, record[m-1][i+1]-dungeon[m-1][i])\\n         for j in range(m-1)[::-1]:\\n             record[j][n-1]=max(1, record[j+1][n-1]-dungeon[j][n-1])\\n         \\n         for i in range(m-1)[::-1]:\\n             for j in range(n-1)[::-1]:\\n                 mina=max(record[i+1][j]-dungeon[i][j], 1)\\n                 minb=max(record[i][j+1]-dungeon[i][j], 1)\\n                 record[i][j]=min(mina,minb)\\n         return record[0][0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not dungeon or not dungeon[0]:\\n             return 1\\n         \\n         m = len(dungeon)\\n         n = len(dungeon[0])\\n         \\n         dp = [None] * n\\n         \\n         for i in range(m - 1, -1, -1):\\n             for j in range(n - 1, -1, -1):\\n                 tmp = []\\n                 if i + 1 < m:\\n                     tmp.append(dp[j])\\n                 \\n                 if j + 1 < n:\\n                     tmp.append(dp[j + 1])\\n                 \\n                 minLeft = 1\\n                 if tmp:\\n                     minLeft = min(tmp)\\n                 \\n                 dp[j] = max(minLeft - dungeon[i][j], 1)\\n         \\n         return dp[0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         n = len(dungeon)\\n         m = len(dungeon[0])\\n \\n         for i in range(n-1, -1, -1):\\n             for j in range(m-1, -1, -1):\\n                 if i+1 < n and j+1 < m:\\n                     temp = max(dungeon[i+1][j], dungeon[i][j+1])\\n                 elif i+1 < n:\\n                     temp = dungeon[i+1][j]\\n                 elif j+1 < m:\\n                     temp = dungeon[i][j+1]\\n                 else:\\n                     temp = 0\\n                 print(temp)\\n                 dungeon[i][j] = dungeon[i][j] + temp\\n                 dungeon[i][j] = min(dungeon[i][j], 0)\\n         return(max(1, 1-dungeon[0][0]))\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(dungeon)\\n         n = len(dungeon[0])\\n         dp = [0] * n\\n         dp[n-1] = max(1, 1 - dungeon[m-1][n-1])\\n         for i in range(m-1, -1, -1):\\n             for j in range(n-1, -1, -1):\\n                 if i == m-1 and j == n-1:\\n                     continue\\n                 if i == m-1:\\n                     dp[j] = max(1, dp[j+1] - dungeon[i][j])\\n                 elif j == n-1:\\n                     dp[j] = max(1, dp[j] - dungeon[i][j])\\n                 else:\\n                     dp[j] = max(1, min(dp[j], dp[j+1]) - dungeon[i][j])\\n         return dp[0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not any(dungeon):\\n             return 1\\n         m = len(dungeon)\\n         n = len(dungeon[0])\\n         memo = {(m - 1, n - 1): 1}\\n         def helper(i, j):\\n             if (i, j) not in memo:\\n                 if i == m - 1:\\n                     ans = max(1, helper(i, j+1) - dungeon[i][j+1])\\n                 elif j == n - 1:\\n                     ans = max(1, helper(i+1, j) - dungeon[i+1][j])\\n                 else:\\n                     ans = min(max(1, helper(i, j+1) - dungeon[i][j+1]),\\n                              max(1, helper(i+1, j) - dungeon[i+1][j]))\\n                 memo[(i, j)] = ans\\n             return memo[(i, j)]\\n         return max(1, helper(0, 0) - dungeon[0][0])\\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    -2,
                    -3,
                    3
                ],
                [
                    -5,
                    -10,
                    1
                ],
                [
                    10,
                    30,
                    -5
                ]
            ]
        ],
        "output": 7,
        "halu_type": "Identification Hallucination",
        "fn_name": "calculateMinimumHP",
        "starter_code": "\nclass Solution:\n    def calculateMinimumHP(self, dungeon: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/dungeon-game/"
    },
    {
        "id": 1312,
        "task_id": 2633,
        "test_case_id": 2,
        "question": "table.dungeon, .dungeon th, .dungeon td {\n  border:3px solid black;\n}\n\n .dungeon th, .dungeon td {\n    text-align: center;\n    height: 70px;\n    width: 70px;\n}\n\nThe demons had captured the princess (P) and imprisoned her in the bottom-right corner of a dungeon. The dungeon consists of M x N rooms laid out in a 2D grid. Our valiant knight (K) was initially positioned in the top-left room and must fight his way through the dungeon to rescue the princess.\n\nThe knight has an initial health point represented by a positive integer. If at any point his health point drops to 0 or below, he dies immediately.\n\nSome of the rooms are guarded by demons, so the knight loses health (negative integers) upon entering these rooms; other rooms are either empty (0's) or contain magic orbs that increase the knight's health (positive integers).\n\nIn order to reach the princess as quickly as possible, the knight decides to move only rightward or downward in each step.\n\n \n\nWrite a function to determine the knight's minimum initial health so that he is able to rescue the princess.\n\nFor example, given the dungeon below, the initial health of the knight must be at least 7 if he follows the optimal path RIGHT-> RIGHT -> DOWN -> DOWN.\n\n\n       \n               \n                       -2 (K)\n                       -3\n                       3\n               \n               \n                       -5\n                       -10\n                       1\n               \n               \n                       10\n                       30\n                       -5 (P)\n               \n       \n\n\n \n\nNote:\n\n\n       The knight's health has no upper bound.\n       Any room can contain threats or power-ups, even the first room the knight enters and the bottom-right room where the princess is imprisoned.",
        "solutions": "[\"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         '''\\n         \\u9006\\u56de\\uff0c\\u5f53\\u662f\\u5c0f\\u4e8e\\u7b49\\u4e8e0\\u65f6\\uff0c\\u662f\\u76f8\\u53cd\\u6570+1.   dp\\u4e3a\\u5230\\u5f53\\u524di,j\\u53ea\\u8981\\u9700\\u8981\\u7684\\u70b9\\u6570\\uff0c \\u6700\\u5c0f\\u4e3a1\\uff0c\\u4fdd\\u8bc1 \\u6d3b\\u7740\\n         \\n         r,  c   r   ,c+1\\n         r+1,c   r+1,c+1\\n         \\n         r,c\\u5904\\u81f3\\u5c11\\u7684\\u70b9\\u6570 - r,c\\u5904\\u60e9\\u7f5a\\u70b9\\u6570  \\u662f  \\u5176\\u4e0b\\u9762\\u548c\\u53f3\\u9762\\u6700\\u5c11\\u9700\\u8981\\u7684\\u70b9\\u6570\\uff0c  \\u4e5f\\u5c31\\u662f\\u7b2c33\\u884c\\n         \\n         '''\\n         row = len(dungeon)\\n         col = len(dungeon[0])\\n         dp = [[0 for  c in range(col)] for r in range(row)]\\n         if dungeon[-1][-1] <= 0:\\n             dp[-1][-1] = -dungeon[-1][-1] + 1\\n         else:\\n             dp[-1][-1] = 1\\n         for r in range(row-2,-1,-1):\\n             dp[r][-1] = dp[r+1][-1]  - dungeon[r][-1]\\n             if dp[r][-1] <= 0:\\n                 dp[r][-1] = 1\\n         for c in range(col-2,-1,-1):\\n             dp[-1][c] = dp[-1][c+1]  - dungeon[-1][c]\\n             if dp[-1][c] <= 0:\\n                 dp[-1][c] = 1\\n         for r in range(row-2,-1,-1):\\n             for c in range(col-2,-1,-1):\\n                 dp[r][c] = min(dp[r+1][c],dp[r][c+1]) - dungeon[r][c]\\n                 if dp[r][c] <= 0:\\n                     dp[r][c] = 1\\n         return dp[0][0]\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n \\n         # sol1: DP, choose the safest path\\n         # http://leetcodesolution.blogspot.kr/2015/01/leetcode-dungeon-game.html\\n         # minInitHealth[i][j] = min(minInitHealth[i+1][j], minInitHealth[i][j+1]) - dungeon[i][j]\\n         # set minInitHealth[i][j] to 1 if < 1\\n         M, N = len(dungeon), len(dungeon[0])\\n         dp = [[0] * N for _ in range(M)]\\n         for i in reversed(list(range(M))):\\n             for j in reversed(list(range(N))):\\n                 v = dungeon[i][j]\\n                 # choose right/down, which requires less\\n                 if i == M - 1 and j == N - 1:\\n                     dp[i][j] = 1 - v\\n                 elif i == M - 1:\\n                     dp[i][j] = dp[i][j + 1] - v\\n                 elif j == N - 1:\\n                     dp[i][j] = dp[i + 1][j] - v\\n                 else:\\n                     dp[i][j] = min(dp[i + 1][j], dp[i][j + 1]) - v\\n                 # always ensure >= 1\\n                 dp[i][j] = max(1, dp[i][j])\\n         return dp[0][0]\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m=len( dungeon)  \\n         n=len( dungeon[0])\\n         \\n         dp=[[float('inf') for _ in range(n+1)] for _ in range(m+1)]\\n         dp[m][n-1]=1\\n         dp[m-1][n]=1\\n         \\n         for i in range(m-1,-1,-1):\\n             for j in range(n-1,-1,-1):\\n                 \\n                 need =min(dp[i+1][j],dp[i][j+1])-dungeon[i][j]\\n                 if need <=0:\\n                     need=1\\n                 dp[i][j]=need \\n         return dp[0][0]        \", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         n = len(dungeon[0])\\n         need = [2**31] * (n-1) + [1]\\n         for row in dungeon[::-1]:\\n             for j in range(n)[::-1]:\\n                 need[j] = max(min(need[j:j+2]) - row[j], 1)\\n         return need[0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         if not dungeon or not dungeon[0]:\\n             return 1\\n         dp = [[0 for i in range(len(dungeon[0]))] for i in range(len(dungeon))]\\n         dp[-1][-1] = 1 - dungeon[-1][-1]\\n         row = len(dungeon)\\n         column = len(dungeon[0])\\n         for i in range(1,row):\\n             dp[row-1-i][-1] = max(1 - dungeon[row-1-i][-1], dp[row-i][-1] - dungeon[row-1-i][-1])\\n         for i in range(1,column):\\n             dp[-1][column-1-i] = max(1 - dungeon[-1][column-1-i], dp[-1][column-i] - dungeon[-1][column-1-i])\\n         for i in range(1,row):\\n             for j in range(1,column):\\n                 dp[row-1-i][column-1-j] = max(1 - dungeon[row-1-i][column-1-j], min(dp[row-i][column-1-j],dp[row-1-i][column-j]) - dungeon[row-1-i][column-1-j])\\n         if dp[0][0] <= 0:\\n             return 1\\n         return dp[0][0]\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         num_rows = len(dungeon)\\n         num_cols = len(dungeon[0])\\n         INF = float('inf')\\n         for ri in range(num_rows - 1, -1, -1):\\n             for ci in range(num_cols - 1, -1, -1):\\n                 if ri == num_rows - 1 and ci == num_cols - 1:\\n                     need = -dungeon[ri][ci] + 1\\n                 else:\\n                     down = INF if ri == num_rows - 1 else dungeon[ri+1][ci]\\n                     right = INF if ci == num_cols - 1 else dungeon[ri][ci + 1]\\n                     need = min(down, right) - dungeon[ri][ci]\\n                 dungeon[ri][ci] = max(need, 1)\\n         return dungeon[0][0]\\n\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         best = [[0] * len(dungeon[0]) for x in range(0, len(dungeon))]\\n         for row in range(-1, -len(dungeon) - 1, -1):\\n             for col in range(-1, -len(dungeon[0]) - 1, -1):\\n                 needed = -dungeon[row][col]\\n                 nextSteps = []\\n                 if row < -1:\\n                     nextSteps.append(best[row + 1][col])\\n                 if col < -1:\\n                     nextSteps.append(best[row][col + 1])\\n                 if len(nextSteps) > 0:\\n                     needed += min(nextSteps)\\n                 needed = max(0, needed)\\n                 best[row][col] = needed\\n                 \\n         return best[0][0] + 1\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m,n =len(dungeon), len(dungeon[0])\\n         record=[[None]*n for i in range(m)]\\n         record[-1][-1]=max(1-dungeon[-1][-1], 1)\\n         for i in range(n-1)[::-1]:\\n             record[m-1][i]=max(1, record[m-1][i+1]-dungeon[m-1][i])\\n         for j in range(m-1)[::-1]:\\n             record[j][n-1]=max(1, record[j+1][n-1]-dungeon[j][n-1])\\n         \\n         for i in range(m-1)[::-1]:\\n             for j in range(n-1)[::-1]:\\n                 mina=max(record[i+1][j]-dungeon[i][j], 1)\\n                 minb=max(record[i][j+1]-dungeon[i][j], 1)\\n                 record[i][j]=min(mina,minb)\\n         return record[0][0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not dungeon or not dungeon[0]:\\n             return 1\\n         \\n         m = len(dungeon)\\n         n = len(dungeon[0])\\n         \\n         dp = [None] * n\\n         \\n         for i in range(m - 1, -1, -1):\\n             for j in range(n - 1, -1, -1):\\n                 tmp = []\\n                 if i + 1 < m:\\n                     tmp.append(dp[j])\\n                 \\n                 if j + 1 < n:\\n                     tmp.append(dp[j + 1])\\n                 \\n                 minLeft = 1\\n                 if tmp:\\n                     minLeft = min(tmp)\\n                 \\n                 dp[j] = max(minLeft - dungeon[i][j], 1)\\n         \\n         return dp[0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         n = len(dungeon)\\n         m = len(dungeon[0])\\n \\n         for i in range(n-1, -1, -1):\\n             for j in range(m-1, -1, -1):\\n                 if i+1 < n and j+1 < m:\\n                     temp = max(dungeon[i+1][j], dungeon[i][j+1])\\n                 elif i+1 < n:\\n                     temp = dungeon[i+1][j]\\n                 elif j+1 < m:\\n                     temp = dungeon[i][j+1]\\n                 else:\\n                     temp = 0\\n                 print(temp)\\n                 dungeon[i][j] = dungeon[i][j] + temp\\n                 dungeon[i][j] = min(dungeon[i][j], 0)\\n         return(max(1, 1-dungeon[0][0]))\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         m = len(dungeon)\\n         n = len(dungeon[0])\\n         dp = [0] * n\\n         dp[n-1] = max(1, 1 - dungeon[m-1][n-1])\\n         for i in range(m-1, -1, -1):\\n             for j in range(n-1, -1, -1):\\n                 if i == m-1 and j == n-1:\\n                     continue\\n                 if i == m-1:\\n                     dp[j] = max(1, dp[j+1] - dungeon[i][j])\\n                 elif j == n-1:\\n                     dp[j] = max(1, dp[j] - dungeon[i][j])\\n                 else:\\n                     dp[j] = max(1, min(dp[j], dp[j+1]) - dungeon[i][j])\\n         return dp[0]\", \"class Solution:\\n     def calculateMinimumHP(self, dungeon):\\n         \\\"\\\"\\\"\\n         :type dungeon: List[List[int]]\\n         :rtype: int\\n         \\\"\\\"\\\"\\n         if not any(dungeon):\\n             return 1\\n         m = len(dungeon)\\n         n = len(dungeon[0])\\n         memo = {(m - 1, n - 1): 1}\\n         def helper(i, j):\\n             if (i, j) not in memo:\\n                 if i == m - 1:\\n                     ans = max(1, helper(i, j+1) - dungeon[i][j+1])\\n                 elif j == n - 1:\\n                     ans = max(1, helper(i+1, j) - dungeon[i+1][j])\\n                 else:\\n                     ans = min(max(1, helper(i, j+1) - dungeon[i][j+1]),\\n                              max(1, helper(i+1, j) - dungeon[i+1][j]))\\n                 memo[(i, j)] = ans\\n             return memo[(i, j)]\\n         return max(1, helper(0, 0) - dungeon[0][0])\\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    0
                ]
            ]
        ],
        "output": 1,
        "halu_type": "Identification Hallucination",
        "fn_name": "calculateMinimumHP",
        "starter_code": "\nclass Solution:\n    def calculateMinimumHP(self, dungeon: List[List[int]]) -> int:\n        ",
        "url": "https://leetcode.com/problems/dungeon-game/"
    },
    {
        "id": 1313,
        "task_id": 2636,
        "test_case_id": 1,
        "question": "A city's skyline is the outer contour of the silhouette formed by all the buildings in that city when viewed from a distance. Now suppose you are given the locations and height of all the buildings as shown on a cityscape photo (Figure A), write a program to output the skyline formed by these buildings collectively (Figure B).\n       \n\nThe geometric information of each building is represented by a triplet of integers [Li, Ri, Hi], where Li and Ri are the x coordinates of the left and right edge of the ith building, respectively, and Hi is its height. It is guaranteed that 0 ≤ Li, Ri ≤ INT_MAX, 0 < Hi ≤ INT_MAX, and Ri - Li > 0. You may assume all buildings are perfect rectangles grounded on an absolutely flat surface at height 0.\n\nFor instance, the dimensions of all buildings in Figure A are recorded as: [ [2 9 10], [3 7 15], [5 12 12], [15 20 10], [19 24 8] ] .\n\nThe output is a list of \"key points\" (red dots in Figure B) in the format of [ [x1,y1], [x2, y2], [x3, y3], ... ] that uniquely defines a skyline. A key point is the left endpoint of a horizontal line segment. Note that the last key point, where the rightmost building ends, is merely used to mark the termination of the skyline, and always has zero height. Also, the ground in between any two adjacent buildings should be considered part of the skyline contour.\n\nFor instance, the skyline in Figure B should be represented as:[ [2 10], [3 15], [7 12], [12 0], [15 10], [20 8], [24, 0] ].\n\nNotes:\n\n\n       The number of buildings in any input list is guaranteed to be in the range [0, 10000].\n       The input list is already sorted in ascending order by the left x position Li.\n       The output list must be sorted by the x position.\n       There must be no consecutive horizontal lines of equal height in the output skyline. For instance, [...[2 3], [4 5], [7 5], [11 5], [12 7]...] is not acceptable; the three lines of height 5 should be merged into one in the final output as such: [...[2 3], [4 5], [12 7], ...]",
        "solutions": "[\"class Solution(object):\\n     def getSkyline(self, buildings):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         cp= set([b[0] for b in buildings]+[b[1] for b in buildings])\\n         i, active, res = 0, [], []\\n         for c in sorted(cp):\\n             while i<len(buildings) and buildings[i][0]<=c:\\n                 heapq.heappush(active, (-buildings[i][2], buildings[i][1]))\\n                 i+=1\\n \\n             while active and active[0][1]<=c:\\n                 heapq.heappop(active)\\n \\n             h= len(active) and -active[0][0]\\n             if not res or h!=res[-1][1]:\\n                 res.append([c, h])\\n         return res\\n \\n\", \"class Solution:\\n     def getSkyline(self, buildings):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n #         def get_height_end(idx, height):\\n #             # print(heightends)\\n #             for left, right in heightends[height]:\\n #                 if left <= idx < right:\\n #                     return right\\n         \\n #         coords1 = []\\n #         for left, right, height in buildings:\\n #             coords1.append(left)\\n #             coords1.append(right)\\n #         coords1.sort()\\n         \\n #         coords = []\\n #         for k, g in itertools.groupby(coords1):\\n #             coords.append(k)\\n             \\n #         index = { num: idx for idx, num in enumerate(coords) }\\n #         events = collections.defaultdict(list)\\n #         heightends = collections.defaultdict(list)\\n         \\n #         for left, right, height in buildings:\\n             \\n #             i = index[left]\\n #             while i < index[right]:\\n #                 # print(i, events, left, right)\\n #                 if events[i] != [] and sorted(events[i])[-1] >= height:\\n #                     i = get_height_end(i, sorted(events[i])[-1])\\n #                     continue\\n \\n #                 events[i].append(height)\\n #                 i += 1\\n #             heightends[height].append([index[left], index[right]])\\n         \\n #         infpoints = []\\n #         prevheight = 0\\n #         for coord in coords:\\n #             coordheights = sorted(events[index[coord]])\\n #             coordheight = 0\\n #             if coordheights != []:\\n #                 coordheight = coordheights[-1]\\n #             if prevheight != coordheight:\\n #                 infpoints.append([coord, coordheight])\\n #                 prevheight = coordheight\\n         \\n #         return infpoints\\n \\n         edges = []\\n         events = collections.defaultdict(list)\\n         for left, right, height in buildings:\\n             edges.append((left, -height, right))\\n             edges.append((right, 0, None))\\n             # heapq.heappush(events[left].append()\\n             # events[right].append((left, right, height))\\n \\n         pq = [(0, float('inf'))]\\n         edges.sort()\\n         skyline = [(0,0)]\\n         \\n         for edge, height, right in edges:\\n \\n             while edge >= pq[0][1]:\\n                 heapq.heappop(pq)\\n             if height:\\n                 heapq.heappush(pq, (height, right))\\n             if skyline[-1][1] != -pq[0][0]:\\n                 skyline.append([edge, -pq[0][0]])\\n         return skyline[1:]\\n \\n\", \"class Solution:\\n     def getSkyline(self, buildings):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         edges = []\\n         for left, right, height in buildings:\\n             edges.append((left, -height, right))\\n             edges.append((right, 0, None))\\n             \\n         edges.sort()\\n         pq = [(0, float('inf'))]\\n         skyline = [(0, 0)]\\n         print(edges)\\n         for left, negheight, right in edges:\\n             while pq[0][1] <= left:\\n                 heapq.heappop(pq)\\n             if negheight:\\n                 heapq.heappush(pq, (negheight, right))\\n             if skyline[-1][1] != -pq[0][0]:\\n                 skyline.append([left, -pq[0][0]])\\n                 \\n         return skyline[1:]\", \"class Solution:\\n     def getSkyline(self, blds):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not blds:\\n             return []\\n         if len(blds) == 1:\\n             return [[blds[0][0], blds[0][2]], [blds[0][1], 0]]\\n         \\n         mid = len(blds) >> 1\\n         left = self.getSkyline(blds[:mid])\\n         right = self.getSkyline(blds[mid:])\\n         return self.merge(left, right)\\n         #conquor how to merge left and right\\n     \\n     def merge(self, left, right):\\n         res = []\\n         h1 = h2 = 0\\n         while left and right:\\n             if left[0][0] < right[0][0]:\\n                 pos, h1 = left.pop(0)\\n             elif right[0][0] < left[0][0]:\\n                 pos, h2 = right.pop(0)\\n             else:\\n                 pos, h1 = left.pop(0)\\n                 h2 = right.pop(0)[1]\\n             h = max(h1, h2)\\n             if not res or res[-1][1] != h:\\n                 res.append([pos, h])\\n                 \\n \\n         if left:\\n             res += left\\n         if right:\\n             res += right\\n         return res\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    2,
                    9,
                    10
                ],
                [
                    3,
                    7,
                    15
                ],
                [
                    5,
                    12,
                    12
                ],
                [
                    15,
                    20,
                    10
                ],
                [
                    19,
                    24,
                    8
                ]
            ]
        ],
        "output": [
            [
                2,
                10
            ],
            [
                3,
                15
            ],
            [
                7,
                12
            ],
            [
                12,
                0
            ],
            [
                15,
                10
            ],
            [
                20,
                8
            ],
            [
                24,
                0
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "getSkyline",
        "starter_code": "\nclass Solution:\n    def getSkyline(self, buildings: List[List[int]]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/the-skyline-problem/"
    },
    {
        "id": 1314,
        "task_id": 2636,
        "test_case_id": 2,
        "question": "A city's skyline is the outer contour of the silhouette formed by all the buildings in that city when viewed from a distance. Now suppose you are given the locations and height of all the buildings as shown on a cityscape photo (Figure A), write a program to output the skyline formed by these buildings collectively (Figure B).\n       \n\nThe geometric information of each building is represented by a triplet of integers [Li, Ri, Hi], where Li and Ri are the x coordinates of the left and right edge of the ith building, respectively, and Hi is its height. It is guaranteed that 0 ≤ Li, Ri ≤ INT_MAX, 0 < Hi ≤ INT_MAX, and Ri - Li > 0. You may assume all buildings are perfect rectangles grounded on an absolutely flat surface at height 0.\n\nFor instance, the dimensions of all buildings in Figure A are recorded as: [ [2 9 10], [3 7 15], [5 12 12], [15 20 10], [19 24 8] ] .\n\nThe output is a list of \"key points\" (red dots in Figure B) in the format of [ [x1,y1], [x2, y2], [x3, y3], ... ] that uniquely defines a skyline. A key point is the left endpoint of a horizontal line segment. Note that the last key point, where the rightmost building ends, is merely used to mark the termination of the skyline, and always has zero height. Also, the ground in between any two adjacent buildings should be considered part of the skyline contour.\n\nFor instance, the skyline in Figure B should be represented as:[ [2 10], [3 15], [7 12], [12 0], [15 10], [20 8], [24, 0] ].\n\nNotes:\n\n\n       The number of buildings in any input list is guaranteed to be in the range [0, 10000].\n       The input list is already sorted in ascending order by the left x position Li.\n       The output list must be sorted by the x position.\n       There must be no consecutive horizontal lines of equal height in the output skyline. For instance, [...[2 3], [4 5], [7 5], [11 5], [12 7]...] is not acceptable; the three lines of height 5 should be merged into one in the final output as such: [...[2 3], [4 5], [12 7], ...]",
        "solutions": "[\"class Solution(object):\\n     def getSkyline(self, buildings):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         cp= set([b[0] for b in buildings]+[b[1] for b in buildings])\\n         i, active, res = 0, [], []\\n         for c in sorted(cp):\\n             while i<len(buildings) and buildings[i][0]<=c:\\n                 heapq.heappush(active, (-buildings[i][2], buildings[i][1]))\\n                 i+=1\\n \\n             while active and active[0][1]<=c:\\n                 heapq.heappop(active)\\n \\n             h= len(active) and -active[0][0]\\n             if not res or h!=res[-1][1]:\\n                 res.append([c, h])\\n         return res\\n \\n\", \"class Solution:\\n     def getSkyline(self, buildings):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         \\n #         def get_height_end(idx, height):\\n #             # print(heightends)\\n #             for left, right in heightends[height]:\\n #                 if left <= idx < right:\\n #                     return right\\n         \\n #         coords1 = []\\n #         for left, right, height in buildings:\\n #             coords1.append(left)\\n #             coords1.append(right)\\n #         coords1.sort()\\n         \\n #         coords = []\\n #         for k, g in itertools.groupby(coords1):\\n #             coords.append(k)\\n             \\n #         index = { num: idx for idx, num in enumerate(coords) }\\n #         events = collections.defaultdict(list)\\n #         heightends = collections.defaultdict(list)\\n         \\n #         for left, right, height in buildings:\\n             \\n #             i = index[left]\\n #             while i < index[right]:\\n #                 # print(i, events, left, right)\\n #                 if events[i] != [] and sorted(events[i])[-1] >= height:\\n #                     i = get_height_end(i, sorted(events[i])[-1])\\n #                     continue\\n \\n #                 events[i].append(height)\\n #                 i += 1\\n #             heightends[height].append([index[left], index[right]])\\n         \\n #         infpoints = []\\n #         prevheight = 0\\n #         for coord in coords:\\n #             coordheights = sorted(events[index[coord]])\\n #             coordheight = 0\\n #             if coordheights != []:\\n #                 coordheight = coordheights[-1]\\n #             if prevheight != coordheight:\\n #                 infpoints.append([coord, coordheight])\\n #                 prevheight = coordheight\\n         \\n #         return infpoints\\n \\n         edges = []\\n         events = collections.defaultdict(list)\\n         for left, right, height in buildings:\\n             edges.append((left, -height, right))\\n             edges.append((right, 0, None))\\n             # heapq.heappush(events[left].append()\\n             # events[right].append((left, right, height))\\n \\n         pq = [(0, float('inf'))]\\n         edges.sort()\\n         skyline = [(0,0)]\\n         \\n         for edge, height, right in edges:\\n \\n             while edge >= pq[0][1]:\\n                 heapq.heappop(pq)\\n             if height:\\n                 heapq.heappush(pq, (height, right))\\n             if skyline[-1][1] != -pq[0][0]:\\n                 skyline.append([edge, -pq[0][0]])\\n         return skyline[1:]\\n \\n\", \"class Solution:\\n     def getSkyline(self, buildings):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         edges = []\\n         for left, right, height in buildings:\\n             edges.append((left, -height, right))\\n             edges.append((right, 0, None))\\n             \\n         edges.sort()\\n         pq = [(0, float('inf'))]\\n         skyline = [(0, 0)]\\n         print(edges)\\n         for left, negheight, right in edges:\\n             while pq[0][1] <= left:\\n                 heapq.heappop(pq)\\n             if negheight:\\n                 heapq.heappush(pq, (negheight, right))\\n             if skyline[-1][1] != -pq[0][0]:\\n                 skyline.append([left, -pq[0][0]])\\n                 \\n         return skyline[1:]\", \"class Solution:\\n     def getSkyline(self, blds):\\n         \\\"\\\"\\\"\\n         :type buildings: List[List[int]]\\n         :rtype: List[List[int]]\\n         \\\"\\\"\\\"\\n         if not blds:\\n             return []\\n         if len(blds) == 1:\\n             return [[blds[0][0], blds[0][2]], [blds[0][1], 0]]\\n         \\n         mid = len(blds) >> 1\\n         left = self.getSkyline(blds[:mid])\\n         right = self.getSkyline(blds[mid:])\\n         return self.merge(left, right)\\n         #conquor how to merge left and right\\n     \\n     def merge(self, left, right):\\n         res = []\\n         h1 = h2 = 0\\n         while left and right:\\n             if left[0][0] < right[0][0]:\\n                 pos, h1 = left.pop(0)\\n             elif right[0][0] < left[0][0]:\\n                 pos, h2 = right.pop(0)\\n             else:\\n                 pos, h1 = left.pop(0)\\n                 h2 = right.pop(0)[1]\\n             h = max(h1, h2)\\n             if not res or res[-1][1] != h:\\n                 res.append([pos, h])\\n                 \\n \\n         if left:\\n             res += left\\n         if right:\\n             res += right\\n         return res\"]",
        "difficulty": "interview",
        "input": [
            [
                [
                    0,
                    2,
                    3
                ],
                [
                    2,
                    5,
                    3
                ]
            ]
        ],
        "output": [
            [
                0,
                3
            ],
            [
                5,
                0
            ]
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "getSkyline",
        "starter_code": "\nclass Solution:\n    def getSkyline(self, buildings: List[List[int]]) -> List[List[int]]:\n        ",
        "url": "https://leetcode.com/problems/the-skyline-problem/"
    },
    {
        "id": 1315,
        "task_id": 2745,
        "test_case_id": 1,
        "question": "You are given a string, s, and a list of words, words, that are all of the same length. Find all starting indices of substring(s) in s that is a concatenation of each word in words exactly once and without any intervening characters.\n\nExample 1:\n\n\nInput:\ns = \"barfoothefoobarman\",\nwords = [\"foo\",\"bar\"]\nOutput: [0,9]\nExplanation: Substrings starting at index 0 and 9 are \"barfoor\" and \"foobar\" respectively.\nThe output order does not matter, returning [9,0] is fine too.\n\n\nExample 2:\n\n\nInput:\ns = \"wordgoodstudentgoodword\",\nwords = [\"word\",\"student\"]\nOutput: []",
        "solutions": "[\"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or words==[]:\\n             return []\\n         lenstr=len(s)\\n         lenword=len(words[0])\\n         lensubstr=len(words)*lenword\\n         times={}\\n         for word in words:\\n             if word in times:\\n                 times[word]+=1\\n             else:\\n                 times[word]=1\\n         ans=[]\\n         for i in range(min(lenword,lenstr-lensubstr+1)):\\n             self.findAnswer(i,lenstr,lenword,lensubstr,s,times,ans)\\n         return ans\\n     def findAnswer(self,strstart,lenstr,lenword,lensubstr,s,times,ans):\\n         wordstart=strstart\\n         curr={}\\n         while strstart+lensubstr<=lenstr:\\n             word=s[wordstart:wordstart+lenword]\\n             wordstart+=lenword\\n             if word not in times:\\n                 strstart=wordstart\\n                 curr.clear()\\n             else:\\n                 if word in curr:\\n                     curr[word]+=1\\n                 else:\\n                     curr[word]=1\\n                 while curr[word]>times[word]:\\n                     curr[s[strstart:strstart+lenword]]-=1\\n                     strstart+=lenword\\n                 if wordstart-strstart==lensubstr:\\n                     ans.append(strstart)\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words: return []\\n         lens, lenw, numw = len(s), len(words[0]), len(words)\\n         res = []\\n         end = lens // lenw * lenw\\n         \\n         for i in range(lenw):\\n             start = sidx = i\\n             record = words[:]\\n             while sidx < end:              \\n                 tmp_sidx = sidx + lenw\\n                 tmp_word = s[sidx: tmp_sidx]\\n                 \\n                 if tmp_word in words:\\n                     if tmp_word in record:\\n                         record.remove(tmp_word)\\n                     elif s[start: start + lenw] != tmp_word:\\n                         sidx = start = start + lenw\\n                         record = words[:]\\n                         continue\\n                     else:\\n                         start = start + lenw\\n                 elif lens - tmp_sidx < lenw * numw:\\n                     break\\n                 else:\\n                     record = words[:]\\n                     start = tmp_sidx\\n                     \\n                 if not record: res.append(start)\\n                 sidx = tmp_sidx  \\n                 \\n         return res\\n\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         lenWord = len(words[0])\\n         numOfWords = len(words)\\n         lenSubstring = lenWord * numOfWords\\n         dic = {}\\n         \\n         for word in words:\\n             if word in dic:\\n                 dic[word] += 1\\n             else:\\n                 dic[word] = 1\\n         \\n         for i in range(min(lenWord, len(s) - lenSubstring + 1)):\\n             curr = {}\\n             strStart = i\\n             wordStart = strStart\\n             while strStart + lenSubstring <= len(s):\\n                 word = s[wordStart: wordStart + lenWord]\\n                 wordStart += lenWord\\n                 if word not in dic:\\n                     strStart = wordStart\\n                     curr.clear()\\n                 else:\\n                     if word in curr:\\n                         curr[word] += 1\\n                     else:\\n                         curr[word] = 1\\n                     while curr[word] > dic[word]:\\n                         curr[s[strStart: strStart + lenWord]] -= 1\\n                         strStart += lenWord\\n                     if wordStart - strStart == lenSubstring:\\n                         result.append(strStart)\\n         return result\\n\", \"from collections import Counter\\n from copy import deepcopy\\n class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words:\\n             return []\\n         \\n         wordCounter = Counter(words)\\n         longest = len(words[0])\\n         lenSubStr = longest * len(words)\\n         n = len(s)\\n \\n         idx = []\\n         for i in range(0, longest):\\n             cnt = {}\\n             j = i\\n             start = i\\n             while start + lenSubStr <= n:\\n                 word = s[j:j+longest]\\n                 j += longest\\n                 if word not in wordCounter:\\n                     start = j\\n                     cnt.clear()\\n                 else:\\n                     if word in cnt:\\n                         cnt[word] += 1\\n                     else:\\n                         cnt[word] = 1\\n                         \\n                     while cnt[word] > wordCounter[word]:\\n                         cnt[s[start: start + longest]] -= 1\\n                         start += longest\\n                         \\n                     if j - start == lenSubStr:\\n                         idx.append(start)\\n \\n         return idx\\n                     \\n                 \\n                 \\n         \\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         lenWord = len(words[0])\\n         numOfWords = len(words)\\n         lenSubstring = lenWord * numOfWords\\n         dic = {}\\n         for word in words:\\n             if word in dic:\\n                 dic[word] += 1\\n             else:\\n                 dic[word] = 1\\n         for i in range(min(lenWord, len(s)-lenSubstring+1)):\\n             curr = {}\\n             strStart = i\\n             wordStart = strStart\\n             while strStart + lenSubstring <= len(s):\\n                 word = s[wordStart: wordStart+lenWord]\\n                 wordStart += lenWord\\n                 if word not in dic:\\n                     strStart = wordStart\\n                     curr.clear()\\n                 else:\\n                     if word in curr:\\n                         curr[word] += 1\\n                     else:\\n                         curr[word] = 1\\n                     while curr[word] > dic[word]:\\n                         curr[s[strStart: strStart+lenWord]] -= 1\\n                         strStart += lenWord\\n                     if wordStart - strStart == lenSubstring:\\n                         result.append(strStart)\\n         return result\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         m, n, k, result, wordbase = len(s), len(words), len(words[0]), [], {}\\n         for value in words:\\n             if value in wordbase:\\n                 wordbase[value] += 1\\n             else:\\n                 wordbase[value] = 1\\n         for i in range(min(k, m - k * n + 1)):\\n             base, starts, startw = {}, i, i\\n             while starts + k * n <= m:\\n                 temp = s[startw:startw + k]\\n                 startw += k\\n                 if temp not in wordbase:\\n                     starts = startw\\n                     base.clear()\\n                 else:\\n                     if temp in base:\\n                         base[temp] += 1\\n                     else:\\n                         base[temp] = 1\\n                     while base[temp] > wordbase[temp]:\\n                         base[s[starts:starts + k]] -= 1\\n                         starts += k\\n                     if startw - starts == k * n:\\n                         result.append(starts)\\n         return result\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         lword = len(words[0])\\n         n = len(words)\\n         lwords = n * lword\\n         if not s or not lwords:\\n             return\\n         \\n         res = []\\n         dic = {} \\n         for key in words:\\n             dic[key] = [0, 0]\\n         for key in words:\\n             dic[key][1] += 1\\n         for k in range(lword):\\n             i = k\\n             while i <= len(s)-lwords:\\n                 if not s[i:i+lword] in list(dic.keys()):\\n                     i += lword\\n                     continue\\n                 j = 0\\n                 start = i\\n                 while s[i:i+lword] in list(dic.keys()) and i<len(s):\\n                     key = s[i:i+lword]\\n                     dic[key][0] += 1\\n                     while dic[key][0] > dic[key][1]:\\n                         dic[s[start:start+lword]][0] -= 1\\n                         j -= 1\\n                         start += lword\\n                     j += 1\\n                     i += lword\\n                     if j==n:\\n                         res.append(start)\\n                         dic[s[start:start+lword]][0] -= 1\\n                         j -= 1\\n                         start += lword\\n                 for key in list(dic.keys()):\\n                     dic[key][0] = 0\\n         return res\\n\", \"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or not words or not words[0]:\\n             return []\\n         n = len(s)\\n         k = len(words[0])\\n         t = len(words) * k\\n         req = {}\\n         for w in words:\\n             req[w] = req[w] + 1 if w in req else 1\\n         ans = []\\n         for i in range(min(k, n - t + 1)):\\n             self._findSubstring(i, i, n, k, t, s, req, ans)\\n         return ans\\n         \\n     def _findSubstring(self, l, r, n, k, t, s, req, ans):\\n         curr = {}\\n         while r + k <= n:\\n             w = s[r:r + k]\\n             r += k\\n             if w not in req:\\n                 l = r\\n                 curr.clear()\\n             else:\\n                 curr[w] = curr[w] + 1 if w in curr else 1\\n                 while curr[w] > req[w]:\\n                     curr[s[l:l + k]] -= 1\\n                     l += k\\n                 if r - l == t:\\n                     ans.append(l)\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         L = len(s)\\n         length = len(words[0])\\n         Length = length * len(words)\\n         result = []\\n         word_dict = {}\\n         for word in words:\\n             word_dict[word] = word_dict[word]+1 if word in word_dict else 1\\n         for i in range(length):\\n             l = i\\n             r = i\\n             curr_dict={}\\n             while r+length<=L:\\n                 word  = s[r:r+length]\\n                 r = r+length\\n                 if word in word_dict:\\n                     curr_dict[word] = curr_dict[word]+1 if word in curr_dict else 1\\n                     while curr_dict[word] > word_dict[word]:\\n                         curr_dict[s[l:l+length]] -=1\\n                         l += length\\n                     if r-l==Length:\\n                         result.append(l)\\n                 else:\\n                     curr_dict.clear()\\n                     l = r\\n             \\n         return result\\n \\n\", \"from collections import Counter\\n from copy import deepcopy\\n class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words:\\n             return []\\n         \\n         wordCounter = Counter(words)\\n         longest = len(words[0])\\n         lenSubStr = longest * len(words)\\n         n = len(s)\\n \\n         idx = []\\n         for i in range(0, min(longest, n - lenSubStr + 1)):\\n             if i in idx:\\n                 continue\\n             cnt = {}\\n             j = i\\n             start = i\\n             while start + lenSubStr <= n:\\n                 word = s[j:j+longest]\\n                 j += longest\\n                 if word not in wordCounter:\\n                     start = j\\n                     cnt.clear()\\n                 else:\\n                     if word in cnt:\\n                         cnt[word] += 1\\n                     else:\\n                         cnt[word] = 1\\n                         \\n                     while cnt[word] > wordCounter[word]:\\n                         cnt[s[start: start + longest]] -= 1\\n                         start += longest\\n                         \\n                     if j - start == lenSubStr:\\n                         idx.append(start)\\n \\n         return list(idx)\\n                     \\n                 \\n                 \\n         \\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \", \"class Solution:\\n     def findSubstring(self, s, words):\\n         if not words:\\n             return []\\n         word_len = len(words[0])\\n         word_set = {}\\n         for word in words:\\n             word_set[word] = word_set.get(word, 0) + 1\\n         scopes = [([], {}) for _ in range(word_len)]\\n         results = []\\n         for i in range(len(s)):\\n             word = s[i:i + word_len]\\n             if word in word_set:\\n                 matched_queue, matched_counts = scopes[i % word_len]\\n                 while matched_counts.get(word, 0) >= word_set[word]:\\n                     matched_counts[matched_queue.pop(0)] -= 1\\n                 matched_queue.append(word)\\n                 matched_counts[word] = matched_counts.get(word, 0) + 1\\n                 if len(matched_queue) == len(words):\\n                     results.append(i - word_len * len(words) + word_len)\\n                     matched_counts[matched_queue.pop(0)] -= 1\\n             else:\\n                 scopes[i % word_len] = ([], {})\\n         return results\", \"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or not words: return []\\n         result, sLen, numW, wLen = [], len(s), len(words), len(words[0])\\n         if sLen < numW*wLen: return []\\n         dic = collections.defaultdict(int)\\n         for i in words:\\n             dic[i] += 1\\n         for i in range(wLen):\\n             left, count = i, 0\\n             tmp = collections.defaultdict(int)\\n             for j in range(i, sLen-wLen+1, wLen):\\n                 s1 = s[j:j+wLen]\\n                 if s1 in dic:\\n                     tmp[s1] += 1\\n                     if tmp[s1] <= dic[s1]: \\n                         count += 1\\n                     else:\\n                         while tmp[s1] > dic[s1]:\\n                             s2 = s[left:left+wLen]\\n                             tmp[s2] -= 1\\n                             if tmp[s2] < dic[s2]:\\n                                 count -= 1\\n                             left += wLen\\n                     if count == numW:\\n                         result.append(left)\\n                         tmp[s[left:left+wLen]] -= 1\\n                         count -= 1\\n                         left += wLen\\n                 else:\\n                     tmp, count, left = collections.defaultdict(int), 0, j+wLen\\n         return result\\n\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         hash = {}\\n         res = []\\n         wsize = len(words[0])\\n         \\n         for str in words:\\n             if str in hash:\\n                 hash[str] += 1\\n             else:\\n                 hash[str] = 1\\n         for start in range(0, len(words[0])):\\n             slidingWindow = {}\\n             wCount = 0\\n             for i in range(start, len(s), wsize):\\n                 word = s[i : i + wsize]\\n                 if word in hash:\\n                     if word in slidingWindow:\\n                         slidingWindow[word] += 1\\n                     else:\\n                         slidingWindow[word] = 1\\n                     wCount += 1\\n                     while hash[word] < slidingWindow[word]:\\n                         pos = i - wsize * (wCount - 1)\\n                         removeWord = s[pos : pos + wsize]\\n                         #print i, removeWord\\n                         slidingWindow[removeWord] -= 1\\n                         wCount -= 1\\n                 else:\\n                     slidingWindow.clear()\\n                     wCount = 0\\n                 if wCount == len(words):\\n                     res.append(i - wsize * (wCount - 1))\\n                     \\n         return res\"]",
        "difficulty": "interview",
        "input": [
            "\"barfoothefoobarman",
            [
                "\"foo",
                "\"bar\""
            ]
        ],
        "output": [
            0,
            9
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "findSubstring",
        "starter_code": "\nclass Solution:\n    def findSubstring(self, s: str, words: List[str]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/substring-with-concatenation-of-all-words/"
    },
    {
        "id": 1316,
        "task_id": 2745,
        "test_case_id": 2,
        "question": "You are given a string, s, and a list of words, words, that are all of the same length. Find all starting indices of substring(s) in s that is a concatenation of each word in words exactly once and without any intervening characters.\n\nExample 1:\n\n\nInput:\ns = \"barfoothefoobarman\",\nwords = [\"foo\",\"bar\"]\nOutput: [0,9]\nExplanation: Substrings starting at index 0 and 9 are \"barfoor\" and \"foobar\" respectively.\nThe output order does not matter, returning [9,0] is fine too.\n\n\nExample 2:\n\n\nInput:\ns = \"wordgoodstudentgoodword\",\nwords = [\"word\",\"student\"]\nOutput: []",
        "solutions": "[\"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or words==[]:\\n             return []\\n         lenstr=len(s)\\n         lenword=len(words[0])\\n         lensubstr=len(words)*lenword\\n         times={}\\n         for word in words:\\n             if word in times:\\n                 times[word]+=1\\n             else:\\n                 times[word]=1\\n         ans=[]\\n         for i in range(min(lenword,lenstr-lensubstr+1)):\\n             self.findAnswer(i,lenstr,lenword,lensubstr,s,times,ans)\\n         return ans\\n     def findAnswer(self,strstart,lenstr,lenword,lensubstr,s,times,ans):\\n         wordstart=strstart\\n         curr={}\\n         while strstart+lensubstr<=lenstr:\\n             word=s[wordstart:wordstart+lenword]\\n             wordstart+=lenword\\n             if word not in times:\\n                 strstart=wordstart\\n                 curr.clear()\\n             else:\\n                 if word in curr:\\n                     curr[word]+=1\\n                 else:\\n                     curr[word]=1\\n                 while curr[word]>times[word]:\\n                     curr[s[strstart:strstart+lenword]]-=1\\n                     strstart+=lenword\\n                 if wordstart-strstart==lensubstr:\\n                     ans.append(strstart)\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words: return []\\n         lens, lenw, numw = len(s), len(words[0]), len(words)\\n         res = []\\n         end = lens // lenw * lenw\\n         \\n         for i in range(lenw):\\n             start = sidx = i\\n             record = words[:]\\n             while sidx < end:              \\n                 tmp_sidx = sidx + lenw\\n                 tmp_word = s[sidx: tmp_sidx]\\n                 \\n                 if tmp_word in words:\\n                     if tmp_word in record:\\n                         record.remove(tmp_word)\\n                     elif s[start: start + lenw] != tmp_word:\\n                         sidx = start = start + lenw\\n                         record = words[:]\\n                         continue\\n                     else:\\n                         start = start + lenw\\n                 elif lens - tmp_sidx < lenw * numw:\\n                     break\\n                 else:\\n                     record = words[:]\\n                     start = tmp_sidx\\n                     \\n                 if not record: res.append(start)\\n                 sidx = tmp_sidx  \\n                 \\n         return res\\n\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         lenWord = len(words[0])\\n         numOfWords = len(words)\\n         lenSubstring = lenWord * numOfWords\\n         dic = {}\\n         \\n         for word in words:\\n             if word in dic:\\n                 dic[word] += 1\\n             else:\\n                 dic[word] = 1\\n         \\n         for i in range(min(lenWord, len(s) - lenSubstring + 1)):\\n             curr = {}\\n             strStart = i\\n             wordStart = strStart\\n             while strStart + lenSubstring <= len(s):\\n                 word = s[wordStart: wordStart + lenWord]\\n                 wordStart += lenWord\\n                 if word not in dic:\\n                     strStart = wordStart\\n                     curr.clear()\\n                 else:\\n                     if word in curr:\\n                         curr[word] += 1\\n                     else:\\n                         curr[word] = 1\\n                     while curr[word] > dic[word]:\\n                         curr[s[strStart: strStart + lenWord]] -= 1\\n                         strStart += lenWord\\n                     if wordStart - strStart == lenSubstring:\\n                         result.append(strStart)\\n         return result\\n\", \"from collections import Counter\\n from copy import deepcopy\\n class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words:\\n             return []\\n         \\n         wordCounter = Counter(words)\\n         longest = len(words[0])\\n         lenSubStr = longest * len(words)\\n         n = len(s)\\n \\n         idx = []\\n         for i in range(0, longest):\\n             cnt = {}\\n             j = i\\n             start = i\\n             while start + lenSubStr <= n:\\n                 word = s[j:j+longest]\\n                 j += longest\\n                 if word not in wordCounter:\\n                     start = j\\n                     cnt.clear()\\n                 else:\\n                     if word in cnt:\\n                         cnt[word] += 1\\n                     else:\\n                         cnt[word] = 1\\n                         \\n                     while cnt[word] > wordCounter[word]:\\n                         cnt[s[start: start + longest]] -= 1\\n                         start += longest\\n                         \\n                     if j - start == lenSubStr:\\n                         idx.append(start)\\n \\n         return idx\\n                     \\n                 \\n                 \\n         \\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         lenWord = len(words[0])\\n         numOfWords = len(words)\\n         lenSubstring = lenWord * numOfWords\\n         dic = {}\\n         for word in words:\\n             if word in dic:\\n                 dic[word] += 1\\n             else:\\n                 dic[word] = 1\\n         for i in range(min(lenWord, len(s)-lenSubstring+1)):\\n             curr = {}\\n             strStart = i\\n             wordStart = strStart\\n             while strStart + lenSubstring <= len(s):\\n                 word = s[wordStart: wordStart+lenWord]\\n                 wordStart += lenWord\\n                 if word not in dic:\\n                     strStart = wordStart\\n                     curr.clear()\\n                 else:\\n                     if word in curr:\\n                         curr[word] += 1\\n                     else:\\n                         curr[word] = 1\\n                     while curr[word] > dic[word]:\\n                         curr[s[strStart: strStart+lenWord]] -= 1\\n                         strStart += lenWord\\n                     if wordStart - strStart == lenSubstring:\\n                         result.append(strStart)\\n         return result\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         m, n, k, result, wordbase = len(s), len(words), len(words[0]), [], {}\\n         for value in words:\\n             if value in wordbase:\\n                 wordbase[value] += 1\\n             else:\\n                 wordbase[value] = 1\\n         for i in range(min(k, m - k * n + 1)):\\n             base, starts, startw = {}, i, i\\n             while starts + k * n <= m:\\n                 temp = s[startw:startw + k]\\n                 startw += k\\n                 if temp not in wordbase:\\n                     starts = startw\\n                     base.clear()\\n                 else:\\n                     if temp in base:\\n                         base[temp] += 1\\n                     else:\\n                         base[temp] = 1\\n                     while base[temp] > wordbase[temp]:\\n                         base[s[starts:starts + k]] -= 1\\n                         starts += k\\n                     if startw - starts == k * n:\\n                         result.append(starts)\\n         return result\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         lword = len(words[0])\\n         n = len(words)\\n         lwords = n * lword\\n         if not s or not lwords:\\n             return\\n         \\n         res = []\\n         dic = {} \\n         for key in words:\\n             dic[key] = [0, 0]\\n         for key in words:\\n             dic[key][1] += 1\\n         for k in range(lword):\\n             i = k\\n             while i <= len(s)-lwords:\\n                 if not s[i:i+lword] in list(dic.keys()):\\n                     i += lword\\n                     continue\\n                 j = 0\\n                 start = i\\n                 while s[i:i+lword] in list(dic.keys()) and i<len(s):\\n                     key = s[i:i+lword]\\n                     dic[key][0] += 1\\n                     while dic[key][0] > dic[key][1]:\\n                         dic[s[start:start+lword]][0] -= 1\\n                         j -= 1\\n                         start += lword\\n                     j += 1\\n                     i += lword\\n                     if j==n:\\n                         res.append(start)\\n                         dic[s[start:start+lword]][0] -= 1\\n                         j -= 1\\n                         start += lword\\n                 for key in list(dic.keys()):\\n                     dic[key][0] = 0\\n         return res\\n\", \"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or not words or not words[0]:\\n             return []\\n         n = len(s)\\n         k = len(words[0])\\n         t = len(words) * k\\n         req = {}\\n         for w in words:\\n             req[w] = req[w] + 1 if w in req else 1\\n         ans = []\\n         for i in range(min(k, n - t + 1)):\\n             self._findSubstring(i, i, n, k, t, s, req, ans)\\n         return ans\\n         \\n     def _findSubstring(self, l, r, n, k, t, s, req, ans):\\n         curr = {}\\n         while r + k <= n:\\n             w = s[r:r + k]\\n             r += k\\n             if w not in req:\\n                 l = r\\n                 curr.clear()\\n             else:\\n                 curr[w] = curr[w] + 1 if w in curr else 1\\n                 while curr[w] > req[w]:\\n                     curr[s[l:l + k]] -= 1\\n                     l += k\\n                 if r - l == t:\\n                     ans.append(l)\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         L = len(s)\\n         length = len(words[0])\\n         Length = length * len(words)\\n         result = []\\n         word_dict = {}\\n         for word in words:\\n             word_dict[word] = word_dict[word]+1 if word in word_dict else 1\\n         for i in range(length):\\n             l = i\\n             r = i\\n             curr_dict={}\\n             while r+length<=L:\\n                 word  = s[r:r+length]\\n                 r = r+length\\n                 if word in word_dict:\\n                     curr_dict[word] = curr_dict[word]+1 if word in curr_dict else 1\\n                     while curr_dict[word] > word_dict[word]:\\n                         curr_dict[s[l:l+length]] -=1\\n                         l += length\\n                     if r-l==Length:\\n                         result.append(l)\\n                 else:\\n                     curr_dict.clear()\\n                     l = r\\n             \\n         return result\\n \\n\", \"from collections import Counter\\n from copy import deepcopy\\n class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words:\\n             return []\\n         \\n         wordCounter = Counter(words)\\n         longest = len(words[0])\\n         lenSubStr = longest * len(words)\\n         n = len(s)\\n \\n         idx = []\\n         for i in range(0, min(longest, n - lenSubStr + 1)):\\n             if i in idx:\\n                 continue\\n             cnt = {}\\n             j = i\\n             start = i\\n             while start + lenSubStr <= n:\\n                 word = s[j:j+longest]\\n                 j += longest\\n                 if word not in wordCounter:\\n                     start = j\\n                     cnt.clear()\\n                 else:\\n                     if word in cnt:\\n                         cnt[word] += 1\\n                     else:\\n                         cnt[word] = 1\\n                         \\n                     while cnt[word] > wordCounter[word]:\\n                         cnt[s[start: start + longest]] -= 1\\n                         start += longest\\n                         \\n                     if j - start == lenSubStr:\\n                         idx.append(start)\\n \\n         return list(idx)\\n                     \\n                 \\n                 \\n         \\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \", \"class Solution:\\n     def findSubstring(self, s, words):\\n         if not words:\\n             return []\\n         word_len = len(words[0])\\n         word_set = {}\\n         for word in words:\\n             word_set[word] = word_set.get(word, 0) + 1\\n         scopes = [([], {}) for _ in range(word_len)]\\n         results = []\\n         for i in range(len(s)):\\n             word = s[i:i + word_len]\\n             if word in word_set:\\n                 matched_queue, matched_counts = scopes[i % word_len]\\n                 while matched_counts.get(word, 0) >= word_set[word]:\\n                     matched_counts[matched_queue.pop(0)] -= 1\\n                 matched_queue.append(word)\\n                 matched_counts[word] = matched_counts.get(word, 0) + 1\\n                 if len(matched_queue) == len(words):\\n                     results.append(i - word_len * len(words) + word_len)\\n                     matched_counts[matched_queue.pop(0)] -= 1\\n             else:\\n                 scopes[i % word_len] = ([], {})\\n         return results\", \"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or not words: return []\\n         result, sLen, numW, wLen = [], len(s), len(words), len(words[0])\\n         if sLen < numW*wLen: return []\\n         dic = collections.defaultdict(int)\\n         for i in words:\\n             dic[i] += 1\\n         for i in range(wLen):\\n             left, count = i, 0\\n             tmp = collections.defaultdict(int)\\n             for j in range(i, sLen-wLen+1, wLen):\\n                 s1 = s[j:j+wLen]\\n                 if s1 in dic:\\n                     tmp[s1] += 1\\n                     if tmp[s1] <= dic[s1]: \\n                         count += 1\\n                     else:\\n                         while tmp[s1] > dic[s1]:\\n                             s2 = s[left:left+wLen]\\n                             tmp[s2] -= 1\\n                             if tmp[s2] < dic[s2]:\\n                                 count -= 1\\n                             left += wLen\\n                     if count == numW:\\n                         result.append(left)\\n                         tmp[s[left:left+wLen]] -= 1\\n                         count -= 1\\n                         left += wLen\\n                 else:\\n                     tmp, count, left = collections.defaultdict(int), 0, j+wLen\\n         return result\\n\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         hash = {}\\n         res = []\\n         wsize = len(words[0])\\n         \\n         for str in words:\\n             if str in hash:\\n                 hash[str] += 1\\n             else:\\n                 hash[str] = 1\\n         for start in range(0, len(words[0])):\\n             slidingWindow = {}\\n             wCount = 0\\n             for i in range(start, len(s), wsize):\\n                 word = s[i : i + wsize]\\n                 if word in hash:\\n                     if word in slidingWindow:\\n                         slidingWindow[word] += 1\\n                     else:\\n                         slidingWindow[word] = 1\\n                     wCount += 1\\n                     while hash[word] < slidingWindow[word]:\\n                         pos = i - wsize * (wCount - 1)\\n                         removeWord = s[pos : pos + wsize]\\n                         #print i, removeWord\\n                         slidingWindow[removeWord] -= 1\\n                         wCount -= 1\\n                 else:\\n                     slidingWindow.clear()\\n                     wCount = 0\\n                 if wCount == len(words):\\n                     res.append(i - wsize * (wCount - 1))\\n                     \\n         return res\"]",
        "difficulty": "interview",
        "input": [
            "\"wordgoodgoodgoodbestword\"",
            [
                "\"word\"",
                "\"good\"",
                "\"best\"",
                "\"word\""
            ]
        ],
        "output": [],
        "halu_type": "Identification Hallucination",
        "fn_name": "findSubstring",
        "starter_code": "\nclass Solution:\n    def findSubstring(self, s: str, words: List[str]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/substring-with-concatenation-of-all-words/"
    },
    {
        "id": 1317,
        "task_id": 2745,
        "test_case_id": 3,
        "question": "You are given a string, s, and a list of words, words, that are all of the same length. Find all starting indices of substring(s) in s that is a concatenation of each word in words exactly once and without any intervening characters.\n\nExample 1:\n\n\nInput:\ns = \"barfoothefoobarman\",\nwords = [\"foo\",\"bar\"]\nOutput: [0,9]\nExplanation: Substrings starting at index 0 and 9 are \"barfoor\" and \"foobar\" respectively.\nThe output order does not matter, returning [9,0] is fine too.\n\n\nExample 2:\n\n\nInput:\ns = \"wordgoodstudentgoodword\",\nwords = [\"word\",\"student\"]\nOutput: []",
        "solutions": "[\"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or words==[]:\\n             return []\\n         lenstr=len(s)\\n         lenword=len(words[0])\\n         lensubstr=len(words)*lenword\\n         times={}\\n         for word in words:\\n             if word in times:\\n                 times[word]+=1\\n             else:\\n                 times[word]=1\\n         ans=[]\\n         for i in range(min(lenword,lenstr-lensubstr+1)):\\n             self.findAnswer(i,lenstr,lenword,lensubstr,s,times,ans)\\n         return ans\\n     def findAnswer(self,strstart,lenstr,lenword,lensubstr,s,times,ans):\\n         wordstart=strstart\\n         curr={}\\n         while strstart+lensubstr<=lenstr:\\n             word=s[wordstart:wordstart+lenword]\\n             wordstart+=lenword\\n             if word not in times:\\n                 strstart=wordstart\\n                 curr.clear()\\n             else:\\n                 if word in curr:\\n                     curr[word]+=1\\n                 else:\\n                     curr[word]=1\\n                 while curr[word]>times[word]:\\n                     curr[s[strstart:strstart+lenword]]-=1\\n                     strstart+=lenword\\n                 if wordstart-strstart==lensubstr:\\n                     ans.append(strstart)\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words: return []\\n         lens, lenw, numw = len(s), len(words[0]), len(words)\\n         res = []\\n         end = lens // lenw * lenw\\n         \\n         for i in range(lenw):\\n             start = sidx = i\\n             record = words[:]\\n             while sidx < end:              \\n                 tmp_sidx = sidx + lenw\\n                 tmp_word = s[sidx: tmp_sidx]\\n                 \\n                 if tmp_word in words:\\n                     if tmp_word in record:\\n                         record.remove(tmp_word)\\n                     elif s[start: start + lenw] != tmp_word:\\n                         sidx = start = start + lenw\\n                         record = words[:]\\n                         continue\\n                     else:\\n                         start = start + lenw\\n                 elif lens - tmp_sidx < lenw * numw:\\n                     break\\n                 else:\\n                     record = words[:]\\n                     start = tmp_sidx\\n                     \\n                 if not record: res.append(start)\\n                 sidx = tmp_sidx  \\n                 \\n         return res\\n\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         lenWord = len(words[0])\\n         numOfWords = len(words)\\n         lenSubstring = lenWord * numOfWords\\n         dic = {}\\n         \\n         for word in words:\\n             if word in dic:\\n                 dic[word] += 1\\n             else:\\n                 dic[word] = 1\\n         \\n         for i in range(min(lenWord, len(s) - lenSubstring + 1)):\\n             curr = {}\\n             strStart = i\\n             wordStart = strStart\\n             while strStart + lenSubstring <= len(s):\\n                 word = s[wordStart: wordStart + lenWord]\\n                 wordStart += lenWord\\n                 if word not in dic:\\n                     strStart = wordStart\\n                     curr.clear()\\n                 else:\\n                     if word in curr:\\n                         curr[word] += 1\\n                     else:\\n                         curr[word] = 1\\n                     while curr[word] > dic[word]:\\n                         curr[s[strStart: strStart + lenWord]] -= 1\\n                         strStart += lenWord\\n                     if wordStart - strStart == lenSubstring:\\n                         result.append(strStart)\\n         return result\\n\", \"from collections import Counter\\n from copy import deepcopy\\n class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words:\\n             return []\\n         \\n         wordCounter = Counter(words)\\n         longest = len(words[0])\\n         lenSubStr = longest * len(words)\\n         n = len(s)\\n \\n         idx = []\\n         for i in range(0, longest):\\n             cnt = {}\\n             j = i\\n             start = i\\n             while start + lenSubStr <= n:\\n                 word = s[j:j+longest]\\n                 j += longest\\n                 if word not in wordCounter:\\n                     start = j\\n                     cnt.clear()\\n                 else:\\n                     if word in cnt:\\n                         cnt[word] += 1\\n                     else:\\n                         cnt[word] = 1\\n                         \\n                     while cnt[word] > wordCounter[word]:\\n                         cnt[s[start: start + longest]] -= 1\\n                         start += longest\\n                         \\n                     if j - start == lenSubStr:\\n                         idx.append(start)\\n \\n         return idx\\n                     \\n                 \\n                 \\n         \\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         lenWord = len(words[0])\\n         numOfWords = len(words)\\n         lenSubstring = lenWord * numOfWords\\n         dic = {}\\n         for word in words:\\n             if word in dic:\\n                 dic[word] += 1\\n             else:\\n                 dic[word] = 1\\n         for i in range(min(lenWord, len(s)-lenSubstring+1)):\\n             curr = {}\\n             strStart = i\\n             wordStart = strStart\\n             while strStart + lenSubstring <= len(s):\\n                 word = s[wordStart: wordStart+lenWord]\\n                 wordStart += lenWord\\n                 if word not in dic:\\n                     strStart = wordStart\\n                     curr.clear()\\n                 else:\\n                     if word in curr:\\n                         curr[word] += 1\\n                     else:\\n                         curr[word] = 1\\n                     while curr[word] > dic[word]:\\n                         curr[s[strStart: strStart+lenWord]] -= 1\\n                         strStart += lenWord\\n                     if wordStart - strStart == lenSubstring:\\n                         result.append(strStart)\\n         return result\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         m, n, k, result, wordbase = len(s), len(words), len(words[0]), [], {}\\n         for value in words:\\n             if value in wordbase:\\n                 wordbase[value] += 1\\n             else:\\n                 wordbase[value] = 1\\n         for i in range(min(k, m - k * n + 1)):\\n             base, starts, startw = {}, i, i\\n             while starts + k * n <= m:\\n                 temp = s[startw:startw + k]\\n                 startw += k\\n                 if temp not in wordbase:\\n                     starts = startw\\n                     base.clear()\\n                 else:\\n                     if temp in base:\\n                         base[temp] += 1\\n                     else:\\n                         base[temp] = 1\\n                     while base[temp] > wordbase[temp]:\\n                         base[s[starts:starts + k]] -= 1\\n                         starts += k\\n                     if startw - starts == k * n:\\n                         result.append(starts)\\n         return result\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         lword = len(words[0])\\n         n = len(words)\\n         lwords = n * lword\\n         if not s or not lwords:\\n             return\\n         \\n         res = []\\n         dic = {} \\n         for key in words:\\n             dic[key] = [0, 0]\\n         for key in words:\\n             dic[key][1] += 1\\n         for k in range(lword):\\n             i = k\\n             while i <= len(s)-lwords:\\n                 if not s[i:i+lword] in list(dic.keys()):\\n                     i += lword\\n                     continue\\n                 j = 0\\n                 start = i\\n                 while s[i:i+lword] in list(dic.keys()) and i<len(s):\\n                     key = s[i:i+lword]\\n                     dic[key][0] += 1\\n                     while dic[key][0] > dic[key][1]:\\n                         dic[s[start:start+lword]][0] -= 1\\n                         j -= 1\\n                         start += lword\\n                     j += 1\\n                     i += lword\\n                     if j==n:\\n                         res.append(start)\\n                         dic[s[start:start+lword]][0] -= 1\\n                         j -= 1\\n                         start += lword\\n                 for key in list(dic.keys()):\\n                     dic[key][0] = 0\\n         return res\\n\", \"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or not words or not words[0]:\\n             return []\\n         n = len(s)\\n         k = len(words[0])\\n         t = len(words) * k\\n         req = {}\\n         for w in words:\\n             req[w] = req[w] + 1 if w in req else 1\\n         ans = []\\n         for i in range(min(k, n - t + 1)):\\n             self._findSubstring(i, i, n, k, t, s, req, ans)\\n         return ans\\n         \\n     def _findSubstring(self, l, r, n, k, t, s, req, ans):\\n         curr = {}\\n         while r + k <= n:\\n             w = s[r:r + k]\\n             r += k\\n             if w not in req:\\n                 l = r\\n                 curr.clear()\\n             else:\\n                 curr[w] = curr[w] + 1 if w in curr else 1\\n                 while curr[w] > req[w]:\\n                     curr[s[l:l + k]] -= 1\\n                     l += k\\n                 if r - l == t:\\n                     ans.append(l)\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         L = len(s)\\n         length = len(words[0])\\n         Length = length * len(words)\\n         result = []\\n         word_dict = {}\\n         for word in words:\\n             word_dict[word] = word_dict[word]+1 if word in word_dict else 1\\n         for i in range(length):\\n             l = i\\n             r = i\\n             curr_dict={}\\n             while r+length<=L:\\n                 word  = s[r:r+length]\\n                 r = r+length\\n                 if word in word_dict:\\n                     curr_dict[word] = curr_dict[word]+1 if word in curr_dict else 1\\n                     while curr_dict[word] > word_dict[word]:\\n                         curr_dict[s[l:l+length]] -=1\\n                         l += length\\n                     if r-l==Length:\\n                         result.append(l)\\n                 else:\\n                     curr_dict.clear()\\n                     l = r\\n             \\n         return result\\n \\n\", \"from collections import Counter\\n from copy import deepcopy\\n class Solution:\\n     def findSubstring(self, s, words):\\n         if not s or not words:\\n             return []\\n         \\n         wordCounter = Counter(words)\\n         longest = len(words[0])\\n         lenSubStr = longest * len(words)\\n         n = len(s)\\n \\n         idx = []\\n         for i in range(0, min(longest, n - lenSubStr + 1)):\\n             if i in idx:\\n                 continue\\n             cnt = {}\\n             j = i\\n             start = i\\n             while start + lenSubStr <= n:\\n                 word = s[j:j+longest]\\n                 j += longest\\n                 if word not in wordCounter:\\n                     start = j\\n                     cnt.clear()\\n                 else:\\n                     if word in cnt:\\n                         cnt[word] += 1\\n                     else:\\n                         cnt[word] = 1\\n                         \\n                     while cnt[word] > wordCounter[word]:\\n                         cnt[s[start: start + longest]] -= 1\\n                         start += longest\\n                         \\n                     if j - start == lenSubStr:\\n                         idx.append(start)\\n \\n         return list(idx)\\n                     \\n                 \\n                 \\n         \\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \", \"class Solution:\\n     def findSubstring(self, s, words):\\n         if not words:\\n             return []\\n         word_len = len(words[0])\\n         word_set = {}\\n         for word in words:\\n             word_set[word] = word_set.get(word, 0) + 1\\n         scopes = [([], {}) for _ in range(word_len)]\\n         results = []\\n         for i in range(len(s)):\\n             word = s[i:i + word_len]\\n             if word in word_set:\\n                 matched_queue, matched_counts = scopes[i % word_len]\\n                 while matched_counts.get(word, 0) >= word_set[word]:\\n                     matched_counts[matched_queue.pop(0)] -= 1\\n                 matched_queue.append(word)\\n                 matched_counts[word] = matched_counts.get(word, 0) + 1\\n                 if len(matched_queue) == len(words):\\n                     results.append(i - word_len * len(words) + word_len)\\n                     matched_counts[matched_queue.pop(0)] -= 1\\n             else:\\n                 scopes[i % word_len] = ([], {})\\n         return results\", \"class Solution(object):\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not s or not words: return []\\n         result, sLen, numW, wLen = [], len(s), len(words), len(words[0])\\n         if sLen < numW*wLen: return []\\n         dic = collections.defaultdict(int)\\n         for i in words:\\n             dic[i] += 1\\n         for i in range(wLen):\\n             left, count = i, 0\\n             tmp = collections.defaultdict(int)\\n             for j in range(i, sLen-wLen+1, wLen):\\n                 s1 = s[j:j+wLen]\\n                 if s1 in dic:\\n                     tmp[s1] += 1\\n                     if tmp[s1] <= dic[s1]: \\n                         count += 1\\n                     else:\\n                         while tmp[s1] > dic[s1]:\\n                             s2 = s[left:left+wLen]\\n                             tmp[s2] -= 1\\n                             if tmp[s2] < dic[s2]:\\n                                 count -= 1\\n                             left += wLen\\n                     if count == numW:\\n                         result.append(left)\\n                         tmp[s[left:left+wLen]] -= 1\\n                         count -= 1\\n                         left += wLen\\n                 else:\\n                     tmp, count, left = collections.defaultdict(int), 0, j+wLen\\n         return result\\n\", \"class Solution:\\n     def findSubstring(self, s, words):\\n         \\\"\\\"\\\"\\n         :type s: str\\n         :type words: List[str]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         hash = {}\\n         res = []\\n         wsize = len(words[0])\\n         \\n         for str in words:\\n             if str in hash:\\n                 hash[str] += 1\\n             else:\\n                 hash[str] = 1\\n         for start in range(0, len(words[0])):\\n             slidingWindow = {}\\n             wCount = 0\\n             for i in range(start, len(s), wsize):\\n                 word = s[i : i + wsize]\\n                 if word in hash:\\n                     if word in slidingWindow:\\n                         slidingWindow[word] += 1\\n                     else:\\n                         slidingWindow[word] = 1\\n                     wCount += 1\\n                     while hash[word] < slidingWindow[word]:\\n                         pos = i - wsize * (wCount - 1)\\n                         removeWord = s[pos : pos + wsize]\\n                         #print i, removeWord\\n                         slidingWindow[removeWord] -= 1\\n                         wCount -= 1\\n                 else:\\n                     slidingWindow.clear()\\n                     wCount = 0\\n                 if wCount == len(words):\\n                     res.append(i - wsize * (wCount - 1))\\n                     \\n         return res\"]",
        "difficulty": "interview",
        "input": [
            "\"barfoofoobarthefoobarman\"",
            [
                "\"bar\"",
                "\"foo\"",
                "\"the\""
            ]
        ],
        "output": [
            6,
            9,
            12
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "findSubstring",
        "starter_code": "\nclass Solution:\n    def findSubstring(self, s: str, words: List[str]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/substring-with-concatenation-of-all-words/"
    },
    {
        "id": 1318,
        "task_id": 2747,
        "test_case_id": 1,
        "question": "Given an array of integers nums sorted in ascending order, find the starting and ending position of a given target value.\n\nYour algorithm's runtime complexity must be in the order of O(log n).\n\nIf the target is not found in the array, return [-1, -1].\n\nExample 1:\n\n\nInput: nums = [5,7,7,8,8,10], target = 8\nOutput: [3,4]\n\nExample 2:\n\n\nInput: nums = [5,7,7,8,8,10], target = 6\nOutput: [-1,-1]",
        "solutions": "[\"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         start = self.firstGreaterEqaul(nums, target)\\n         if start==len(nums) or nums[start]!=target:\\n             return [-1, -1]\\n         return [start, self.firstGreaterEqaul(nums, target+1)-1]\\n     def firstGreaterEqaul(self, nums, target):\\n         lo, hi = 0, len(nums)\\n         while lo<hi:\\n             mid = (hi+lo)//2\\n             if nums[mid]<target:\\n                 lo = mid + 1\\n             else:\\n                 hi = mid\\n         return lo\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         import bisect\\n         lo=bisect.bisect_left(nums,target)\\n         if target in nums[lo:lo+1]:\\n             hi=bisect.bisect(nums,target)-1\\n             return [lo, hi]\\n         else:\\n             return [-1, -1]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         left, right = 0, len(nums)-1\\n         while left <= right:\\n             mid = (left+right) // 2\\n             print(('mid:', mid))\\n             if nums[mid] == target:\\n                 print('hello')\\n                 if mid == 0 or nums[mid-1] < target:\\n                     result.append(mid)\\n                     break\\n                 else:\\n                     right = mid -1\\n             elif nums[mid] < target:\\n                 left = mid+1\\n             else:\\n                 right = mid-1\\n         left, right = 0, len(nums)-1\\n         while left <= right:\\n             mid = (left+right) // 2\\n             if nums[mid] == target:\\n                 if mid == len(nums)-1 or nums[mid+1] > target:\\n                     result.append(mid)\\n                     break\\n                 else:\\n                     left = mid + 1\\n             elif nums[mid] < target:\\n                 left = mid+1\\n             else:\\n                 right = mid-1\\n         if len(result) == 0:\\n             return [-1, -1]\\n         return result\\n         \\n\", \"\\n \\n class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if len(nums) == 0:\\n             return [-1, -1]\\n \\n         nums = [target - 1] + nums + [target + 1]\\n \\n         lbound = -1\\n \\n         l, r = 1, len(nums) - 2\\n \\n         while l <= r:\\n \\n             m = (l + r)//2\\n \\n             if nums[m] == target and nums[m - 1] < target:\\n                 lbound = m\\n                 break\\n \\n             if nums[m] < target:\\n                 l = m + 1\\n             else: # nums[m] >= target:\\n                 r = m - 1\\n \\n         if lbound == -1:\\n             return [-1, -1]\\n \\n         rbound = -1\\n         l, r = 1, len(nums) - 2\\n \\n         while l <= r:\\n \\n             m = (l + r)//2\\n \\n             if nums[m] == target and nums[m + 1] > target:\\n                 rbound = m\\n                 break\\n \\n             if nums[m] <= target:\\n                 l = m + 1\\n             else: # target < nums[m]\\n                 r = m - 1\\n \\n         return [lbound - 1, rbound - 1]\\n \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return [-1,-1]\\n         a=self.findLeft(nums,target)\\n         b=self.findRight(nums,target)\\n         if nums[a]!=target:\\n             return [-1,-1]\\n         return [a,b]\\n         \\n     def findLeft(self,nums,target):\\n         low,high=0,len(nums)-1\\n         while low<high:\\n             mid=(low+high)//2\\n             \\n             if nums[mid]>=target:\\n                 high=mid\\n             else:\\n                 low=mid+1\\n         return low\\n     \\n     def findRight(self,nums,target):\\n         low,high=0,len(nums)-1\\n         while low<high:\\n             mid=(low+high)//2+1\\n             if nums[mid]>target:\\n                 high=mid-1\\n             else:\\n                 low=mid\\n         return high\\n     \\n     \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         #m = int(len(nums)/2)\\n         #upper, lower = nums[:m], nums[m:]\\n         s, e = -1, -1\\n         l, u = 0, len(nums)-1\\n         if not nums or target > nums[u] or target < nums[l]:\\n             return [s, e]\\n         m = int((l+u)/2)\\n         while u >= l:\\n             if nums[m] > target:\\n                 if m == u:\\n                     break\\n                 u = m\\n                 #if int((l+u)/2) == u:\\n                 #    break\\n                 m = int((l+u)/2)\\n             elif nums[m] < target:\\n                 l = m\\n                 if int((l+u)/2) == l:\\n                     m = l+1\\n                 else:\\n                     m = int((l+u)/2)\\n             else:\\n                 s = e = m\\n                 while 0 < s and nums[s-1] == nums[s]:\\n                     s-=1\\n                 while e < len(nums)-1 and nums[e+1] == nums[e]:\\n                     e+=1\\n                 break\\n         return [s, e]\\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         s, e = -1, -1\\n         l, u = 0, len(nums)\\n         if not nums or target > nums[u-1] or target < nums[l]:\\n             return [s, e]\\n         m = (l+u)//2\\n         while u > l:\\n             if nums[m] > target:\\n                 u = m\\n                 m = (l+u)//2\\n             elif nums[m] < target:\\n                 l = m+1\\n                 m = (l+u)//2\\n             else:\\n                 s = e = m\\n                 while 0 < s and nums[s-1] == nums[s]:\\n                     s-=1\\n                 while e < len(nums)-1 and nums[e+1] == nums[e]:\\n                     e+=1\\n                 break\\n         return [s, e]\\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         left = 0; right = len(nums) - 1\\n         while left <= right:\\n             mid = int((left + right) / 2)\\n             if nums[mid] > target:\\n                 right = mid - 1\\n             elif nums[mid] < target:\\n                 left = mid + 1\\n             else:\\n                 list = [0, 0]\\n                 if nums[left] == target: list[0] = left\\n                 if nums[right] == target: list[1] = right\\n                 for i in range(mid, right+1):\\n                     if nums[i] != target: list[1] = i - 1; break\\n                 for i in range(mid, left-1, -1):\\n                     if nums[i] != target: list[0] = i + 1; break\\n                 return list\\n         return [-1, -1]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"           \\n         def findBoundary(nums, left, right, target):\\n             [l_boundary, r_boundary] = [len(nums), -1]\\n             middle = int((left+right)/2)\\n             if left <= right and nums[left] <= target and nums[right] >= target:\\n                 if nums[left] == target:\\n                     l_boundary = left\\n                 if nums[right] == target:\\n                     r_boundary = right\\n                 if nums[middle] < target:\\n                     [l_boundary, r_boundary] = findBoundary(nums, middle+1, right, target)\\n                 elif nums[middle] > target:\\n                     [l_boundary, r_boundary] = findBoundary(nums, left, middle, target)\\n                 else:\\n                     l_boundary = min(findBoundary(nums, left, middle-1, target)[0], middle)\\n                     r_boundary = max(findBoundary(nums, middle+1, right, target)[1], middle)\\n             return [l_boundary, r_boundary]\\n         \\n         boundary = findBoundary(nums, 0, len(nums)-1, target)\\n         if boundary[1] == -1:\\n             return [-1, -1]\\n         else:\\n             return boundary\\n \\n                 \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         # self made, two times binary search, beats 93%\\n         if not nums:\\n             return [-1, -1]\\n         # find left most index\\n         l, r = 0, len(nums) - 1\\n         while l + 1 < r:\\n             mid = l + (r - l) // 2\\n             if nums[mid] < target:\\n                 l = mid\\n             else:\\n                 r = mid\\n         left = l if nums[l] == target else r\\n         # find right most index\\n         l, r = 0, len(nums) - 1\\n         while l + 1 < r:\\n             mid = l + (r - l) // 2\\n             if nums[mid] <= target:\\n                 l = mid\\n             else:\\n                 r = mid\\n         right = r if nums[r] == target else l\\n         if nums[left] != target:\\n             return [-1, -1]\\n         \\n         return [left, right]\\n\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index - 1 \\n         after = index + 1\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if nums == []: return [-1, -1]\\n         m = int(len(nums)/2)\\n         if target == nums[0]:   return self.getRange(nums, 0)\\n         elif target == nums[-1]: return self.getRange(nums, len(nums) -1)\\n         elif target == nums[m]: return self.getRange(nums, m)\\n         else:\\n             if target < nums[0] or target > nums[-1]:   return [-1,-1]\\n             elif target < nums[m]:\\n                 return self.searchRange(nums[: m], target)\\n             else:\\n                 tempRange = self.searchRange(nums[m:], target)\\n                 if tempRange[0] == -1:  return [-1, -1]\\n                 else: return [r + m  for r in tempRange]\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index - 1 \\n         after = index + 1\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def getIndex(self, nums, target):\\n         if nums == [] or target < nums[0] or target > nums[-1]:  return -1\\n         if target == nums[0]:  return 0\\n         if target == nums[-1]:  return len(nums) -1\\n         \\n         m = int(len(nums)/2)\\n         \\n         if target < nums[m]:\\n             return self.getIndex(nums[:m], target) \\n         else:\\n             return -1 if self.getIndex(nums[m: ], target) < 0 else self.getIndex(nums[m: ], target) + m\\n             \\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         index = self.getIndex(nums, target)\\n         if index == -1: return [-1, -1]\\n         else:\\n             return self.getRange(nums, index)\\n\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index\\n         after = index\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if nums == []: return [-1, -1]\\n         m = int(len(nums)/2)\\n         if target == nums[0]:   return self.getRange(nums, 0)\\n         elif target == nums[-1]: return self.getRange(nums, len(nums) -1)\\n         elif target == nums[m]: return self.getRange(nums, m)\\n         else:\\n             if target < nums[0] or target > nums[-1]:   return [-1,-1]\\n             elif target < nums[m]:\\n                 return self.searchRange(nums[: m], target)\\n             else:\\n                 tempRange = self.searchRange(nums[m:], target)\\n                 if tempRange[0] == -1:  return [-1, -1]\\n                 else: return [r + m  for r in tempRange]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         start = 0\\n         end = len(nums) - 1\\n         mid = 0\\n         ans = [-1, -1]\\n         while (start <= end):\\n             mid = int((start + end) / 2)\\n             if (nums[mid] == target):\\n                 ans[0] = mid  \\n                 ans[1] = mid  \\n                   \\n                 i = mid - 1  \\n                 while i >= 0 and nums[i] == target:  \\n                     ans[0] = i  \\n                     i -= 1  \\n                   \\n                 i = mid + 1  \\n                 while i < len(nums) and nums[i] == target:  \\n                     ans[1] = i  \\n                     i += 1  \\n                       \\n                 break  \\n             elif (nums[mid] > target):\\n                 end = mid - 1\\n             else:\\n                 start = mid + 1\\n             \\n         return ans\\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                5,
                7,
                7,
                8,
                8,
                10
            ],
            [
                8
            ]
        ],
        "output": [
            3,
            4
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "searchRange",
        "starter_code": "\nclass Solution:\n    def searchRange(self, nums: List[int], target: int) -> List[int]:\n",
        "url": "https://leetcode.com/problems/search-for-a-range/"
    },
    {
        "id": 1319,
        "task_id": 2747,
        "test_case_id": 2,
        "question": "Given an array of integers nums sorted in ascending order, find the starting and ending position of a given target value.\n\nYour algorithm's runtime complexity must be in the order of O(log n).\n\nIf the target is not found in the array, return [-1, -1].\n\nExample 1:\n\n\nInput: nums = [5,7,7,8,8,10], target = 8\nOutput: [3,4]\n\nExample 2:\n\n\nInput: nums = [5,7,7,8,8,10], target = 6\nOutput: [-1,-1]",
        "solutions": "[\"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         start = self.firstGreaterEqaul(nums, target)\\n         if start==len(nums) or nums[start]!=target:\\n             return [-1, -1]\\n         return [start, self.firstGreaterEqaul(nums, target+1)-1]\\n     def firstGreaterEqaul(self, nums, target):\\n         lo, hi = 0, len(nums)\\n         while lo<hi:\\n             mid = (hi+lo)//2\\n             if nums[mid]<target:\\n                 lo = mid + 1\\n             else:\\n                 hi = mid\\n         return lo\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         import bisect\\n         lo=bisect.bisect_left(nums,target)\\n         if target in nums[lo:lo+1]:\\n             hi=bisect.bisect(nums,target)-1\\n             return [lo, hi]\\n         else:\\n             return [-1, -1]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         left, right = 0, len(nums)-1\\n         while left <= right:\\n             mid = (left+right) // 2\\n             print(('mid:', mid))\\n             if nums[mid] == target:\\n                 print('hello')\\n                 if mid == 0 or nums[mid-1] < target:\\n                     result.append(mid)\\n                     break\\n                 else:\\n                     right = mid -1\\n             elif nums[mid] < target:\\n                 left = mid+1\\n             else:\\n                 right = mid-1\\n         left, right = 0, len(nums)-1\\n         while left <= right:\\n             mid = (left+right) // 2\\n             if nums[mid] == target:\\n                 if mid == len(nums)-1 or nums[mid+1] > target:\\n                     result.append(mid)\\n                     break\\n                 else:\\n                     left = mid + 1\\n             elif nums[mid] < target:\\n                 left = mid+1\\n             else:\\n                 right = mid-1\\n         if len(result) == 0:\\n             return [-1, -1]\\n         return result\\n         \\n\", \"\\n \\n class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if len(nums) == 0:\\n             return [-1, -1]\\n \\n         nums = [target - 1] + nums + [target + 1]\\n \\n         lbound = -1\\n \\n         l, r = 1, len(nums) - 2\\n \\n         while l <= r:\\n \\n             m = (l + r)//2\\n \\n             if nums[m] == target and nums[m - 1] < target:\\n                 lbound = m\\n                 break\\n \\n             if nums[m] < target:\\n                 l = m + 1\\n             else: # nums[m] >= target:\\n                 r = m - 1\\n \\n         if lbound == -1:\\n             return [-1, -1]\\n \\n         rbound = -1\\n         l, r = 1, len(nums) - 2\\n \\n         while l <= r:\\n \\n             m = (l + r)//2\\n \\n             if nums[m] == target and nums[m + 1] > target:\\n                 rbound = m\\n                 break\\n \\n             if nums[m] <= target:\\n                 l = m + 1\\n             else: # target < nums[m]\\n                 r = m - 1\\n \\n         return [lbound - 1, rbound - 1]\\n \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return [-1,-1]\\n         a=self.findLeft(nums,target)\\n         b=self.findRight(nums,target)\\n         if nums[a]!=target:\\n             return [-1,-1]\\n         return [a,b]\\n         \\n     def findLeft(self,nums,target):\\n         low,high=0,len(nums)-1\\n         while low<high:\\n             mid=(low+high)//2\\n             \\n             if nums[mid]>=target:\\n                 high=mid\\n             else:\\n                 low=mid+1\\n         return low\\n     \\n     def findRight(self,nums,target):\\n         low,high=0,len(nums)-1\\n         while low<high:\\n             mid=(low+high)//2+1\\n             if nums[mid]>target:\\n                 high=mid-1\\n             else:\\n                 low=mid\\n         return high\\n     \\n     \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         #m = int(len(nums)/2)\\n         #upper, lower = nums[:m], nums[m:]\\n         s, e = -1, -1\\n         l, u = 0, len(nums)-1\\n         if not nums or target > nums[u] or target < nums[l]:\\n             return [s, e]\\n         m = int((l+u)/2)\\n         while u >= l:\\n             if nums[m] > target:\\n                 if m == u:\\n                     break\\n                 u = m\\n                 #if int((l+u)/2) == u:\\n                 #    break\\n                 m = int((l+u)/2)\\n             elif nums[m] < target:\\n                 l = m\\n                 if int((l+u)/2) == l:\\n                     m = l+1\\n                 else:\\n                     m = int((l+u)/2)\\n             else:\\n                 s = e = m\\n                 while 0 < s and nums[s-1] == nums[s]:\\n                     s-=1\\n                 while e < len(nums)-1 and nums[e+1] == nums[e]:\\n                     e+=1\\n                 break\\n         return [s, e]\\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         s, e = -1, -1\\n         l, u = 0, len(nums)\\n         if not nums or target > nums[u-1] or target < nums[l]:\\n             return [s, e]\\n         m = (l+u)//2\\n         while u > l:\\n             if nums[m] > target:\\n                 u = m\\n                 m = (l+u)//2\\n             elif nums[m] < target:\\n                 l = m+1\\n                 m = (l+u)//2\\n             else:\\n                 s = e = m\\n                 while 0 < s and nums[s-1] == nums[s]:\\n                     s-=1\\n                 while e < len(nums)-1 and nums[e+1] == nums[e]:\\n                     e+=1\\n                 break\\n         return [s, e]\\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         left = 0; right = len(nums) - 1\\n         while left <= right:\\n             mid = int((left + right) / 2)\\n             if nums[mid] > target:\\n                 right = mid - 1\\n             elif nums[mid] < target:\\n                 left = mid + 1\\n             else:\\n                 list = [0, 0]\\n                 if nums[left] == target: list[0] = left\\n                 if nums[right] == target: list[1] = right\\n                 for i in range(mid, right+1):\\n                     if nums[i] != target: list[1] = i - 1; break\\n                 for i in range(mid, left-1, -1):\\n                     if nums[i] != target: list[0] = i + 1; break\\n                 return list\\n         return [-1, -1]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"           \\n         def findBoundary(nums, left, right, target):\\n             [l_boundary, r_boundary] = [len(nums), -1]\\n             middle = int((left+right)/2)\\n             if left <= right and nums[left] <= target and nums[right] >= target:\\n                 if nums[left] == target:\\n                     l_boundary = left\\n                 if nums[right] == target:\\n                     r_boundary = right\\n                 if nums[middle] < target:\\n                     [l_boundary, r_boundary] = findBoundary(nums, middle+1, right, target)\\n                 elif nums[middle] > target:\\n                     [l_boundary, r_boundary] = findBoundary(nums, left, middle, target)\\n                 else:\\n                     l_boundary = min(findBoundary(nums, left, middle-1, target)[0], middle)\\n                     r_boundary = max(findBoundary(nums, middle+1, right, target)[1], middle)\\n             return [l_boundary, r_boundary]\\n         \\n         boundary = findBoundary(nums, 0, len(nums)-1, target)\\n         if boundary[1] == -1:\\n             return [-1, -1]\\n         else:\\n             return boundary\\n \\n                 \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         # self made, two times binary search, beats 93%\\n         if not nums:\\n             return [-1, -1]\\n         # find left most index\\n         l, r = 0, len(nums) - 1\\n         while l + 1 < r:\\n             mid = l + (r - l) // 2\\n             if nums[mid] < target:\\n                 l = mid\\n             else:\\n                 r = mid\\n         left = l if nums[l] == target else r\\n         # find right most index\\n         l, r = 0, len(nums) - 1\\n         while l + 1 < r:\\n             mid = l + (r - l) // 2\\n             if nums[mid] <= target:\\n                 l = mid\\n             else:\\n                 r = mid\\n         right = r if nums[r] == target else l\\n         if nums[left] != target:\\n             return [-1, -1]\\n         \\n         return [left, right]\\n\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index - 1 \\n         after = index + 1\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if nums == []: return [-1, -1]\\n         m = int(len(nums)/2)\\n         if target == nums[0]:   return self.getRange(nums, 0)\\n         elif target == nums[-1]: return self.getRange(nums, len(nums) -1)\\n         elif target == nums[m]: return self.getRange(nums, m)\\n         else:\\n             if target < nums[0] or target > nums[-1]:   return [-1,-1]\\n             elif target < nums[m]:\\n                 return self.searchRange(nums[: m], target)\\n             else:\\n                 tempRange = self.searchRange(nums[m:], target)\\n                 if tempRange[0] == -1:  return [-1, -1]\\n                 else: return [r + m  for r in tempRange]\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index - 1 \\n         after = index + 1\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def getIndex(self, nums, target):\\n         if nums == [] or target < nums[0] or target > nums[-1]:  return -1\\n         if target == nums[0]:  return 0\\n         if target == nums[-1]:  return len(nums) -1\\n         \\n         m = int(len(nums)/2)\\n         \\n         if target < nums[m]:\\n             return self.getIndex(nums[:m], target) \\n         else:\\n             return -1 if self.getIndex(nums[m: ], target) < 0 else self.getIndex(nums[m: ], target) + m\\n             \\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         index = self.getIndex(nums, target)\\n         if index == -1: return [-1, -1]\\n         else:\\n             return self.getRange(nums, index)\\n\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index\\n         after = index\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if nums == []: return [-1, -1]\\n         m = int(len(nums)/2)\\n         if target == nums[0]:   return self.getRange(nums, 0)\\n         elif target == nums[-1]: return self.getRange(nums, len(nums) -1)\\n         elif target == nums[m]: return self.getRange(nums, m)\\n         else:\\n             if target < nums[0] or target > nums[-1]:   return [-1,-1]\\n             elif target < nums[m]:\\n                 return self.searchRange(nums[: m], target)\\n             else:\\n                 tempRange = self.searchRange(nums[m:], target)\\n                 if tempRange[0] == -1:  return [-1, -1]\\n                 else: return [r + m  for r in tempRange]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         start = 0\\n         end = len(nums) - 1\\n         mid = 0\\n         ans = [-1, -1]\\n         while (start <= end):\\n             mid = int((start + end) / 2)\\n             if (nums[mid] == target):\\n                 ans[0] = mid  \\n                 ans[1] = mid  \\n                   \\n                 i = mid - 1  \\n                 while i >= 0 and nums[i] == target:  \\n                     ans[0] = i  \\n                     i -= 1  \\n                   \\n                 i = mid + 1  \\n                 while i < len(nums) and nums[i] == target:  \\n                     ans[1] = i  \\n                     i += 1  \\n                       \\n                 break  \\n             elif (nums[mid] > target):\\n                 end = mid - 1\\n             else:\\n                 start = mid + 1\\n             \\n         return ans\\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [
                5,
                7,
                7,
                8,
                8,
                10
            ],
            [
                6
            ]
        ],
        "output": [
            -1,
            -1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "searchRange",
        "starter_code": "\nclass Solution:\n    def searchRange(self, nums: List[int], target: int) -> List[int]:\n",
        "url": "https://leetcode.com/problems/search-for-a-range/"
    },
    {
        "id": 1320,
        "task_id": 2747,
        "test_case_id": 3,
        "question": "Given an array of integers nums sorted in ascending order, find the starting and ending position of a given target value.\n\nYour algorithm's runtime complexity must be in the order of O(log n).\n\nIf the target is not found in the array, return [-1, -1].\n\nExample 1:\n\n\nInput: nums = [5,7,7,8,8,10], target = 8\nOutput: [3,4]\n\nExample 2:\n\n\nInput: nums = [5,7,7,8,8,10], target = 6\nOutput: [-1,-1]",
        "solutions": "[\"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         start = self.firstGreaterEqaul(nums, target)\\n         if start==len(nums) or nums[start]!=target:\\n             return [-1, -1]\\n         return [start, self.firstGreaterEqaul(nums, target+1)-1]\\n     def firstGreaterEqaul(self, nums, target):\\n         lo, hi = 0, len(nums)\\n         while lo<hi:\\n             mid = (hi+lo)//2\\n             if nums[mid]<target:\\n                 lo = mid + 1\\n             else:\\n                 hi = mid\\n         return lo\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         import bisect\\n         lo=bisect.bisect_left(nums,target)\\n         if target in nums[lo:lo+1]:\\n             hi=bisect.bisect(nums,target)-1\\n             return [lo, hi]\\n         else:\\n             return [-1, -1]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         result = []\\n         left, right = 0, len(nums)-1\\n         while left <= right:\\n             mid = (left+right) // 2\\n             print(('mid:', mid))\\n             if nums[mid] == target:\\n                 print('hello')\\n                 if mid == 0 or nums[mid-1] < target:\\n                     result.append(mid)\\n                     break\\n                 else:\\n                     right = mid -1\\n             elif nums[mid] < target:\\n                 left = mid+1\\n             else:\\n                 right = mid-1\\n         left, right = 0, len(nums)-1\\n         while left <= right:\\n             mid = (left+right) // 2\\n             if nums[mid] == target:\\n                 if mid == len(nums)-1 or nums[mid+1] > target:\\n                     result.append(mid)\\n                     break\\n                 else:\\n                     left = mid + 1\\n             elif nums[mid] < target:\\n                 left = mid+1\\n             else:\\n                 right = mid-1\\n         if len(result) == 0:\\n             return [-1, -1]\\n         return result\\n         \\n\", \"\\n \\n class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if len(nums) == 0:\\n             return [-1, -1]\\n \\n         nums = [target - 1] + nums + [target + 1]\\n \\n         lbound = -1\\n \\n         l, r = 1, len(nums) - 2\\n \\n         while l <= r:\\n \\n             m = (l + r)//2\\n \\n             if nums[m] == target and nums[m - 1] < target:\\n                 lbound = m\\n                 break\\n \\n             if nums[m] < target:\\n                 l = m + 1\\n             else: # nums[m] >= target:\\n                 r = m - 1\\n \\n         if lbound == -1:\\n             return [-1, -1]\\n \\n         rbound = -1\\n         l, r = 1, len(nums) - 2\\n \\n         while l <= r:\\n \\n             m = (l + r)//2\\n \\n             if nums[m] == target and nums[m + 1] > target:\\n                 rbound = m\\n                 break\\n \\n             if nums[m] <= target:\\n                 l = m + 1\\n             else: # target < nums[m]\\n                 r = m - 1\\n \\n         return [lbound - 1, rbound - 1]\\n \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not nums:\\n             return [-1,-1]\\n         a=self.findLeft(nums,target)\\n         b=self.findRight(nums,target)\\n         if nums[a]!=target:\\n             return [-1,-1]\\n         return [a,b]\\n         \\n     def findLeft(self,nums,target):\\n         low,high=0,len(nums)-1\\n         while low<high:\\n             mid=(low+high)//2\\n             \\n             if nums[mid]>=target:\\n                 high=mid\\n             else:\\n                 low=mid+1\\n         return low\\n     \\n     def findRight(self,nums,target):\\n         low,high=0,len(nums)-1\\n         while low<high:\\n             mid=(low+high)//2+1\\n             if nums[mid]>target:\\n                 high=mid-1\\n             else:\\n                 low=mid\\n         return high\\n     \\n     \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         #m = int(len(nums)/2)\\n         #upper, lower = nums[:m], nums[m:]\\n         s, e = -1, -1\\n         l, u = 0, len(nums)-1\\n         if not nums or target > nums[u] or target < nums[l]:\\n             return [s, e]\\n         m = int((l+u)/2)\\n         while u >= l:\\n             if nums[m] > target:\\n                 if m == u:\\n                     break\\n                 u = m\\n                 #if int((l+u)/2) == u:\\n                 #    break\\n                 m = int((l+u)/2)\\n             elif nums[m] < target:\\n                 l = m\\n                 if int((l+u)/2) == l:\\n                     m = l+1\\n                 else:\\n                     m = int((l+u)/2)\\n             else:\\n                 s = e = m\\n                 while 0 < s and nums[s-1] == nums[s]:\\n                     s-=1\\n                 while e < len(nums)-1 and nums[e+1] == nums[e]:\\n                     e+=1\\n                 break\\n         return [s, e]\\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         s, e = -1, -1\\n         l, u = 0, len(nums)\\n         if not nums or target > nums[u-1] or target < nums[l]:\\n             return [s, e]\\n         m = (l+u)//2\\n         while u > l:\\n             if nums[m] > target:\\n                 u = m\\n                 m = (l+u)//2\\n             elif nums[m] < target:\\n                 l = m+1\\n                 m = (l+u)//2\\n             else:\\n                 s = e = m\\n                 while 0 < s and nums[s-1] == nums[s]:\\n                     s-=1\\n                 while e < len(nums)-1 and nums[e+1] == nums[e]:\\n                     e+=1\\n                 break\\n         return [s, e]\\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         left = 0; right = len(nums) - 1\\n         while left <= right:\\n             mid = int((left + right) / 2)\\n             if nums[mid] > target:\\n                 right = mid - 1\\n             elif nums[mid] < target:\\n                 left = mid + 1\\n             else:\\n                 list = [0, 0]\\n                 if nums[left] == target: list[0] = left\\n                 if nums[right] == target: list[1] = right\\n                 for i in range(mid, right+1):\\n                     if nums[i] != target: list[1] = i - 1; break\\n                 for i in range(mid, left-1, -1):\\n                     if nums[i] != target: list[0] = i + 1; break\\n                 return list\\n         return [-1, -1]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"           \\n         def findBoundary(nums, left, right, target):\\n             [l_boundary, r_boundary] = [len(nums), -1]\\n             middle = int((left+right)/2)\\n             if left <= right and nums[left] <= target and nums[right] >= target:\\n                 if nums[left] == target:\\n                     l_boundary = left\\n                 if nums[right] == target:\\n                     r_boundary = right\\n                 if nums[middle] < target:\\n                     [l_boundary, r_boundary] = findBoundary(nums, middle+1, right, target)\\n                 elif nums[middle] > target:\\n                     [l_boundary, r_boundary] = findBoundary(nums, left, middle, target)\\n                 else:\\n                     l_boundary = min(findBoundary(nums, left, middle-1, target)[0], middle)\\n                     r_boundary = max(findBoundary(nums, middle+1, right, target)[1], middle)\\n             return [l_boundary, r_boundary]\\n         \\n         boundary = findBoundary(nums, 0, len(nums)-1, target)\\n         if boundary[1] == -1:\\n             return [-1, -1]\\n         else:\\n             return boundary\\n \\n                 \\n\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         # self made, two times binary search, beats 93%\\n         if not nums:\\n             return [-1, -1]\\n         # find left most index\\n         l, r = 0, len(nums) - 1\\n         while l + 1 < r:\\n             mid = l + (r - l) // 2\\n             if nums[mid] < target:\\n                 l = mid\\n             else:\\n                 r = mid\\n         left = l if nums[l] == target else r\\n         # find right most index\\n         l, r = 0, len(nums) - 1\\n         while l + 1 < r:\\n             mid = l + (r - l) // 2\\n             if nums[mid] <= target:\\n                 l = mid\\n             else:\\n                 r = mid\\n         right = r if nums[r] == target else l\\n         if nums[left] != target:\\n             return [-1, -1]\\n         \\n         return [left, right]\\n\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index - 1 \\n         after = index + 1\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if nums == []: return [-1, -1]\\n         m = int(len(nums)/2)\\n         if target == nums[0]:   return self.getRange(nums, 0)\\n         elif target == nums[-1]: return self.getRange(nums, len(nums) -1)\\n         elif target == nums[m]: return self.getRange(nums, m)\\n         else:\\n             if target < nums[0] or target > nums[-1]:   return [-1,-1]\\n             elif target < nums[m]:\\n                 return self.searchRange(nums[: m], target)\\n             else:\\n                 tempRange = self.searchRange(nums[m:], target)\\n                 if tempRange[0] == -1:  return [-1, -1]\\n                 else: return [r + m  for r in tempRange]\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index - 1 \\n         after = index + 1\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def getIndex(self, nums, target):\\n         if nums == [] or target < nums[0] or target > nums[-1]:  return -1\\n         if target == nums[0]:  return 0\\n         if target == nums[-1]:  return len(nums) -1\\n         \\n         m = int(len(nums)/2)\\n         \\n         if target < nums[m]:\\n             return self.getIndex(nums[:m], target) \\n         else:\\n             return -1 if self.getIndex(nums[m: ], target) < 0 else self.getIndex(nums[m: ], target) + m\\n             \\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         index = self.getIndex(nums, target)\\n         if index == -1: return [-1, -1]\\n         else:\\n             return self.getRange(nums, index)\\n\", \"class Solution:\\n     def getRange(self, nums, index):\\n         prev = index\\n         after = index\\n         while prev >= 0 and nums[prev] == nums[index]:\\n             prev -= 1\\n         while after < len(nums) and nums[after] == nums[index]:\\n             after += 1\\n         return [prev + 1, after -1]\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if nums == []: return [-1, -1]\\n         m = int(len(nums)/2)\\n         if target == nums[0]:   return self.getRange(nums, 0)\\n         elif target == nums[-1]: return self.getRange(nums, len(nums) -1)\\n         elif target == nums[m]: return self.getRange(nums, m)\\n         else:\\n             if target < nums[0] or target > nums[-1]:   return [-1,-1]\\n             elif target < nums[m]:\\n                 return self.searchRange(nums[: m], target)\\n             else:\\n                 tempRange = self.searchRange(nums[m:], target)\\n                 if tempRange[0] == -1:  return [-1, -1]\\n                 else: return [r + m  for r in tempRange]\", \"class Solution:\\n     def searchRange(self, nums, target):\\n         \\\"\\\"\\\"\\n         :type nums: List[int]\\n         :type target: int\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         start = 0\\n         end = len(nums) - 1\\n         mid = 0\\n         ans = [-1, -1]\\n         while (start <= end):\\n             mid = int((start + end) / 2)\\n             if (nums[mid] == target):\\n                 ans[0] = mid  \\n                 ans[1] = mid  \\n                   \\n                 i = mid - 1  \\n                 while i >= 0 and nums[i] == target:  \\n                     ans[0] = i  \\n                     i -= 1  \\n                   \\n                 i = mid + 1  \\n                 while i < len(nums) and nums[i] == target:  \\n                     ans[1] = i  \\n                     i += 1  \\n                       \\n                 break  \\n             elif (nums[mid] > target):\\n                 end = mid - 1\\n             else:\\n                 start = mid + 1\\n             \\n         return ans\\n         \\n\"]",
        "difficulty": "interview",
        "input": [
            [],
            [
                0
            ]
        ],
        "output": [
            -1,
            -1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "searchRange",
        "starter_code": "\nclass Solution:\n    def searchRange(self, nums: List[int], target: int) -> List[int]:\n",
        "url": "https://leetcode.com/problems/search-for-a-range/"
    },
    {
        "id": 1321,
        "task_id": 4536,
        "test_case_id": 1,
        "question": "Given a non-empty array of digits representing a non-negative integer, plus one to the integer.\n\nThe digits are stored such that the most significant digit is at the head of the list, and each element in the array contain a single digit.\n\nYou may assume the integer does not contain any leading zero, except the number 0 itself.\n\nExample 1:\n\n\nInput: [1,2,3]\nOutput: [1,2,4]\nExplanation: The array represents the integer 123.\n\n\nExample 2:\n\n\nInput: [4,3,2,1]\nOutput: [4,3,2,2]\nExplanation: The array represents the integer 4321.",
        "solutions": "[\"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         carry=1\\n         for i in range(len(digits)-1, -1, -1):\\n             digits[i]+=carry\\n             if digits[i] > 9:\\n                 digits[i]-=10\\n                 carry=1\\n             else:\\n                 carry=0    \\n             if carry == 0:\\n                 break    \\n         if carry == 1:\\n             digits.insert(0, 1)\\n         return digits    \", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         l = len(digits)\\n         i = l-1\\n         while i >= 0:\\n             if digits[i] != 9:\\n                 digits[i] += 1\\n                 break\\n             else:\\n                 digits[i] = 0\\n                 i -= 1\\n         if i == -1:\\n             digits = [1] + digits\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         i = len(digits) - 1\\n         carry = 1\\n         while carry!=0 or i >=0:\\n             temp = digits[i] + carry\\n             digits[i] = temp % 10\\n             carry = temp // 10\\n             i-=1\\n             if i < 0 and carry > 0:   \\n                 digits.insert(0,carry)\\n                 carry = 0\\n         return digits    \\n\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         l = [str(i) for i in digits]\\n         l =  int(''.join(l))\\n         l +=1\\n         l = list(str(l))\\n         l = [int(i) for i in l]\\n         return l\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         flag = 1\\n         i = len(digits)-1\\n         while i >=0:\\n             if digits[i]+flag <10:\\n                 digits[i] = digits[i]+flag\\n                 flag = 0\\n             else:\\n                 digits[i] = (digits[i]+flag)%10\\n             i-=1\\n         if flag == 1:\\n             digits.insert(0,1)\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return []\\n         \\n         i = len(digits) - 1\\n         while i >= 0:\\n             if digits[i] < 9:\\n                 digits[i] += 1\\n                 return digits\\n             else:\\n                 digits[i] = 0\\n                 i -= 1\\n         if digits[0] == 0:\\n             return [1]+digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = [str(x) for x in digits]\\n         num = int(''.join(d)) + 1\\n         \\n         return list(map(int, str(num)))\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         sum = 0\\n         for i in range(0, len(digits)):\\n             sum = sum * 10 + digits[i]\\n             \\n         return [int(i) for i in str(sum+1)]\\n\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         carry = 1\\n         for i in range(len(digits)-1,-1,-1):\\n                 print(i)\\n                 if digits[i] + carry == 10:\\n                     digits[i] = 0\\n                 else:\\n                     digits[i] += 1\\n                     return digits\\n         digits.insert(0,1)\\n         return digits      \", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return [1]\\n         carry = 1\\n         for i in reversed(range(len(digits))):\\n             if carry == 0:\\n                 return digits\\n             carry, digits[i] = (digits[i] + carry) // 10, (digits[i]+carry) % 10\\n         if carry:\\n             return [1] + digits\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         plus = 1\\n         for i in range(len(digits)-1, -1, -1):\\n             if digits[i] + plus > 9:\\n                 digits[i] = 0\\n                 plus = 1\\n             else:\\n                 digits[i] = digits[i] + plus\\n                 plus = 0\\n         if plus == 1:\\n             digits.insert(0, 1)\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if digits[-1] != 9:\\n             digits[-1] += 1\\n         else:\\n             for i in range(-1, -len(digits)-1, -1):\\n                 if digits[i] == 9:\\n                     digits[i] = 0\\n                 else:\\n                     digits[i] += 1\\n                     break\\n             else:\\n                 digits.insert(0, 1)\\n                 \\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         num = 0\\n         for i in range(len(digits)):\\n             num += 10**i * digits[len(digits)-i-1]\\n         return(list(map(int, list(str(num+1)))))\\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                1,
                2,
                3
            ]
        ],
        "output": [
            1,
            2,
            4
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "plusOne",
        "starter_code": "\nclass Solution:\n    def plusOne(self, digits: List[int]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/plus-one/"
    },
    {
        "id": 1322,
        "task_id": 4536,
        "test_case_id": 2,
        "question": "Given a non-empty array of digits representing a non-negative integer, plus one to the integer.\n\nThe digits are stored such that the most significant digit is at the head of the list, and each element in the array contain a single digit.\n\nYou may assume the integer does not contain any leading zero, except the number 0 itself.\n\nExample 1:\n\n\nInput: [1,2,3]\nOutput: [1,2,4]\nExplanation: The array represents the integer 123.\n\n\nExample 2:\n\n\nInput: [4,3,2,1]\nOutput: [4,3,2,2]\nExplanation: The array represents the integer 4321.",
        "solutions": "[\"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         carry=1\\n         for i in range(len(digits)-1, -1, -1):\\n             digits[i]+=carry\\n             if digits[i] > 9:\\n                 digits[i]-=10\\n                 carry=1\\n             else:\\n                 carry=0    \\n             if carry == 0:\\n                 break    \\n         if carry == 1:\\n             digits.insert(0, 1)\\n         return digits    \", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         l = len(digits)\\n         i = l-1\\n         while i >= 0:\\n             if digits[i] != 9:\\n                 digits[i] += 1\\n                 break\\n             else:\\n                 digits[i] = 0\\n                 i -= 1\\n         if i == -1:\\n             digits = [1] + digits\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         i = len(digits) - 1\\n         carry = 1\\n         while carry!=0 or i >=0:\\n             temp = digits[i] + carry\\n             digits[i] = temp % 10\\n             carry = temp // 10\\n             i-=1\\n             if i < 0 and carry > 0:   \\n                 digits.insert(0,carry)\\n                 carry = 0\\n         return digits    \\n\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         l = [str(i) for i in digits]\\n         l =  int(''.join(l))\\n         l +=1\\n         l = list(str(l))\\n         l = [int(i) for i in l]\\n         return l\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         flag = 1\\n         i = len(digits)-1\\n         while i >=0:\\n             if digits[i]+flag <10:\\n                 digits[i] = digits[i]+flag\\n                 flag = 0\\n             else:\\n                 digits[i] = (digits[i]+flag)%10\\n             i-=1\\n         if flag == 1:\\n             digits.insert(0,1)\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return []\\n         \\n         i = len(digits) - 1\\n         while i >= 0:\\n             if digits[i] < 9:\\n                 digits[i] += 1\\n                 return digits\\n             else:\\n                 digits[i] = 0\\n                 i -= 1\\n         if digits[0] == 0:\\n             return [1]+digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = [str(x) for x in digits]\\n         num = int(''.join(d)) + 1\\n         \\n         return list(map(int, str(num)))\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         sum = 0\\n         for i in range(0, len(digits)):\\n             sum = sum * 10 + digits[i]\\n             \\n         return [int(i) for i in str(sum+1)]\\n\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         carry = 1\\n         for i in range(len(digits)-1,-1,-1):\\n                 print(i)\\n                 if digits[i] + carry == 10:\\n                     digits[i] = 0\\n                 else:\\n                     digits[i] += 1\\n                     return digits\\n         digits.insert(0,1)\\n         return digits      \", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return [1]\\n         carry = 1\\n         for i in reversed(range(len(digits))):\\n             if carry == 0:\\n                 return digits\\n             carry, digits[i] = (digits[i] + carry) // 10, (digits[i]+carry) % 10\\n         if carry:\\n             return [1] + digits\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         plus = 1\\n         for i in range(len(digits)-1, -1, -1):\\n             if digits[i] + plus > 9:\\n                 digits[i] = 0\\n                 plus = 1\\n             else:\\n                 digits[i] = digits[i] + plus\\n                 plus = 0\\n         if plus == 1:\\n             digits.insert(0, 1)\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if digits[-1] != 9:\\n             digits[-1] += 1\\n         else:\\n             for i in range(-1, -len(digits)-1, -1):\\n                 if digits[i] == 9:\\n                     digits[i] = 0\\n                 else:\\n                     digits[i] += 1\\n                     break\\n             else:\\n                 digits.insert(0, 1)\\n                 \\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         num = 0\\n         for i in range(len(digits)):\\n             num += 10**i * digits[len(digits)-i-1]\\n         return(list(map(int, list(str(num+1)))))\\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                4,
                3,
                2,
                1
            ]
        ],
        "output": [
            4,
            3,
            2,
            2
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "plusOne",
        "starter_code": "\nclass Solution:\n    def plusOne(self, digits: List[int]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/plus-one/"
    },
    {
        "id": 1323,
        "task_id": 4536,
        "test_case_id": 3,
        "question": "Given a non-empty array of digits representing a non-negative integer, plus one to the integer.\n\nThe digits are stored such that the most significant digit is at the head of the list, and each element in the array contain a single digit.\n\nYou may assume the integer does not contain any leading zero, except the number 0 itself.\n\nExample 1:\n\n\nInput: [1,2,3]\nOutput: [1,2,4]\nExplanation: The array represents the integer 123.\n\n\nExample 2:\n\n\nInput: [4,3,2,1]\nOutput: [4,3,2,2]\nExplanation: The array represents the integer 4321.",
        "solutions": "[\"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         carry=1\\n         for i in range(len(digits)-1, -1, -1):\\n             digits[i]+=carry\\n             if digits[i] > 9:\\n                 digits[i]-=10\\n                 carry=1\\n             else:\\n                 carry=0    \\n             if carry == 0:\\n                 break    \\n         if carry == 1:\\n             digits.insert(0, 1)\\n         return digits    \", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         l = len(digits)\\n         i = l-1\\n         while i >= 0:\\n             if digits[i] != 9:\\n                 digits[i] += 1\\n                 break\\n             else:\\n                 digits[i] = 0\\n                 i -= 1\\n         if i == -1:\\n             digits = [1] + digits\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         i = len(digits) - 1\\n         carry = 1\\n         while carry!=0 or i >=0:\\n             temp = digits[i] + carry\\n             digits[i] = temp % 10\\n             carry = temp // 10\\n             i-=1\\n             if i < 0 and carry > 0:   \\n                 digits.insert(0,carry)\\n                 carry = 0\\n         return digits    \\n\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         l = [str(i) for i in digits]\\n         l =  int(''.join(l))\\n         l +=1\\n         l = list(str(l))\\n         l = [int(i) for i in l]\\n         return l\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         flag = 1\\n         i = len(digits)-1\\n         while i >=0:\\n             if digits[i]+flag <10:\\n                 digits[i] = digits[i]+flag\\n                 flag = 0\\n             else:\\n                 digits[i] = (digits[i]+flag)%10\\n             i-=1\\n         if flag == 1:\\n             digits.insert(0,1)\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return []\\n         \\n         i = len(digits) - 1\\n         while i >= 0:\\n             if digits[i] < 9:\\n                 digits[i] += 1\\n                 return digits\\n             else:\\n                 digits[i] = 0\\n                 i -= 1\\n         if digits[0] == 0:\\n             return [1]+digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         d = [str(x) for x in digits]\\n         num = int(''.join(d)) + 1\\n         \\n         return list(map(int, str(num)))\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         sum = 0\\n         for i in range(0, len(digits)):\\n             sum = sum * 10 + digits[i]\\n             \\n         return [int(i) for i in str(sum+1)]\\n\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         carry = 1\\n         for i in range(len(digits)-1,-1,-1):\\n                 print(i)\\n                 if digits[i] + carry == 10:\\n                     digits[i] = 0\\n                 else:\\n                     digits[i] += 1\\n                     return digits\\n         digits.insert(0,1)\\n         return digits      \", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         if not digits:\\n             return [1]\\n         carry = 1\\n         for i in reversed(range(len(digits))):\\n             if carry == 0:\\n                 return digits\\n             carry, digits[i] = (digits[i] + carry) // 10, (digits[i]+carry) % 10\\n         if carry:\\n             return [1] + digits\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         plus = 1\\n         for i in range(len(digits)-1, -1, -1):\\n             if digits[i] + plus > 9:\\n                 digits[i] = 0\\n                 plus = 1\\n             else:\\n                 digits[i] = digits[i] + plus\\n                 plus = 0\\n         if plus == 1:\\n             digits.insert(0, 1)\\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         \\n         if digits[-1] != 9:\\n             digits[-1] += 1\\n         else:\\n             for i in range(-1, -len(digits)-1, -1):\\n                 if digits[i] == 9:\\n                     digits[i] = 0\\n                 else:\\n                     digits[i] += 1\\n                     break\\n             else:\\n                 digits.insert(0, 1)\\n                 \\n         return digits\", \"class Solution:\\n     def plusOne(self, digits):\\n         \\\"\\\"\\\"\\n         :type digits: List[int]\\n         :rtype: List[int]\\n         \\\"\\\"\\\"\\n         num = 0\\n         for i in range(len(digits)):\\n             num += 10**i * digits[len(digits)-i-1]\\n         return(list(map(int, list(str(num+1)))))\\n\"]",
        "difficulty": "introductory",
        "input": [
            [
                0
            ]
        ],
        "output": [
            1
        ],
        "halu_type": "Identification Hallucination",
        "fn_name": "plusOne",
        "starter_code": "\nclass Solution:\n    def plusOne(self, digits: List[int]) -> List[int]:\n        ",
        "url": "https://leetcode.com/problems/plus-one/"
    }
]